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Models in Biology
Modelling Biology
Basic Applications of Mathematics and Statistics in the Biological Sciences
Part I: Mathematics
Script C
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
UMK Toruń 2008
2 Models in Biology
Contents Introduction .................................................................................................................................................. 3
1: How to build a model ............................................................................................................................... 4
2. From Euclidean to fractal geometry ....................................................................................................... 10
3: Biological growth processes ................................................................................................................... 18
4: Models of competition and predation ..................................................................................................... 26
5: Models in biochemistry .......................................................................................................................... 34
6. Markov chains ........................................................................................................................................ 41
7. The Weibull function and life table analysis .......................................................................................... 48
8. Basic models in genetics ........................................................................................................................ 53
Literature .................................................................................................................................................... 59
Online archives and textbooks .................................................................................................................... 60
Mathematical software ............................................................................................................................... 61
Important internet pages ............................................................................................................................. 62
Latest update: 10.01.2008
Models in Biology 3
Introduction
The following text is the third part of a lecture in basic mathematics biologists. The whole lecture con-
tains what might be considered an international standard of basic knowledge although many readers will surely
miss important branches. This part deals with the application of mathematics in biology. It focuses on model
building and interpretation. Again, many examples are included that show how to program simple tasks with a
spreadsheet program and how to use advanced mathematics software.
The following text in not a textbook. It is intended as a script to present the contents of the lecture in a
condensed form. There is no need to write a textbook again. Today, the internet took over many former tasks
textbooks had. The end of this text contains therefore a small overview over important internet pages where
students can find mathematics glossaries, textbooks, and program collections.
4 Models in Biology
1. How to build a model
In this lecture we will learn how to build simple biological models using a spreadsheet program. We will
see why this is necessary and how to use such models.
Why is it necessary to build mathematical models using experimental or observational data? There are
several reasons for this. Biology has transformed from natural history (history!) to an explanatory science.
It not only tries to describe phenomena in nature it tries to understand causes and relations. For this task we
have to structure our observations and to look for relations between them. This is exactly the modelling proc-
ess: we use the science of structures, mathematics, to uncover hidden patterns and relations. Modelling is
therefore more than finding out whether sample means differ or whether we have simple correlations between
data. We have to parameterize these relations. But models have many other tasks. First of all, they generate
new predictions about nature, predictions that then have to be verified or falsified. This prediction generating
feature is of course also a method to verify our model. Secondly, good models allow predictions to be make
about the future. This is a main aim for all environmental models. They are designed to predict the future of
populations, ecosystems and biodiversity. At last models reduce the chaos in our data and allow the develop-
ment of new theories and concepts.
Models can be classified into certain classes. On one end of a continuum we have verbal models stating
more or less precisely relations between a set of variables. These verbal statements may be incorporated into
diagrams where the variables are connected by arrows. Then, we have a qualitative model. On the other end,
there are explicit mathematical models that formalize relations. These relations may be fully parameterized
and then we have a quantitative model that gives quantitative predictions about variable states. At last, models
may contain exact parameter values at all stages of computation. We speak of deterministic models because
all future states of the models can in principle be computed by the initial set of values. On the other hand, the
model might contain more or less stochastic variables, variables that are driven by random events. In this case,
future parameter values are less sure or even chaotic. In this case we speak of stochastic models.
This short discussion indicates already what we need to build a model. This discussion is visualized in
Fig. 1.1
• The first step is that we have a theory. Shortly speaking, a theory is a set of hypotheses stated in a
formal language. We need hypotheses about nature and the relations between certain variables.
Modelling without a priori theoretical reasoning will lead to nothing. Our a priori experience
must lead to a selection of variables, so-called drivers, of the model. These drivers might have
explicit or stochastic values. They might be parameterized (characterized by explicit values or
functions) or not. In the latter case the model itself should assign values or value ranges to these
parameters.
• Then, we have to collect the necessary data. These data have to match the requirements of our
theory. Making experiments or observations without an explicit theory in mind will very often
result in large sets of data without any value because afterwards we (suddenly) notice that one or
another important variable had been ignored and not measured or that our method was inappro-
priate to incorporate the variable values of the model. This latter case occurs very often if we
took to few replicates and the variability in measurement is too high. Problems also arise if we
Models in Biology 5
used different methods for observations and we
later notice that these differences make it impos-
sible to compare the data (for instance because
they differ in the degree of quantitativeness).
• In a next step we have to confirm assumed rela-
tionships between these drivers. We might as-
sign qualitative or quantitative relations. If we
quantify the relations (for instance from a re-
gression analysis) we parametrize these rela-
tions.
• Then, we have to formalize the relations. This is
best done by a flow diagram or flow chart.
The flow diagram forces us to write each rela-
tion and each step of the model explicitly. This
step often uncovers smaller or larger errors in
our initial model that would have remained undiscovered in a purely verbal model formulation.
Making flow diagrams learns us thinking hard!
• The following step is then a technical one. Rewriting our flow diagram into a computer algo-
rithm. For more complicated models this should be a done using a common computer language
like C++, Pascal, R or Fortran, simple models can be written via a spreadsheet program like Ex-
cel.
• Our model will generate a set of output variables or whole classes of relations. We have to check
these parameters, whether their values are realistic, whether they correctly predict real values and
whether they are able to predict the future.
• At the end we have to modify our model in the light of its predictions and variable states.
Let’s exemplify the above steps of modelling from a simple example with real data. We measured the
population densities of a parasitic wasp species of the hymenopteran genus Aspilota during a series of genera-
tions. Aspilota is a group of small braconid wasps that predominantly develop as internal parasitoids of necro-
phagous flies of the family Phoridae. It is a very abundant and species rich genus. Our initial assumption is that
the population densities should be influenced by the densities of its host species and by a set of weather vari-
ables. Additionally, we assume that the densities of the previous generation should also influence wasp densi-
ties because high or low previous densities should find their expression in reproductive rates. By this, we ver-
bally stated an initial theory and pointed to a set of interesting variables. These variables are Dwasp, Dhost, and
climatic variables. What climatic variables? To allow a model to be constructed we must specify the variables
and their way of influencing. From previous studies and a literature survey we decide to recognize five climatic
input variables, precipitation CP, cloudiness CC, air temperature CT, relative atmospheric humidity CH, and Year Year Hosts
CT CH CP CS CC CT CH CP CS CC previous Gen. following Gen.1980 15 77 111 157.2 71.25 116 7 1
1981 1.9667 80 58.9333 51.8 77.5 1981 15.4 76 165.3 133.8 75 120 3.8 0.21982 -0.9 79 32.9667 96.6 60 1982 16.7 71 54.5 193.9 62.5 260 3.2 0.31983 2.967 78.667 52.9333 67.7333 76.25 1983 16.8 65 34.9 192.2 62.5 191 3.8 2.81984 1.7667 76.333 43.5666 75.6 67.5 1984 14.2 72 52.4 146.9 76.25 148 0.1 0.11985 -1.3889 77.667 29.2667 77.5 71.6667 1985 13.4 76 139.7 135.8 76.25 56 0.8 0.11986 -0.467 78 54 62.3 74.1667 1986 16.6 70 50.6 227.8 57.5 178 14.5 12.31987 -2.489 81.333 62.9333 79.4 70 1987 14.2 78 100.5 125.2 80 70 10.3 8
Mean values of climatic factors (January to March) Mean values of climatic factors (June) Aspilota
Fig. 1.1 Theory
DriversParameters
Data
Quantifi-cation
Functions
Flow chart
ComputeralgorithmOutput
New predictions
6 Models in Biology
total hours of sunshine CS. We incorporate into our theory that these climatic drivers affect larval mortality
mainly during the periods of activity of the insect and in the winter. By this we state a preliminary theory about
main factors influencing wasp densities.
Now, we stated our initial hypotheses sufficiently precise and are able to gather the necessary data. A
part of the data are shown in the next Table. Immediately we notice one problem of the data set. We have in
total 12 input variables but only 8 years of observation (16 generations). In reality the number of observations
should always be much larger than the number of input variables, but in any case at least as large. In our case,
we will deal with generations and the data set is slightly larger than the number of input variables.
We will try (with caution) to find out whether the climatic variables and the previous generation deter-
mine the following wasp generation. In a first step we develop a series of equations that try to describe the in-
terdependencies of our variables. This can either be done by a try and error methods, by predefined hypothesis
taken from other studies or by special statistical techniques which we will discuss in the statistics part
We have now two functions, one for the first (Gen1) and one for the second generation (Gen2) of the
wasp species. We find that probably cloudiness, hours of sunshine and rel. humidity are of minor influence.
The most probable hypothesis concerning the influencing variables includes temperature and precipitation dur-
ing the activity period and the winter, the number of hosts, and the wasp density of the previous generation.
This is of course a first model. We reduced the whole set of data into two
equations that show us that probably the main influencing factors for wasp densi-
ties are temperatures and precipitations during the activity period as well as pre-
vious wasp densities. All other variables and astonishingly host densities (Pho)
seem to have only minor effects and should be left out. The coefficient of deter-
mination is high indicating that about 80 to 90 percent of the observed variance
in wasp density could be explained by the included variables. We also infer that
high temperatures obviously hamper
wasp development or activity. Pre-
cipitation has once a positive and
once a negative effect. Our regression
model provides us therefore immedi-
ately with some important and not
foreseeable hypotheses about our
wasp population.
We identified the main input drivers
of our future model. We can now try
to develop an explicit flow chart. For
flow charts apply several conventions
of which the most important are
shown beside. They allow construct-
ing charts that explicitly show every
1 97.6 7.0 0.61 0.92 2G 2 55.7 3.5 0.05 0.99 1
So Wi
Ju Ju
Gen CT CP Genen CT CP Gen
= − + −= − − +
Get data
Counter
Start
Y=f(x)
n > max
Writeoutput
Y = value?
Z=g(x) H=h(x)
Stop
Start main
Compute
Stop main
Case option
Writeoutput
Loop test
Casestatement
Get data
Counter
Fig. 1.2 Fig. 1.3
Models in Biology 7
Start
GetGen2, Pho
Ran(CT, CP, Pho)Gen2=f(Gen2)
Gen1 = f(CT,CP,Gen2,Pho)
[IF (Gen1 < 0) minIF (Gen1 > Pho) Pho]
Ran(CT, CP, Pho)Gen1=f(Gen1)
Gen2 = f(CT,CP,Gen1,Pho)
[IF (Gen1 < 0) minIF (Gen1 > Pho) Pho]
Loops
StopFig. 1.4
AB
CD
EF
GH
1A
spilo
ta m
odel
2G
en2
Pho
CT(
Sum
)C
T(Ju
)C
P(W
in)
CP
(Jun
)3
Star
ting
cond
ition
s10
100
4m
ax17
.716
.882
785
min
1513
.478
656
Itera
tion
7C
ondi
tions
+C3
=+($
D$3
-C7)
/$D
$3=+
LOS
()*(
E$4-
E$5)
+E$5
=+LO
S()*
(F$4
-F$5
)+F$
5=+
LOS
()*(
G$4
-G$5
)+G
$5=+
LOS(
)*(H
$4-H
$5)+
H$5
81
Com
pute
=97.
6-7*
E7+
0.61
*G7-
0.92
*C7*
D7
9C
heck
=JEŻ
ELI(C
8<$D
$3;C
8;$D
$3)
10C
heck
=JEŻ
ELI
(C9>
0;C
9;1)
=+($
D$3
-C10
)/$D
$311
1.5
Com
pute
=55.
7-3.
5*F7
-0.0
5*H
7+0.
99*C
10*D
1012
Che
ck=J
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LI(C
11<$
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heck
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8+1
Con
ditio
ns;=
+C13
=+($
D$3
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)/$D
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)*(E
$4-E
$5)+
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=+LO
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(F$4
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)+F$
5=+
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()*(
G$4
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)*(H
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ompu
te=9
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G14
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ck=J
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heck
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ompu
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)
8 Models in Biology
step of the program. Now, we develop such a chart for our problem. We begin with a start button. Next, we set
Gen2 to a start value. Weather conditions are unforeseeable and we model this by letting these conditions fluc-
tuate at random between observed upper and lower limits. Next we compute the density of the second genera-
tion according to the multiple regression equation above. But we have to modify the equation. Wasp densities
can’t be higher than a certain upper limit, the carrying capacity. This limit is set by the number of host species,
the Phoridae (Pho). For simplicity, we assume these densities to be more or less constant. Additionally, the
reproductive rates should decrease at very high densities (according to the logistic growth equation). To model
this we use this logistic growth equation instead of Gen1 or Gen2. We set
Now, we look at the result of our computation. We use
Gen2 for further computation or we first adjust Gen2 if
the values are below 0 or above the carrying capacity. The
whole procedure is repeated for the 2. generation. At the
end of the model should be a loop that controls the num-
ber of iterations to be done. The whole flow chart is
shown in Fig. 1.4. The Table on the previous side shows
an Excel solution for this problem. We have the random
weather conditions (green) and the data checks (blue).
Computing the densities is shown in the yellow cells. Ad-
ditionally, we have a generation counter. This counter
helps us later sorting our data and making a plot.
Now we have to analyze our data. First of all, does our
model give realistic results? To check for this, we plot the result of 1000 model runs against the generation
counters (separately for the first and the second generation). We detect three main features (Figs. 1.5 and 1.6).
First of all, the model predicts densities and variability of the spring generations to be higher than that of the
summer generations (20 ± 7 to 15 ± 5 individuals m-2). Because we used real data we can compare this predic-
tion with reality. Indeed the measured densities of the spring generation of Aspilota were about 20% higher
than that of the summer generation and the variability about two times higher. However, the predicted mean
densities are in both cases about two times too high.
The model predicts correctly upper density boundaries of about 25 to 35 ind. m-2. Indeed, the real Aspi-
lota densities were during an 8 year study period always below 30 individuals per m2.
We also note three occasions were the predicted density fall below zero. Our model population would
1, 21, 2 1,2 Pho GenGen GenPho−⎛ ⎞= ⎜ ⎟
⎝ ⎠
0
5
10
15
20
25
30
0 500 1000Summer generation
Den
sity
05
1015202530354045
0 500 1000Spring generation
Den
sity
-505
101520253035
0 500 1000
Summer generation
Den
sity
-505
1015202530354045
0 500 1000Spring generation
Den
sity
Fig. 1.5
Fig. 1.6
Fig. 1.7 Fig. 1.8
Models in Biology 9
die out. From this we can predict an extinction probability of 3 / 1000 = 0.003. In reality extinction probabili-
ties were estimated by two different methods and expected to be above 1% per year. Our model predicts too
low extinction probabilities. Why? To answer this question we have to look to our model drivers. We simpli-
fied our model by holding the host density Pho constant. How do the model predictions change if we let Pho
changing in a similar way as the climatic factors, randomly between measured upper and lower limits. This is
shown in Figs. 1.7 and 1.8. Suddenly, the situation changes. First of all, predicted mean densities are lower than
before. Now, they resemble the observed densities but are still about 30% too high. The number of zero counts
increased markedly. The Figure shows an example with 12 such counts but from about 10000 generations com-
puted I inferred a mean of about 30 extinctions per 1000 generations. We expect therefore a local extinction
rate (just by chance) of about 3%. This prediction is again in line with the estimate obtained by other methods.
The latter modification also points to the main driver of our model. It is the host density. This was not
to foresee from our initial equations. However, remember that it is only a model. The next and decisive step is
to obtain a series of exact predictions that can be tested with field data. For such a test more independent data
on population densities of both wasp and host species would be necessary.
10 Models in Biology
2. From Euclidean to fractal geometry
Look at the following figure 2.1. I tried to measure the length of the boundary of Europe. This seems a
silly task but let’s try. We take a map and measure the length of the countries’ boundaries and all the coastlines.
How exact can we measure? This depends of course on our unit of measurement. If we take 1000 km as a basic
unit our estimate will be quite misleading. A unit of 1 m would give a very exact result but is impossible to
obtain. But in theory it would even be possible to use 1 mm as a base. Then we would have to consider every
small pebble that influences the boundary length. Now, let’s plot the boundary length of Europe against our
unit of measurement. Measurements are given as length / unit (unit–1). To plot unit against unit we take the re-
ciprocal. Instead of 1000 m-1 as the unit we use 1/1000m-1 = 0.001m, instead of 1 cm we use 1 / 0.01m-1 = 100
m and so on. Our reciprocal is therefore a measure of magnification or exactness of our measures. After some
time and work we get a picture that is shown below (Fig. 2.2). We detect an allometric relation between our
magnification (often also termed scaling factor) and the boundary length measured. The slope of this allomet-
ric relation is 0.3. This seems to be a curious example. It means that if we would reduce the unit of measure-
ment to very smalls values (even to infinity), our perceived boundary length of Europe would become larger
and larger up to infinity.
Let’s try another example. We take a circle
of radius 1 and do the same exercise (Fig.
2.3). Archimedes once computed the cir-
cumference of a circle by a series of lines
that inscribe and circumscribe the circle.
He used triangles with angle α and
summed up all the length d. The smaller α
is, the more such triangles inside a circle
exist and the better will be our estimate of
the circumference. The circumference is of
1000 km500 km
y = 157000x0.30
1000
10000
100000
1000000
0.001 0.01 0.1 1
Scaling factor
Leng
th o
f Eur
ope
[km
]
Scaling factor = 1 / unit of measurement
Fig. 2.1
Fig. 2.2
Models in Biology 11
course 2πr. To compute d for each
angle α we use the already known
cosine law. In this case we simply
get
We plot the circumference Σd
against the scaling factor α (Fig.
2.3). Now, the allometric relationship that appeared in the Europe example vanished. Perceived length and scal-
ing factor are for simple geometric objects nearly independent. Where is the difference between the ‘natural
object’ and the geometric one? Or, must natural objects be treated with a different geometry?
To answer this question we have to deal with a modern branch of geometry, with fractal geometry.
Indeed the French mathematician Benoit Mandelbrot (born 1924 in Warsaw) developed fractal geometry just
by asking “How long is the coast of Britain?”
Consider again a power law. It has the general form y = axz. If we compare two values y1 and y2 that,
say, stem from two different observations, we find that the quotient y1 / y2 is
(2.1)
The relation is therefore independent of the initial settings and the unit of measurement. These initial
conditions contains the factor a that is cancelled out by the division. Only the exponent z remained, the scaling
exponent. This is a very important property of power functions. One interpretation of this feature is that the
structure of the process the power function describes is independent of the data points we consider. We get the
same pattern at different degrees of resolution. A spatial or temporal pattern that looks always similar inde-
pendent of the (spatial or temporal) scale at which we look at it is called a self-similar pattern. A general de-
scription of such self-similar patterns is the power function.
A good example is the Figure 2.4 on the left side. A very simple geometric object is repeated in the
same way at different scales. The result is an object that looks similar to the branching patterns of our nervous
or circular system.
One of the first mathematicians, who studied self replicating geometric objects was Wacław Sierpiński
(Polish mathematician, 1882-1969). His Sierpiński triangle (Fig. 2.5) is the
basis for many artworks and computer graphics. Building these objects is
very simple. A computer program that constructs Sierpiński objects
looks as follows
1. Start with a triangle (or another simple object)
2. Shrink this triangle to 1/2 (or another scale)
3. Make three copies (or n copies)
4. Arrange these copies in quadrants 2, 3, and 4 (or other
fixed points)
5. Go to step 2
2(1 cos( ))d α= −
1 1 1
2 2 2
z zy x xay a x x
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
αα
d
d
y = 3 .10x0.002
1
10
1 10 100 1000
Scaling factor
Circ
umfe
renc
e
Fig. 2.3
Fig. 2.4
12 Models in Biology
By this simple instruction many so-called self-replicating
(or self-similar) objects (fractals) can be produced (Fig.
2.6). But Fig. 2.7 is also a self replicate. A simple instruc-
tion is repeated again and again resulting in a complicated
pattern. I choose this object because it looks very similar
to the villi of a vertebrate intestine. Many programs gener-
ate self-similar objects or fractals. Fig. 2.7 is a so-called
Julia set (after the French mathematician Gaston Julia,
1893-1973), generated by a very simple iterative instruc-
tion f(x) = z2 –0.75, where z is a so-called complex num-
ber. I used the program ChaosPro for computing the Fig-
ures above and beside.
Indeed many complex biological patterns stem from a self
replicating process. A nice example for self-similarity is a
fern (Fig. 2.8). The same structure is repeated at all scales
of resolution, from large leafs to the smallest leaflets. But
not all self-replicating objects are self-similar. Fig. 2.9
shows the Australian giant earthworm Megascolides
australis (Picture from Kästner, Lehrbuch der speziellen
Zoologie, Part I,3. Stuttgart 1982). Annelida have the
ancestral seriate (metameric) body plan of the metazoan.
But they are not self-similar. The body rings are not rep-
licated at different scales, different levels of resolution.
Fig. 2.5
Fig. 2.6
Fig. 2.7
Fig.2.8
Models in Biology 13
Now, we look at the problem of dimension. A point has the
Euclidean dimension 0, a line the dimension 1, an area 2, and a
cube 3.
In our case length L and resolution s are connected by a power
function. In general total length L should be the ruler length l
multiplied with the number of ruler lengths needed. This is a
function n(l) of l. Because of s = 1 /l we get
We assumed n(l) as being a power function of s with exponent d.
The length of an object is therefore not a fixed
value but is related to the magnification (the
resolution) we look at it. In this case we better
speak of perceived length. For a simple
Euclidean object (d -1) = 0 as in our circle ex-
ample. Hence, d = 1. In other words, length is
independent of the resolution we look at it. The
maximum value of d is of course 1 because the
perceived length L can’t grow faster than the
scaling factor s.
What about the constant a? Let’s consider a
simple geometric object (Fig. 2.10). In this case
d = 1 and L = a. If we have a square we can
divide this square into 9 smaller squares without overlapping. A cube can be divided into 27 such subcubes and
so on. 3 = 31, 9 = 32, 27=33 and so on. Therefore, the length of a line, the area, or the volume appears to be the
ruler length (in our example s = 3) to the power of the Euclidean dimension, for a line 1, for an area 2, for a
volume 3, and so on. Therefore a = sD. We can now combine our last two equations and get
(2.2)
This equation is a fundamental one. It tells us how the length of an object depends on the magnification
we look at it. To see this we return to our example of the boundary length of Europe. The length was L =
157000s0.3. s, the inverse of the ruler length l, is a measure of the magnification under which we measure the
boundary. At the highest ruler length of 1000 km the magnification is lowest. For our Europe example equation
2.3 becomes L = 157000s(1+ 0.3) - 1. The Euclidean dimension we consider is a length. D, the dimension of a
length, takes therefore the value 1.
How to interpret the value D+d. Because of 0 ≤ d ≤ 1 D+d takes always values between the actual
Euclidean dimension and the next higher one. It is commonly termed the fractal dimension of an object. The
boundary of Europe has therefore the fractal dimension of 1.3, a perimeter of a circle has a fractal dimension
that equals its Euclidean one (D = 1).
Why is it important to know about fractal geometry? Because it provides the clue for understanding
1 11( ) d d dL n l l s s ass
− −∝ ∝ ∝ =
1 ( ) 1D d D dL s s s− + −= =
a = 31
a = sD
a = 32
a = 33
Fig. 2.10
Fig. 2.9
14 Models in Biology
many relations of organismic growth patterns, body size relations, patterns in ontogeny, gene expression, or
ecology. I shall give two examples. The ferns shown above leads us to the problem of ontogenetic growth.
Consider a primordial blood or xylem vessel. Genetic activity generates at a certain time t a specified branching
pattern. If the genetic activity would remain unchanged over time (this is the simplest possible pattern) branch-
ing would occur at each time window t in a similar manner. The whole process would be self-similar. Recent
investigations showed indeed that many ontogenetic processes can be understood in this way. A nice example
are colour bands in butterflies that are generated by only a few (sometimes only 2) enzymes that act similar at
different times t.
Assume that the Figure below (Fig. 2.11) shows the branching pattern of blood vessels or a plant’s
phloem or xylem system. The process repeats in a similar manner at each stage. It is therefore a self-similar
process. From this notion we immediately derive basic relations between vessel parts. For instance, total vessel
volume vn (the sum of all vessels) at each stage k must fulfil a power function of the form
where VK and rK are the respective volumes and radia at the beginning of the branching process. Similar power
laws hold for vessel length and cross-sectional areas.
Vessel volume is related to many other physiological variables like nutrient flow, metabolic rate, con-
ductivity, heat production etc. For instance tissue area must scale to the quotient of vessel radia as
A similar proportion holds of course for the total vessel length. Because tissue area is linear proportional to the
maximum metabolic rate and vessel volume is V = πr2L we can introduce these relations into the above equa-
tion and get
In other words, metabolic rate should scale allometrically to
vessel volume. Vessel volume is proportional to total body weight.
Simple scaling laws tell us therefore that metabolic rate should scale
allometrically to total body mass. This is our already known law of
Kleiber. Of course, such reasoning does not provide values of our
scaling exponents z and x. However, recent investigations showed that
with reasonable starting conditions values similar to the ones observed
in nature result. That means, that ’simple’ geometric reasoning is able
to explain morphological and physiological patterns in nature.
A second example. Animals of different body size perceive
their environment in a different manner. What is for us a small
meadow is for a mouse a large wood of grasses and for an even
smaller insect a universe. A meadow has a fractal dimension. We can
an n
K K
V rV r
⎛ ⎞= ⎜ ⎟
⎝ ⎠
z
n n
K K
A rA r
⎛ ⎞= ⎜ ⎟
⎝ ⎠
x
n n
K K
M VM V
⎛ ⎞=⎜ ⎟
⎝ ⎠
Fig. 2.11
πrK2LK
rK
LKπrK2LK
rK
LK
Models in Biology 15
measure this as in our previous example. But now we use a slightly different method. We take a series of pho-
tographs of the meadow as shown in the next Figure (Fig.
2.12). Each photo was taken at a different magnification.
We measure the darker areas between the blades (by lay-
ing a grid upon the photos and counting all the cells not or
only in part occupied by blades) as an estimate of free
space and plot them against magnification (Fig. 2.13). Of
course, this is only a simple example and we have only
three data points but our method (the so-called grid
method) results in a power function. The program Excel
automatically fits such a function to the data points (we will see later how) and gives the respective equation.
The slope is 0.28. From the above fundamental function of self-similar processes
we interpret that the blades of a meadow form a fractal landscape with a fractal length dimension of (D+d) =
1.28.
Let’s look through the eyes of an animal. An animal of body length, say, 10 cm perceives its environ-
ment from a certain point of view. How will an animal of 1cm perceive the same environment. If the habitat
has fractal properties perceived lengths of habitat structures should follow the above function. For an animal of
1 cm body length habitat boundaries (like grass blades) will be 100.28 = 3.63 times as large. The area on which
these animals might live would be 2* 3.63 = 7.26 times larger. Let’s formalize our argument, because this is
always the first step for a mathematical treatment. We assumed that area A scales to the measurement of length
as
Now, we assume two other things. First of all we change length through species body weight. Body
weight should scale to length to an exponent of 3 (W ∝ L 3). Next, we assume that the number of individuals
per area of a species is proportional to the area in which they live (N ∝ Α). In our case the area is the perceived
area of a species. We get
( ) 1( ) D dL s bs + −=
2*0.28 0.56A s L−∝ ∝
Fig. 2.12
y = 6.2x0.28
1
10
100
1 10 100Magnification
Are
a
Fig. 2.13
16 Models in Biology
It seems that we got a first simple ecological rule, that the number of individuals of a species is approxi-
mately independent of its body length. This seems to be a silly result. You have 1000 insects on a squre meter
grassland, but 1000 elephants are seldom found. But this is not the end of the argument. Individual number
(species abundance) depends on a second important variable. The available food. We know already Kleiber’s
rule that metabolic rate scales to body weight to the power of 0.75. If food intake is proportional to metabolic
rate and limits the abundance, the body weight should be inverse proportional to the number of individuals (N∝
W-1) of a species and should scale to body weight to a power of -0.75. Individual number is therefore limited by
two independent factors, perceived habitat and available
energy for metabolism. We can now combine both argu-
ments. To do this we have to multiply both scaling laws
and get
(2.3)
We expect therefore that an animal of 1 gram body
weight should be 1000.94 =76 times less abundant than a
species of only 0.01 g. We can rescale to body length and argue that we expect that an animal of 1 cm body
length should be 102.82 = 661 times more abundant than an animal of 10 cm body length. Now, we have a gen-
eral ecological rule how abundances of animal species and their body weights should be related. We got this
rule by a combination of fundamental scaling laws and from fractal geometry. The crucial variable in our equa-
tion is the fractal dimension D of the habitat under study. Fig. 2.14 shows a real example. The figure gives a
plot of mean body weights versus mean densities of 18 guilds of animals (from Testacea through nematodes,
various arthropods to vertebrates) of a German beech forest on limestone (data from Schaefer M. 1991. Fauna
of the European temperate deciduous forest. In: Temperate Deciduous Forests, Eds. E. Röhrig, B. Ulrich. Am-
sterdam, pp. 503-525). We detect a nice allometric relation and the power function describing it has a slope of -
0.89, which is nearly exactly the theoretical value we inferred above.
At the end we once again compare Euclidean and fractal geometry. Such a comparison was provided by
three American ecologists, G. West, B. Enquist and J. Brown, in 1999. They wondered why in organisms quar-
ter power laws dominate and termed this pattern the fourth dimension of life. The table below shows the dif-
ferences in power law exponents between ordinary Euclidean and fractal geometry. In fractal tissues length L
(or l for fractals) scales to area A (a) by the power of 1/3 and to volume V (v) by the power of 1/4. Volume in
Euclidean bodies if proportional to A3/2 , in fractal bodies this relation takes v = a4/3. But in every case volume
is linearly proportional to body mass M.
How to derive these proportionalities. West, Brown and Enquist proposed one solution in the scientific
journal Science. L ∝ A1/2 and L ∝
V1/3. Hence A ∝ V2/3. West, Brown
and Enquist now assumed that liv-
ing organisms have at least to a cer-
tain degree self-similar structures of
energy and metabolite transporting
0.56 / 3 0.19 N W W− −∝ ∝
0.75 0.19 0.94N W W W− − −∝ ∝
Variable Conventional Euclidean Fractal Length L∝A1/2∝V1/3∝W1/3 l∝a1/3∝v1/4∝W1/4
Area A∝L2∝V2/3∝W2/3 a∝l3∝v3/4∝W3/4
Volume V∝L3∝A3/2∝W v∝l4∝a4/3∝W
y = 4.1223x-0.89
0.00010.001
0.010.1
110
1001000
10000100000
100000010000000
0.000001 0.001 1 1000
Mean body weight [mg]
Mea
n de
nsity
[m-2
]
Fig. 2.14
Models in Biology 17
organs and tissues. In such tissues our simple scaling rules
have to be modified. According to eq. 2.3 we should have
modified equations.
(2.4)
Of course ε and φ both have values between 0 and 1. What is φ? By the same logic as above it should be
possible to express a volume by the product of area and length. Considering again a fractal object and using eq.
2.4 we get
with η being between 0 and 1. We see that the scaling exponents (3 + φ) in eq. 2.4 equals (3 + ε + η).
We introduce this in eq. 2.4 and get eq. 2.5
(2.5)
The second crucial assumption in the whole argument is now that evolution has maximised the meta-
bolic rate of organisms with respect to body mass. Body mass scales linearly to volume. Metabolic rate as a
catalytic process is proportional to tissue surface. Hence under this argumentation the surface (area) volume
ratio should be maximised. Hence we are looking for the solution of
Later we will learn how to solve this problem systematically. Now we simply simulate the function us-
ing various values of ε and η (Fig. 2.11). We get a maximum at ε = 1 and η = 0. Hence
(2.6)
Now, all other scaling laws for fractal tissues of the last table follow. Because a ∝ M (the metabolic rate)
and v ∝ W (the body weight) we get immediately
(2.7)
This is Kleiber’s metabolic rule. Eq. 2.6 and 2.7 form the basis for many quarter power scaling laws in
biology. These are recently intensively studied and the field of biological scaling is one of the most dynami-
cally developing.
We detect another important thing. ε = 1, that means that in bio-
logically optimized organisms tissue surfaces should have a frac-
tal dimension near 3 rather than the Euclidean value of 2. If you
look at intestine villi as in the Figure beside or at our brain you
see why.
1223
13
l aa v
l v
εεφ
φ
+++
+
⎫∝ ⎪
∝⎬⎪
∝ ⎭
2 1 3v al l l lε η ε η+ + + +∝ ∝ =
23a v
εε η+
+ +∝
2 max3
εε η+
→+ +
2 1 33 1 4a v v
++∝ =
34M W∝
00.10.20.30.40.50.60.70.8
0 0.2 0.4 0.6 0.8 1 1.2
ε
k
η = 0
η = 1η = 0.5
Fig. 2.15
Photo by Gwen V. Child, UTMB
18 Models in Biology
3. Biological growth processes
In lecture 11 we heard already about growth processes, about exponential growth and the Pearl-Verhulst
model of logistic growth. Now we will deal with these processes in more detail. Growth models can be given in
two ways, a discrete form and a continuous form. Discrete forms are generally given as recursive functions of
the form
or
(3.1)
This latter version is a difference equation that depends on growth per discrete time steps Δt.
For instance the exponential growth model in discrete forms is
and
In continuous form the latter difference equation is given as a differential equation of the form
Difference equations are appropriate for discrete populations where generations do not overlap. They are
also more easy to model in computer algorithms, for instance when using Excel to model population growth.
Now we look closer to the logistic growth process. It was given by the differential equation
(3.2)
The biological interpretation of this equation is that the rate of change results from an exponential
growth process and a damping process that reduces the population growth. This damping acts immediately on
the population. What happens if this damping sets on later. Hence if the exponential process precedes the
damping we expect to find first a high population size which is afterwards reduced by mortality factors. In this
case we speak of time lags. For instance, assume a viral or bacterial disease. Viruses have typical exponential
growth rates at initial stages of an epidemic. Later, more and more hosts are attacked and the viruses do not
find appropriate hosts. Infection rates decrease, but they decrease with a certain time lag. Such time lags can be
modelled by a simple modification of our initial logistic growth model.
(3.3)
Eq. 3.3 contains a time lag τ in the damping term. Let’s study the behaviour of this modified model.
We use the discrete version of eq. 3.3 and plot the generations using Excel. We approximate N(t+1) by
the sum of N(t) plus the difference ΔN.
1 ( )t tN f N+ =
1 ( )t t tN N f N t+ − = Δ
1 0t tN rN N+ = +
1t tt
N N N rNt t
+ − Δ= =
Δ Δ
( )dN f Ndt
=
2( ) ( )( )( ) ( ) ( )dN t K N t rrN t rN t N tdt K K
−= = −
( ) ( )( )( ) ( ) ( ) ( )dN t K N t rrN t rN t N t N tdt K K
τ τ− −= = − −
Models in Biology 19
(3.4)
Our Excel model looks as follows. We have N(t) in column B. ΔN is computed in column C. To do this
we also need a column that gives us N(t-τ).
N(t) is now the sum of N(t) and ΔN.
Similar models were studied by the
British biologist Robert May in the first half
of the 1970th and had enormous influence
( )( 1) ( ) ( ) ( ) K N tN t N t N N t rN tK
τ− −+ = + Δ = +
Fig.3.1
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=0.7; τ = 2;K =500
E
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=0.2; τ = 0;K =500
A
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=2.099; τ = 0;K =500
B
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=1; τ = 1;K =500
C
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=2.7; τ = 0;K =500
F
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=2.95; τ = 0;K =500
G
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=3.05; τ = 0;K =500
H
A B C D E12 Parameters r K Tau K3 1 500 1 1004 t N(t) Delta N N(t-tau)5 0 10 9.296851659
6 +A5+1max(0,B5
+C5)+$B$3*B5*(1-
(D6/$C$3) +B5
0100200300400500600700800900
1000
0 20 40 60 80 100 120
TimeN
(t)r = 0.3; τ = 5; K = 500
D
20 Models in Biology
on ecological and parasitological research.
This simple model gives a number of predictions.
1. At low rates of r the population increases logistically until a maximum value is reached (Fig. 3.1 A).
This maximum can be inferred by setting the differential equation of the logistic growth to zero.
It equals the so called carrying capacity K. The biological interpretation is that irrespective of parameter
value the logistic growth model has one point of equilibrium (stationary point), the carrying capacity K.
2. Higher growth rates r result in regular cycles, which at even higher rates appear more and more ir-
regular. This means that the resulting cycles have longer and longer periods (B, F, and G)
3. At some growth rates the system initially appears to be relatively stable, but then larger amplitudes
(higher population fluctuations) appear (B).
4. Even small time lags force the system towards population fluctuations. The larger the time lag is, the
smaller r has to be to leave the system inside a stable range (D and E).
5. At higher growth rates the system looks more an more chaotic. We speak of pseudo chaos or deter-
ministic chaos, because it is not really chaotic since its generating function is strict deterministic. Again we
notice that a very simple deterministic function is able to generate unforeseeable (pseudo)-chaotic patterns.
They are called pseudo-chaotic because they are still deterministic but the resulting pattern is so complicated
that we have difficulties to find a posteriori, that means without knowledge of the generating function, any
regularities. May showed that at certain parameter combinations it is impossible to infer the generating function
from its output. Hence the analysis of biological time series might suggest very complicated structuring forces
whereas the series is in reality generated by a very simple deterministic process.
6. Too high rates of reproduction lead to the extinction of the population due to too high population fluc-
tuations.
May was not the first to observe that seemingly simple models can generate very unforeseeable output.
One important recursive model is the so-called Ricker
function, after W. Ricker, who introduced it in the 1950th
to model fish population dynamics, to estimate fish densi-
ties and to establish fishery quotes. This model is defined
by
(3.5)
( ) 0 ( ) 0 ( )dN t rr N t N t Kdt K
= → − = → =
( )1
kNk kN rN e α β− +
+ =
0
5
10
15
20
25
30
0 10 20 30 40 50 60
k
Nk
Fig.3.2
0100200300400500600700800900
1000
0 200 400 600 800 1000
N(t)
N(t)
+1
r=0.7; τ = 2;K =500
B
0100200300400500600700800900
1000
0 200 400 600 800 1000
N(t)
N(t)
+1
r=0.7; τ = 1;K =500
A Fig.3.3
Models in Biology 21
Fig. 3.2 shows an example for this model with r =
5, α = 0.5 and β = 2. We notice a very irregular
pattern. Indeed, the above function generates either
very regular patterns or, for certain parameter com-
binations, unforeseeable irregular ones. In this it
resembles the logistic growth model. It also produces a pseudo-chaotic behaviour. Models like the Ricker
model, or the closely related Nicholson-Bailey model, are important because due to their flexibility they allow
irregular fluctuations in animal populations to be modelled. We learn later how to do this.
An important tool in the study of discrete recursive models are plots of N(t+1) against N(t) as shown in
Fig. 3.3. The Fig. shows the effects of a time lag. In A a stable point is reached after a few generations. In B no
stable point is reached. The systems fluctuates around a point that equals K but it will never reach it.
When using differential equations we have the problem to solve them. In the last lecture we learned
already how to solve the logistic equation. But in most cases it is very difficult to do this. However, again Math
programs like Maple of Mathematica can do these technical things for us. We have to interpret their results.
Above the Mathematica solution of the logistic differential equation without time lag is given. The result looks
different from our solution on page 80.
This is now the same form as before.
For x = 0 C becomes
Of course our model is still highly simplified. It
treats populations as living in a homogeneous world without external influences. Such models might apply for
viral or bacterial growth. For animals or plants other variables that influence population growth have to be con-
sidered. Additionally, many populations have low recruiting rates at low densities. Our logistic growth model
instead predicts high growth rates at low densities. A simple possibility to make the model more realistic is to
introduce a lower density limit M below which the population goes extinct. Our model now becomes
(3.6)
This modified model has no simple closed solution. A numerical solution as before shows that for N
>> M this modified model behaves similar to the simple logistic growth model. For low N population growth
becomes very slow (Fig. 3.4).
(1)0 1
rx rx
rx f rx rxKe Ke Ky
e e e N Ce−= = =− − −
0
0
K NCN−
=−
( )( )dN K N N MrNdt K N
− −=
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r = 1; K = 500; M = 100
Fig. 3.4
22 Models in Biology
Random offshots Environmental factors can be very different. But we can combine them and modify our initial logistic
growth model by a simple additive term.
(3.7)
What is ran(a,b). It is a random number that affects the change in population size. We speak in this case
of a random offshot. Simply speaking random numbers are defined by a certain instruction that generates them
in such a way that they are not foreseeable inside the
range the instruction defines. Most often used are linear
random numbers. Give a range (a,b). Inside this range
every real number has the same probability to appear.
Computer programs generate linear random numbers in
the range (0,1). For number in other ranges we transform
them. For instance to get random numbers in the range
between 0 and 10 we simply have to multiply ran(0,1)
with 10. To get random numbers between 3 and 7 we
( ) ( , )dN K NrN ran a bdt K
−= +
Fig.3.5
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
+1
r=1; τ = 1;K =500, RAN(-5,5)
A
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
+1
r=1; τ = 1;K =500, RAN(-50,50)
B
0
2
4
6
8
10
12
14
0 20 40 60 80 100 120
Time
N(t)
Fig. 3.6
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
A
r=2.5;K=500;m=0
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
B
r=2.5;K=500;m=100
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
C
r=2.5;K=500;m=201
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
D
r=4.5;K=500;m=350
Fig.3.7
Models in Biology 23
have to add 3 and then multiply with 4. In general
(3.8)
For model 3.7 being realistic our random offshoot has sometimes to be above and sometimes below
zero. Hence the range might be (-a,a). If ran in eq. 3.7 is small in relation to dN the model still remains largely
deterministic (Fig. 3.5 A). At a high range (-a,a) dN becomes more and more unforeseeable and our model
transforms to a stochastic one (Fig. 3.5 B).
What is if the random offshoot solely determines next generations. Then our model looks as follows.
(3.9)
This is the simplest version of a so called random walk. An example gives Fig. 3.6. A random walk is
a process where the next step starts with the previous value but the direction and the amount of change is un-
foreseeable. Random walk models have wide application in biology and in the next part we will deal with them
in more detail.
Constant rates of foraging Now we consider another modification of the logistic growth model. Assume our model deals with a
fish population that is reduced by fishery. The reduction is independent of the actual population density. Hence
we use the discrete version of the model and modify eq. 3.1 by adding a constant term m, denoting the mortal-
ity by fishing.
(3.10)
Fig. 3.7 on the previous page shows plots of this model. We get four main predictions
1. Small to moderate fishery might stabilize population densities (A and B). This surprising result stems
from the reduction in density and hence in the more moderate increase in fish density.
2. Above a certain rate of fishery, the fish population inevitably dies out (overfishing). This result seems
rather trivial. However, it would be surprising if our model would not predict this outcome (C).
3. The extinction threshold depends on r, the rate of reproduction. The higher r is the more fish can be
gathered (B, D).
4. At high r and m fish populations should become more and more unpredictable. However, higher m
values further tend to stabilize populations.
Now we look a little bit closer at eq. 3.10 and transform it into a difference equation. We modify eq.
3.10 and introduce the time difference Δt
We transform and get a differential equation of the form
Has this equation stationary points, where the population remains unchanged? Our derivative function is
obviously quadratic. To study the behaviour of this function we compute the roots. Mathematica gives the fol-
( , ) ( ) (0,1)ran a b b a ran a= − +
1 ( , )t tN N ran a a+ = + −
( )( 1) ( ) ( ) ( ) K N tN t N t N m N t rN t mK
−+ = + Δ − = + −
( )( ) ( ) ( ) ( ) K N tN t t N t N t m t N t rN t t m tK
−+Δ = +Δ Δ − Δ = + Δ − Δ
( ) ( ) ( ) ( )( ) ( ) ( )(1 )N t t N t K N t dN K N tN t rN t m N t r mt K dt K
+Δ − − −= + − → = + −
Δ
24 Models in Biology
lowing solution. However, our problem has no simple solu-
tion. Look at Fig. 3.8. When applying differential equations
we are not dealing with a single solution but with a class of equations. To interpret our results we have to dif-
ferentiate between these classes. We analyse a so–called phase diagram as given in Fig. 3.8. In our case we can
differentiate three cases. Remember that the functions denote change in population size. Hence, negative values
denote a decrease, positive values an increase.
First dN / dt has no root (the blue line) . Because all values are negative the change in density is always
negative and the population inevitable dies out. This occurs when (-4m + rK) < 0. In this case the root of our
function has no solution in R. The value m = rK / 4 is termed the critical harvesting rate. If m exceeds this
value the population dies out.
Second, there is one root (the red line). In this case (-4m + rK) = 0 and the equilibrium point is N(t) =
K / 2. However any small disturbance of this highly fragile system would also lead to the extinction of the fish
population. This is an often observed phenomenon, in models like this and in nature. Balanced systems (system
in equilibrium) can be highly unstable.
Third, there are two roots. In this case populations with densities below N1 and above N2 go extinct.
Populations having densities between N1 and N2 increase until a stable (and maximum) density at N2 is
reached.
At last we consider a case where the rate of fishery is not stable but itself a function of fish density. This
is a more realistic case because harvest is of course a function of encounter with the fishes. For simplicity we
assume that harvest is proportional to fish density. Our new model looks therefore as follows
(3.11)
We consider only one example (Fig. 3.9). Even at high rates of fishery (m = 1, hence the total initial
population is fished) the system remains stable if only the rate of increase is high enough. Low rates of increase
instead will drive the population towards ex-
tinction. Too high values of r again result in a
pseudo chaotic behaviour of the system. We
solve eq. 3.11 again for N(t) to obtain the roots.
Now our solution looks very different. There is
one trivial stable equilibrium at N(t) = zero.
Another equilibrium point is at (m-r-1) = 0. The
case when this term is larger than zero does not
make, of course, sense. Hence for the popula-
tion to survive must hold m < r + 1. Our simple
( ) ( )( 1) ( ) ( ) ( ) ( ) ( ) ( )(1 )K N t K N tN t N t N mN t N t rN t mN t N t m rK K
− −+ = +Δ − = + − = − +
-35-30-25-20-15-10
-505
10
0 0.5 1 1.5 2 2.5 3
N(t)
dN/d
t
N1 N2
Fig. 3.8
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r = 3;τ = 1;K = 500;m = 1
Fig. 3.9
Models in Biology 25
model predicts therefore that for the population to
be stable the rate of fishery must be less than the
rate of reproduction + 1.
Of course, all of these models are very simplistic.
But they form still the base for more sophisticated models that incorporate much more density influencing vari-
ables. Additionally, logistic growth models like the ones above had an enormous influence on the development
of ecological, parasitological and economic theories. You find very nice descriptions of various growth models
at http://www.math.duke.edu/education/postcalc/growth/contents.html.
26 Models in Biology
4. Models of competition and predation
Competition Now we extend our logistic growth model and deal with two populations that compete for a common
resource. Suppose two parasite species A and B infect a common host. The infection by one species prohibits
the infection by another. How to model this system? We assume that both species follow a logistic growth
equation as studied before. As a simple modification we add to the logistic term (1 - N/ K) another term that
mimics the effect of competition, hence that further reduces the increase in population size. Our simple model
looks as follows
(4.1)
In this model KA, and KB denote the carrying capacities for A and B. The terms αΑ NA and αΒ NB are the
competition reduction terms and the α-values are the competition coefficients that denote the strength of the
effects of B on A and vice versa. This type of model was independently introduced by Lotka (1925) and
Volterra (1926) (Alfred James Lotka, 1880—1949, American demographer and mathematician; Vito Volterra,
1860—1940, Italian mathematician) and later (in a slightly different version) intensively studied by the Russian
mathematician George F. Gause. It heavily influenced biological modelling and lead to the competition para-
digm in ecology.
We have two coupled differential equations and our task is to study the behaviour of this system and to
interpret the results. We again study the
sign of dN(t) / dt and first compute the
equilibrium points of the system (dN / dt
= 0). Hence
We solve these linear algebraic equations for the equilibrium densities NA and NB. and get
(4.2)
We divide both equations through another and get
Coexistence is possible if NA and NB are both positive We have two possibilities to obtain positive val-
ues for NA/NB
( ) ( ) ( )( ) 1
( ) ( ) ( )( ) 1
A A B BA A
A
B B A AB B
B
dN t N t N tr N tdt K
dN t N t N tr N tdt K
α
α
⎛ ⎞+= −⎜ ⎟
⎝ ⎠⎛ ⎞+
= −⎜ ⎟⎝ ⎠
( ) ( )( ) ( )
A B B A
B A A B
N t N t KN t N t K
αα
+ =+ =
1
1
B B AA
A B
A A BB
A B
K KN
K KN
αα α
αα α
−=
−−
=−
A B B A
B A A B
N K KN K K
αα
−=
−
Models in Biology 27
1.αBKB > KA and αAKA > KB. We multiply both inequalities and get αBαB > 1. Both competition coeffi-
cients are by definition smaller than 1. Our condition is therefore impossible. The interpretation is that both
species should go extinct. However, there is always one stable point, when one species dies out and the other
survives. Hence, in this case the outcome depends on which species is the first to go extinct. The other sur-
vives.
2.αBKB < KA and αAKA < KB. In this case both values of NA and NB become positive. A stable (also
fragile) equilibrium exists and both can species survive.
In two other cases one species dies out
1.αBKB > KA and αAKA < KB Species B survives, A goes extinct.
2.αBKB < KA and αAKA > KB Species A survives, B goes extinct.
How to simulate this model? The Table below shows an Excel solution. I used again the discrete logistic
growth model and introduced the competition term.
Fig. 4.1 shows one solution of case four where no stable equilibrium exists. It depends on growth rates
and, as in this case, on the difference in carrying capacity which species survives. Even small differences may
lead to opposite outcomes. However, nearly always one species dies out. Hence the Lotka - Volterra competi-
tion model was one of the main arguments
for the long hold view that species that
share the same set of resources cannot coex-
ist. This is the principle of competitive
exclusion, first formulated by the G. F.
Gause in 1934.
But look at the following modification of
our model. The original model is strictly
deterministic although the outcome is not
( ) ( )( 1) ( ) ( ) 1
( ) ( )( 1) ( ) ( ) 1
A B BA A A A
A
B A AB B B B
B
N t N tN t N t r N tK
N t N tN t N t r N tK
α
α
⎛ ⎞++ = + −⎜ ⎟
⎝ ⎠⎛ ⎞+
+ = + −⎜ ⎟⎝ ⎠
A B C D E F G H I1 Species A αA Species B αB2 Parameters r K Tau r K Tau3 2.5 500 0 0.5 2.6 500 0 14 t N(t) Delta N N(t-tau) N(t) Delta N N(t-tau)5 0
6 +A5+1max(0,B5
+C5)
+$B$3*B5*(1-(D6+$I$3*E5)/$C
$3) +B5 max(0,E5+F5)
*(1-(G6+$E$3*B5)/$G$3 +E5
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
Species B: r = 2.5; K = 500; α = 1.0; τ = 0
Species A: r = 2.6; K = 500; α = 0.5; τ = 0
Fig. 4.1
A B C D E F G H I1 Species A αA Species B αB2 Parameters r K Tau r K Tau3 2.5 500 0 0.5 2.6 500 0 14 t N(t) Delta N N(t-tau) Stochasticity N(t) Delta N N(t-tau) Stochasticity5 0 0.1 0.01
6 +A5+1max(0,B5
+C5)
((D6+$I$3*E5)/$C$3)+$E$6*B
5*(los()-0.5*B5) +B5 max(0,E5+F5)
+$F$3*F5*(1-(G6+$E$3*B5)/$G$3)+$I$6*B5*(l
os()-0.5*B5) +E5
28 Models in Biology
always foreseeable. However, more realistic
are stochastic models where via random
numbers other factors are incorporated that
influence growth rates. Here I simply added
a linear random number term (aN(t)ran (-
1,1)) to the model. The same parameter val-
ues as above now allow for a stable coexis-
tence of both species (Fig. 4.2). Hence,
random effects might allow coexistence.
Recent investigation with more elaborate models have demonstrated that chaotic model behaviour is an impor-
tant element in maintaining and even enhancing species richness of natural habitats.
Predation Assume we have one protein that is produced proportionally to its concentration, it est it triggers its own
production. Viral infections are typical examples for this. Assume further we have an enzyme that counteracts
and degrades this protein. Its production is proportional to its own concentration and to the concentration of the
protein. Viral and bacterial infections and the reaction of the immune system are examples for such a process.
In ecology predator - prey or parasitoid - host system are examples. Indeed the following class of models was
first developed for predator - prey systems. Again Lotka and independently Volterra proposed it as the follow-
ing pair of quadratic differential equations
(4.3)
The logic behind this model is very simple. The rate in prey increase dH / dt is proportional to prey
abundance (as in the exponential growth model) and it is proportional to the number of prey taken by the preda-
tors. The latter is given as HP, the rate of prey - predator encounters. The change in predator density dP / dt
follows the same simple logic. Note the negative sign of rP. This simply notifies the fact that the predator popu-
lation would die out without prey present.
We study this system in the same way as before. Equilibrium points are at
Contrary to the previous example the model pre-
dicts that both populations should survive. A plot of
prey and predator densities versus time shows Fig. 4.3.
We observe an irregular cycling of both populations.
Although in theory the system should be stable, small
H H
P P
dH r H HPdt
dP r P HPdt
α
α
= −
= − +
H
H
P
P
rP
rH
α
α
=
=
0
50
100
150
200
250
300
0 20 40 60 80 100 120
Time
N(t)
Prey
Predator
Fig. 4.3
050
100150200250300350400450500
0 20 40 60 80 100 120
Time
N(t)
Species B: r = 2.5; K = 500; α = 1.0; τ = 0
Species A: r = 2.6; K = 500; α = 0.5; τ = 0
Fig. 4.2
Models in Biology 29
disturbances might force prey or predators to become
extinct.
This original version of the model is not very
realistic. We modify our model and assume for both
populations a logistic growth with time delay. Hence
we introduce the logistic growth terms and get
(4.4)
To evaluate the stability point of this system we consider a model without time delay and solve for dH /
dt = 0 and dP / dt = 0 and obtain the solution shown below. Instead of one single solution we have four. The
first two and the fourth contain trivial solutions and are not realistic except the case when the prey survives and
the (single) predator dies out. The third solution instead is quite complicated. But we observe that a positive
(and therefore realistic) values for H only appear if rP < αP KH. Hence KH > rP / αP . Similar, for the predator
population to be stable we need KP > rH / αH. The discrete model version in Excel looks as below. Fig. 4.4
shows again a cycling behaviour and low predator densities. Our stability criterion is met and we expect both
populations to persist. An often observed feature in
such models is that predator populations are more
stable than prey populations. Mean densities of
predators are lower than mean prey densities.
These are quite realistic model features.
At last we introduce a stochastic element
into our predator - prey model. As in the logistic
growth model before we add a simple random
number (ran(-2,2)) to the change terms of our
model. Using the same parameter values as above
we get a picture as in Fig. 4.5. Prey densities cycle
as before although less regularly and often damped.
But predator densities are now decoupled from the
prey cycling and more stable than before. Again
we see that introducing a stochastic element into
deterministic models might change model behav-
( ) ( )( ) 1 ( ) ( )
( ) ( )( ) 1 ( ) ( )
H HH
P PP
dH t H tr H t H t P tdt K
dP t P tr P t H t P tdt K
τ α
τ α
⎛ ⎞−= − −⎜ ⎟
⎝ ⎠⎛ ⎞−
= − − +⎜ ⎟⎝ ⎠
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
Time
N(t)
Prey
Predator
Fig. 4.4
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
Time
N(t)
Prey
Predator
Fig. 4.5
A B C D E F G H I1 Prey Predator2 Parameters r K Tau αΗ r K Tau αP3 0.9 500 0 0.04 0.05 200 2 0.00184 t N(t) Delta N N(t-tau)5 0 100 89.2832788 20 0.883662456
6 +a5+1max(0,B5
+C5)
=($B$3*B5)*(1-D6/$C$3)-
$E$3*D6*E5 +B5max(0,E5+
F5)
$F$3*E5)*(1-G6/$G$3)+$I$
3*B5*E5 +E5
30 Models in Biology
iour significantly. Additionally, random effects might stabilize patterns as in the case of the predators in Fig.
4.5.
Parasite—host models Now we consider a special case of predation,
parasitism. The aim is to develop a specific model
that describes the dynamics of parasites and their
hosts. Parasitism is a special case of predation
because the parasite does not kill its host immedi-
ately but allows him further reproduction. Hence
our simple Lotka - Volterra predator - prey ap-
proach does not work. The first step in develop-
ing a model is to study the biology of hosts and
parasites. From this we make a conceptional
model in form of a flow diagram that shows us
life history stages and potential parameters of our
future model. The next step is then to quantify
our model. In 1985 J. P. Hudson (In Rollinson &
Anderson (1985), Ecology and Genetics of Host-Parasite Interactions, Acad. Press London) gave a model of
infections of red grouse (Lagopus scoticus) by the nematode Trichostrongylus tenuis. This model can readily
be extended to deal with a general class of infections. Such an extended conceptual model serves as the basis
for further quantifications. It looks as follows (Fig. 4.6). We have host and parasite populations. Hosts have
natural mortality and parasite induced mortality. Additionally many parasites reduce host fecundity. Parasites
have certain birth and mortality rates. Additionally we need a parameter that describes infection rates.
In 1978 Robert May and Roy Anderson developed a set of models to describe such host parasite interac-
tions. They became the standard models for research in parasitology.
Model development looks as follows. Host density is given by
(4.5)
The logic behind these assumptions is simple. A change in host density is the difference between natural
fecundity and mortality multiplied with actual host density minus parasite induced mortality and reduction and
fecundity multiplied with actual parasite density. Hence we again assume that increase and decrease are line-
arly proportional to actual host and parasite density.
( ) ( )dH b a H Pdt
α ρ= − − +
Fig. 4.6
b A B C D E F G H I
1 Prey Predator2 Parameters r K Tau αΗ r K Tau αP3 0.9 500 0 0.04 0.05 200 2 0.00184 t N(t) Delta N N(t-tau)5 0 100 90.01137617 20 1.527903287
6 +a5+1max(0,B5
+C5)
=($B$3*B5)*(1-D6/$C$3)-
$E$3*D6*E5+4*LOS()-2 +B5
max(0,E5+F5)
=(-$F$3*E5)*(1-G6/$G$3)+$I$3*B5*E54*LOS()-2 +E5
HostPopulation
H
ParasitePopulation
P
Naturalhost mortality
Parasiteinduced
host mortality
Naturalhost fecundity
Parasiteinduced
reductionin fecundity
Infectionrate
Parasitebirth rate
Parasitemortality
a
α
bρ
μ
λ
β
Models in Biology 31
Parasite densities take a more complicated form
(4.6)
Again the change in density is the sum of different influencing factors. Increase in density is given by
the number of encounters of parasites and hosts (the product of P and H, the so-called mass effect) multiplied
by the parasite birth rate and the infection rate. The first term defines the proportionality of increase with para-
site density and with host density. However, not the actual density but the density of available hosts is taken.
This number is modelled by the term H / (H0 - H) , with H0 being the initial host density. The next term con-
tains parasite decrease. This again is assumed to be proportional to actual density with the parameters natural
mortality and increase in host density (a parameter that in fact leads to a reduction in the parasites). The last
term contains parasite mortality due to host mortality. The last multiplicative term P / (1+1/k) intends to mimic
parasite aggregation, the fact that multiple parasites occur in one host. The term is taken from a special statisti-
cal distribution, the negative binomial, with which we will deal in the statistics part.
First, we try to evaluate stability conditions. We apply Mathematica and solve both equations for H and
P (for dH / dt = 0 and dP / dt = 0) The program returns an undigestable output.
We try another way. Eqs. 4.5 and 4.6 can be simplified to
where the constants c and C contain all the constant parameter values. H* denotes the equilibrium den-
sity of H, where no further change in density occurs. Hence we approximated the term containing H0 by the
equilibrium densities. All parameters are positive. Now a very simple solution exists with positive values for H
0
( )1/
dP H PP b P Pdt H H k k
λβ μ α= − + −− +
1 2
3 4 5 50
( ) ( )
dH c H c PdtdP HP c c c P P C c Pdt H H
∗
∗
= −
= − − = −−
32 Models in Biology
and P. We conclude that our model has an equilib-
rium. We further conclude that at this point the quo-
tient of parasite to host density should be P* / H* =
c1 / c2. Hence or model predicts that for parasite -
host systems equilibrium densities should mainly be determined by host fecundity and mortality.
Lastly, we turn to one of the actual standard models in parasitology, proposed by Robert May and Roy
Anderson in 1980. They assumed again that changes in host densities are the sum of host fecundity that is pro-
portional to actual host density and host mortality that is again a mass effect, hence the product of H and P, the
parasite density. However, now total host density is divided into a population part H that is not infected and a
part J that is infected. Hence H + J = Htotal. The model looks as followed
(4.7)
r(H+J) is the reproduction rate of not infected hosts. δPJ is the reduction in host reproduction induced by
parasite infection. (α + a)J is the mortality of infected parasites. λJ denotes the increase in parasitism due to
infection. The most complicated term is the last. It formalizes the assumption that the decrease in parasite den-
sity is proportional to actual parasite density multi-
plied with natural mortality rate λ and the destruction
of pathogens by hosts assumed to be proportional to
total host density.
This model is able to generate host parasite cycles in
density and is widely used in the study of co-
evolution of parasites and hosts. The analysis of
stability gives (by setting the model equations to
zero) a very complicated result. But the equilibrium
density for P is simple. We see that for the parasite
population to be stable (a - r + α) > 0. Hence
The biological interpretation is simple. For the para-
sites to persist the total mortality rate of the infected
hosts has to be larger than the fecundity of the host
population. Additionally, we look at the term for H.
Because all rates are positive the term is positive if
again
Hence, hosts only have a stable equilibrium if para-
( )
( )
( ( )
dH r H J PHdtdJ PH a JdtdP JP H J Pdt
δ
δ α
λ μ γ
= + −
= − +
= − + +
a rα+ >
a rα+ >
Models in Biology 33
sites have a stable equilibrium. This is only possi-
ble if either host and parasite populations are con-
stant (very improbable) or if the populations ex-
hibit a cyclic behaviour around the equilibrium
points. Additionally we have a solution for δ, the
parasite induced mortality rate
This inequality tells that for the hosts to have stable populations r must be smaller than a + α, the mor-
tality rates of infected hosts. This is the same condition as before.
2 2( ) ( 2 ) a aa r a aa r
αα δ α δα
+− + > + → >
+ −
34 Models in Biology
5. Models in biochemistry
In leture three we dealt already with one important
model in biochemistry. The Michaelis Menten model of
enzyme kinetics. In this lecture we will discuss other
important models. Our starting point is a general enzyme
substrate reaction where an enzyme E binds to a substrate S
to form a compound ES. This is a reversible process and SE can dissociate to E and S. The concentrations of E,
S, and ES depend on the initial concentrations of E and S. Our question is what are the concentrations at
equilibrium? To answer this question we have to develop a model that gives us the concrentrations. Our
reaction equation looks as follows
E + S ↔ ES (5.1)
We denote the concentrations of E, S, and ES with [E], [S], and [ES}. Because the system is closed the
total amount of material must be constant. For every compound of ES one E and one S is lost. From this we get
two conservation equations
[E] + [ES] = [E0] and
[S] + [ES] = [S0]
where E0 and S0 denote the initial concentrations. Hence
[E] - [S] = [E0] - [S0] (5.2)
We assume that the speed of the associative reaction is proportional to both reactants, the concentrations
of [E] and [S]. This is a general assumption in kinetics and we have to understand why. The chance for one
molecule of E to bind to S is linearly proportional to [S]. The propbability of n molecules of E to bind is
therefore n[S]. N is nothing more than the concentration of E, [E]. Hence the chance to bind is proportional to
[E][S]. The higher this chance is, the higher is the speed. This assumption is similar to the predator - prey
encounter probability in the Lotka Volterra predation model. It is termed the mass effect. Hence
( 5 . 3 )
Remember that if a process is proportional to several variables
then we have to take the product of these variables. kES
denotes the proportionality constant. The dissociaton process
can be described in the same way
(5.4)
The change in the concentration of ES should be proportional to the association and dissociation speeds.
Hence
(5.5)
[ ][ ]
[ ][ ]
ES
ES
ES ES
v Ev Sv k E S
∝∝=
;
; ;
[ ][ ]
E S
E S E S
v ESv k ES
∝
=
; ;[ ][ ] [ ]ES E S ES E Sv v v k E S k ES− = Δ = −
y = 3.3851x2 + 0.7265x + 0.5071
00.5
11.5
22.5
33.5
44.5
5
0 0.2 0.4 0.6 0.8 1[S]
[S]
Fig. 5.2
Enzyme Enzyme Substratecomplex
Enzyme andproducts
Fig. 5.1
Models in Biology 35
We introduce equation 5.1 into 5.2 and get
kES([S0]+[E0]) and kE;S[S0] are constants. We denote them k1 and k2 and get
(5.6)
Based on very simple assumption we got a simple model of reaction kinetics. We see that the change in
reaction speed depends solely on the change in substrate concentration. Our model enables us to predict reac-
tion times in dependence of substrate concentrations. These have to be measured. We might also estimate the
constants by simultaneous measuring Δv and [S] and fitting our model to a plot of Δv against [S]. This is shown
in Fig. 5.2. Δv cannot be measured directly. Again we make an approximation and measure the change of [S]
and use this as a measure Δv. Hence we assume that the change in concentration is proportional to the speed.
Programs like Excel provide automatically a so-called fit to our data from the type of model we predefined.
This is in our case a quadratic function. In other words we assume that
This can be rewritten into a quadratic first order differential equation
(5.7)
Now we rearrange
We know this equation already. It is the differential equation for logistic growth with constant predation
(eq. 3.10 in lecture 3). Indeed this similarity is not accidental. In the case of ecology we deal with individuals
of a population. In chemistry we deal with molecules belonging to a certain chemical. If both entities
(individuals and molecules) obey similar mass proportion laws the mathematical description should also be
similar. Indeed many ecological, chemical and also genetic models are very similar. They deal with entities that
behave according to certain probabilistic laws. Their environment (ecosystems, membrane surfaces or chromo-
somes) can be described by the same geometry (often a fractal geometry). Then the mathematical description of
these very different entities looks nearly identical.
Now we look at the substrate enzyme reaction from a slightly different perspective. Enzymes often have
a very small concentration with respect to the substrate. Because [S] is assumed to be constant eq. 5.3 changes
to
(5.8)
The mass conservation law is now
;
; ;
[ ]([ ] [ 0] [ 0]) ([ 0] [ ])[ ][ ] [ ] ([ 0] [ 0]) [ 0] [ ])
ES E S
ES ES E S E S
v k S S S E k S Sv k S S S k S E k S k S
Δ = − + − −
Δ = − + − +
2; 1 2[ ] ( )[ ]ES E Sv k S k k S kΔ = + − −
2; 1 2[ ] [ ] ( )[ ]ES E SS k S k k S kΔ = + − −
2; 1 2
[ ] [ ] ( )[ ]ES E Sd S k S k k S kdt
= + − −
2; ; ;
;
[ ] [ ][ ] ( ( ([ 0] [ 0])[ ]) [ 0] [ ](1 ) [ 0]1([ 0] [ 0])ES E S ES E S ES E S
E S
d S Sk S k k S E S k S k S k Sdt S E
k
= + − + − = − −+ −
; ;
[ ][ ]
ES ES
E S E S
v k Ev k ES
==
36 Models in Biology
[E0] - [E] = [ES] (5.9)
The change in speed dv is given by
(5.10)
Now the change in speed is propor-
tional to the enzyme concentration and we
get a first order linear differential equation
The solution is
This is also a Michaelis Menten process but in a different form. In lecture 3 we dealt with speed against
substrate concentration here we looked at enzyme concentration against time. The maximum reaction speed is
of course at [E] = kE;S / (kES + kE;S)
Collision theory Above we modelled a chemical reaction via a mass effect. This can be generalized. For any reaction
n1A + n2B ↔n3C +n4D
we can establish the equilibrium equation of the form
(5.11)
With K being the reaction constant. The speed of the forward reaction is according to the mass effect
(5.12)
The sum of n1 and n2 determines the order of the reaction (in eq. 5.11; in 5.12 of course n3 + n4)). To
solve these equations we have to determine the concentration of the reactants in dependence of time. Let’s
consider three basic types of reactions. A simple reaction has the form A→B. The speed of this reaction is
proportional to the concentration of A. This is a first order reaction. Hence
The concentration time function is an exponential function. Integration gives
(5.13)
Next we consider reactions of the second order.
A + B ↔C +D
; ; ;([ 0] [ ]) [ ] ( )[ ] [ 0]E S ES ES E S E Sdv k E E k E k k E k Edt
= − − = − + +
1 2[ ] [ ]d E k E kdt
= − +
;( ); ;
; ;
[ 0] [ 0][ ] [ 0] ES E Sk k tE S E S
ES E S ES E S
k E k EE E e
k k k k− +⎛ ⎞
= + −⎜ ⎟⎜ ⎟+ +⎝ ⎠
3 4
1 2
[ ] [ ][ ] [ ]
n n
n n
C D KA B
=
1 2
3 4
[ ] [ ]
[ ] [ ]
n nAB
n nCD
A B v
C D v
∝
∝
[ ] [ ]Ad k Adt
= −
0
[ ]
0[ 0]
[ ] /[ ] [ ] [ ]A t
kt
A t
d A A kdt A A e−= − → =∫ ∫
Models in Biology 37
We define the value of x that
diminishes A and B due to reaction.
Hence if the initial concentration of
A is [A0] the concentration after
time t is [A0-x]. Th reaction speed
is then v = dx / dt. We can therefore
describe the process by
The backward reaction is given by
These are quadratic first order differen-
tial equations. The solutions are again logistic
growth equation. For higher order reactions solu-
tions become very complicated or do even not exit
as closed functions. The above solution is quite
complicated. The Excel solution is shown above
and Fig. 5.2 shows that the concentration of A de-
creases asymptotically by an hyperbolic function.
We can do simpler and solve by hand!
(5.14)
Now we apply a small trick. It holds
(5.15)
This is a very important equation with which many quadratic differential equations can be solved by
hand. It is a special case of a class of equations by which quotients can be simplified, the method of dividing
into partial fractions.
We solve eq. 5.13 as follows
Setting x = 0 at t = 0 and combining both integration constants gives
At the end we get
0 0[ ][ ]ABdxv k A x B xdt
= = − − −
0 0[ ][ ]CDdyv k C y D ydt
= = − −
0 0[ ][ ] ABdx k dt
A x B x= −
− −
0 0 0 0 0 0
1 1 1 1[ ][ ] [ ] [ ] [ ] [ ]A x B x A B A x B x
⎛ ⎞= −⎜ ⎟− − − − −⎝ ⎠
( ) ( )( ) 00 0 1 1 2
0 0 0 0 0 0 0 0 0
[ ]1 1 1ln [ ] ln [ ] ln[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] AB AB
A xdx dx A x B x C C k dt k t CA B A x B x A B A B B x
⎛ ⎞⎛ ⎞ ⎛ ⎞−− = − − − + = + = − =− +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− − − − − −⎝ ⎠ ⎝ ⎠⎝ ⎠
∫ ∫
0
0 0 0
[ ]1 ln[ ] [ ] [ ]
ACA B B
⎛ ⎞= ⎜ ⎟− ⎝ ⎠
0
10
20
30
40
50
60
0 0.2 0.4 0.6
Time
x(t)
A(t)
Fig. 5.3
A B C D E1 Constants A B C[1] k2 50 49 0.020203 2
3
=LN($B$2/$C$2)/($
B$2-$C$2)
4 Time x(t) A(t)
5
0
+($B$2*EXP($C$2*$E$2*A5+$D$2)-$C$2*EXP($B$2*$E$2*A5+$D$2))/(
EXP($C$2*$E$2*A5+$D$2)-EXP($B$2*$E$2*A5+$D$2))
=$b$2-B5
38 Models in Biology
(5.16)
This is a linear equation. Plotting ln ([A0 - x]) / [B0 - x]) against time t yields a straight line with slope -k
([A0] - [B0]) and intercept ln ([A0] / [B0]). From these values x and k can be determined if we only know the
initial concentrations of A and B.
Note that the above method cannot be used when the initial concentrations of A and B are equal. Then
our integration by the partial fractions method looks different.
Setting x = 0 at t = 0 and combining both integration constants gives
Our linear equation now becomes
(5.17)
Fig. 5.4 shows typical functions of the concentra-
tions of one of the products [c] against time. The plot
was computed using eq. 5.13 and the Mathematica solution above. Second order reaction tend to have steeper
initial slopes.
Again we notice that our models of reaction kinetics have a similar structure to simple population
growth models. In the case of kinetics molecules and atoms are treated as being billiard balls flying randomly
in space and colliding with one another. Recent ecological models treat individuals of populations in a very
similar manner. Host—prey encounter are simulated solely according to a simple mass effect with encounter
probabilities being proportional to the densities of the interacting individuals and the migration rates (their
‘temperature’). Under these assumptions kinetic and ecological models look very similar.
One way to model our reaction kinetics or ecological laws is via the collision theory. Molecules or indi-
viduals interact when they collide (or meet). Now take the law for ideal gases. A gas that is compressed
changes its internal energy. The energy you need to compress is transformed into kinetic energy leading to
higher collision rates of the gas molecules. Hence the change in internal energy due to changing pressure
should be proportional to its volume. Hence from pV = nRT and referring to one mol gives
Integrating gives
We do not need integration constants because we are working with definite integrals. Pressure is nothing
more than concentration per unit volume. Hence our theory should also refer to solutions and chemical reac-
0 00 0
0 0
[ ] [ ]ln ([ ] [ ]) ln[ ] [ ]A x Ak A B tB x B
⎛ ⎞ ⎛ ⎞−= − − +⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
1 2200
1[ ][ ]
dx C kdt kt CA xA x
−= + = − = − +
−−∫ ∫
0
1[ ]
CA−
=
0 0
1 1[ ] [ ]
ktA x A
= +−
( )dG RTV pdp p
= =
2 2
1 1
22 1
1ln
G p
G p
pnRTdG dp G G G nRTp p
= → − = Δ =∫ ∫
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
Time
[C] First order
Second order
Fig. 5.4
Models in Biology 39
tions. There p1 and p2 denote the concentrations before and after the reaction of the type of eq. 5.11. The speed
is therefore
Therefore
or according to eq. 5.11
(5.18)
This is the well known Gibbs - Helmoltz equation (after the German chemist Herman von Helmholtz,
1821-1894, and Josiah W. Gibbs, American chemist, 1839-1903) which describes the change in free enthalpy
in dependence of temperature and concentration of reactants. Our initial mass law assumed that reaction speed
should be proportional to the number of encounters. This is here assumed to be proportional to the free en-
thalpy of the system. We solve for K and get the important equation.
(5.19)
The Swedish Nobel winner Svante Arrhenius (1859-1927) now assumed that reaction speed is propor-
tional to the number of activated particles in the system, hence the number of particles having an energy higher
than necessary for the reaction: v ∝ nA. This number should be proportional to the equilibrium constant K. We
get an equation that describes the dependence of reaction speed on temperature
(5.20)
with ΔG = ln(A)ΔE, with ΔE being the energy necessary to initiate the reaction (activation energy). A
defines the maximum speed at high temperature.
At the end we look at the dependence of the reaction constant K on the temperature T. From the Gibbs
Helmholtz equation we get
Therefore, the change in K at changing t is given by
(5.21)
This is the well known equation of van’t Hoff (after the Dutch chemist Jacobus H. van’t Hoff, 1852-
1911). To estimate a difference in K from a temperature difference T1 to T2 we have to integrate
1 2
3 4
1
2
ln([ ] [ ] )
ln([ ] [ ] )
n nAB
n nCD
v G nRT A B
v G nRT C B
= Δ =
= Δ =
31 2 4
3 1 241 2
[ ] [ ] [ ] [ ]ln ln[ ] [ ][ ] [ ]
nn n n
n n nnA B C DG G G nRT nRT
A BC DΔ = Δ − Δ = = −
lnG nRT KΔ = −
Gn R TK e− Δ
=
EnRTv K v Ae− Δ
∝ → =
G ln( ) ln( )nRT
G nRT K K ΔΔ = − → = −
2ln( )d K Gdt nRT
Δ=
2 2
1 1
22 1 2
1 2 1
1 1 1ln( ) ln( ) ln( ) ln( )K T
K T
K G Gd K K K dtK nR nR T TT
⎛ ⎞Δ Δ= − = = = − −⎜ ⎟
⎝ ⎠∫ ∫
40 Models in Biology
Note the change in sign at the end.
What has all of this to do with ecology? At the
beginning I argued that biological models might look
very similar independent for what they were originally
designed. Consider a virus disease or a new phyto-
phagous insect on plants immigrating into a new re-
gion. We can reinterpret our collision theory. The tem-
perature of the individuals is now the speed of migra-
tion (remember that temperature is the speed of molecules or atoms). The activation energy might be inter-
preted as the level of immune system (the health) or the level of plant defence against phytophagous insects
infection rate. N is the population density of the insect or the number of viruses in the air. Hence we expect that
the speed of new infections, the infection rate per time, is a first order reaction and should follow an exponen-
tial function of migration potential. In Fig. 5.5 the infection rate k according to this model is plotted against
migration rate M. D denotes the density and c the constant. We also expect the infection speed v, to be propor-
tional to the number of insects or viruses present. Hence v ∝ D = kD with k being the infection rate. The num-
ber of infected plants or host I should follow an exponential function
Of course our analogy has limitations. We did not consider reproduction rates of the insects or viruses.
For our model to be realistic we would have to add productions rates of the ‘reactants’. Nevertheless for short
term changes in infections (inside one generation) or for rapid colonisations our simple analogy model appears
to be quite appropriate. Indeed in 2001 the American plant ecologist Stephen Hubbell published an intensively
discussed book, where he based major parts of community ecology solely on a so-called ecological drift model,
that is in its core nothing more than an extended model of reaction kinetics. In the statistics part we will deal
with such models in detail.
At the end I should notice that
today for medical and biochemi-
cal enzyme kinetics a huge num-
ber of different software is avail-
able that analyse automatically
many of the previous models and
provide numerical solutions.
( )dIv k D Idt
= = −
00.10.20.30.40.50.60.70.80.9
0 0.2 0.4 0.6Migration rate
Infe
ctio
n ra
te
Fig. 5.5
m a x
−
=c
D Mv v e
Models in Biology 41
6. Markov chains
Beside equation solving matrices have by far more biological applications. Assume you are studying a
contagious disease. You identified as small group of 4 persons infected by the disease. These 4 persons con-
tacted in a given time another group of 5 persons. The latter 5 persons had contact with other persons, say with
6, and so on. How fast does the disease spread in the population? To answer this question we first define a ma-
trix describing the first contacts. You have four infected persons and 5 contact persons of the second group.
Hence person 1 of the first group contacted with person 2 of the second group. No. 2 of the first group
contacted with No. 1, 2, and 4 of the second group and so on. Now you describe the second order contacts of
group three with group two.
To find the number of persons in group three that had (via group two) contact with infected persons of
group one you have to multiply both matrices. We get as the result
How to interpret this result? From the computational scheme of a dot product of two matrices follows
that the new elements of C result from all combinations of respective rows and columns of A and C. Hence the
ones and twos denote indirect contacts of a person in the third group with a person of the first group. Person 1
of the third group had 6 indirect contacts (1+1+2+2), the persons 2 and 4 only one.
However, we can also use probabilities of infection instead of contacts. Say that any contact gives a
probability of 0.3 that a person will be infected. We have to replace the one with 0.3 to get the probability that
persons in contact with infected persons get infected. Our model becomes
0 1 0 01 1 1 00 0 1 10 1 0 00 0 0 1
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
A
0 1 1 0 10 0 0 0 11 1 0 0 11 0 0 0 00 1 1 0 01 1 0 0 1
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
B
1 1 2 20 0 0 11 2 1 10 1 0 01 1 2 11 2 1 1
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
= • = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
C B A
42 Models in Biology
Hence person 1 of the third group has a probability of 0.54 of being infected. Of course this method can
be applied to further groups or generations (if we interpret the groups as generations). By this we get probabili-
ties of occurrences of initial events in subsequent time windows. The matrix multiplication allows for the pre-
diction of infections during epidemics.
Markov chains The above discussion leads immediately to the concept of Markov chains (after the Russian mathemati-
cian Andrei Markov, 1856-1922). A Markov chain is a sequences of random
variables in which the future variable is determined by the present variable but
is independent of the way in which the present state arose from its predeces-
sors. Hence if we have a series of process states the value of state n is deter-
mined by only two things. The value of state n-1 and by a rule that tells how
state n-1 might transform into state n. Most often these rules contain probabili-
ties. Then state n-1 goes with probability pi into any state i of n. Hence in a
Markov chain states prior than the previous do not influence the future fate of the chain. This is why Markov
chains are often said to be without memory.
Take for instance a gene that has three alleles A, B, and C. These can mutate into each other with prob-
abilities that are given in Fig. 6.1. A mutates into B with probability 0.12 and into C with probability 0.2.
Hence with probability 1 - 0.12 - 0.2 = 0.68 nothing happens. We can these so-called transition probabilities
write in a matrix form.
This matrix that gives the transition probabilities is called the transition matrix. The sum of all matrix
rows must add to 1, the sum of all probabilities. This is a general feature of all probability (stochastic) matri-
ces. If we now take the initial allele frequencies we can compute the frequencies of the alleles in the next gen-
eration. Assume we have initial frequencies of A = 0.2, B = 0.5, and C = 0.3. This gives a vector of the form X0
= {0.2, 0.5, 0.3}. The frequencies in the next generation are computed from
Again, the new frequencies of A = 0.201, B = 0.429, and C = 0.37 add up to zero. If we multiply two
probability matrices the resulting matrix is again a probability matrix.
If we continue the process we get X3=PX2, X4=PX3… In general the frequencies of a Markov chain after
0.09 0.09 0.18 0.180 0 0 0.09
0.09 0.18 0.09 0.090 0.09 0 0
0.09 0.09 0.18 0.090.09 0.18 0.09 0.09
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
= • = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
C B A
0.68 0.07 0.10.12 0.78 0.050.2 0.15 0.85
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
P
1 0
0.68 0.07 0.1 0.2 0.2010.12 0.78 0.05 0.5 0.4290.2 0.15 0.85 0.3 0.37
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= • = =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
X P X
A B
C
0.10.2
0.050.15
0.07
0.12
Fig. 6.1
Models in Biology 43
n states starting from the initial conditions X0 and
determined by the transition matrix P is given by
(6.1)
Of cause this law is very similar to recursive
processes leading to exponential distributions. It is
a generalization. Equation 6.1 defines the simplest
form of a Markov chain process.
We also see that state n+1 is only dependent
on state n. This property serves even as the general
definition of a Markov process. The probability i of
a state Xn with respect to the previous states X1 to
Xn-1 is the same as the probability of Xn with re-
spect to state Xn-1 only. The previous states have no
influence any more. Mathematically written
(6.2)
Does our mutation process above reach in stable allele frequencies or do they change forever? This
question can be answered twofold. Does the frequency distribution remains constant or does the process elimi-
nate one or more alleles? The first
question is whether the frequencies
of the alleles remain constant. In this
case the following condition must
hold
This can be written in terms of eigen-
vectors
(6.3)
Pn is called the stationary state. This state is defined
by the eigenvector U of the transition matrix P with
the largest eigenvalue. This is scaled to λ = 1. Xn is
called the steady-state or equilibrium vector. The
Excel example beside shows the transition matrix of
three alleles. λ3 = 1 and the third eigenvector U3 de-
fines the stationary state, that is the frequency distribu-
10 0
−= • = •Λ • •n nnX P X U U X
n n 1 n 2 n 3 n 1 n n 1p(X i | X ,X ,X ...X ) p(X i | X )− − − − −= = =
1 •+ = =n n nX X P X
n n nP X 1X (P 1I) X 0• = → − • =
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12 14
Steps
Freq
uenc
y
Stationary frequenciesdefined by the eigenvector
Fig. 6.2
44 Models in Biology
tion the process will end in. Note that the eigenvectors are nor-
malized to have the length one. To get the frequencies
(probabilities) we have to divide U3 through the sum of it’s val-
ues. Fig. 6.2 shows that in our case the allele frequencies quickly
converge to the steady state.
Do all Markov chains converge? We look at several im-
portant special cases. Fig. 6.3 shows a graphical representation
of a Markov chain with four states. Given are the transition
probabilities. The missing probabilities can be inferred from the
scheme. We see that state D cannot be reached from any other
state. It forms a closed part of the whole chain. If a chain does
not contain closed subsystems it is called irreducible. In such a
system all states can be reached. Fig.
6.4 now shows a simple example of a
periodic chain. The whole chain
forms a circle. Fig. 6.3 shows an
aperiodic chain. Fig. 6.3 shows also
two other important concept. First
the states A, B, and C are recurrent that means it is sure that that a finite time (even a very long time) the
process returns to the initial state. State D is not recurrent. In some chains there is only a certain probability that
the chain returns to a previous state. These chains are called transient. There is no way back to D. An impor-
tant class of finite Markov chains are now recurrent and aperiodic chains. These are called ergodic. The chain
of Fig. 6.3 is ergodic (except of state D), chain of Fig. 6.4 not (it is periodic). The next table shows the transi-
tion matrices, the eigenvalues, and the eigenvectors of both chains. For both chains λ = 1 exists, but only chain
6.3 converges. The probability matrix theorem now tells that every irreducible ergodic transition matrix (that
is the matrix containing only probabilities) has a steady state vector T to which the process converges.
(6.4)
This steady state vector is defined by the eigenvector of the matrix. Hence for ergodic matrices (these
are by far the most important) eq. 6.3 has always a solution. The theorem also implies that every transition
matrix has an eigenvector λ = 1.
Now look at the following matrix
It defines a transition matrix. Once state B or state D is reached the probability of change to B and D is 1.
These states cannot be left. The matrix has two absorbing states. In general a transition matrix has as many
absorbing states as it has ones on its diagonal.
Fig. 6.2 points also to another problem. How fast do Markov processes converge to the steady state. The
kk 0lim P X T→ ∞ =
0.5 0 0.2 00.2 1 0.4 00.2 0 0.1 00.1 0 0.3 1
⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠
P
A
BD
C
0.3
0.9
0.6
0.3
0.4
0.1
Fig. 6.3
Fig. 6.4
A B C
0.6
0.8 0.7
A B C D Eigenvalues Eigenvector 4A 0 0.3 0.3 0 -0.3 0.384111B 0.4 0.7 0 0 0.1 0.512148C 0.6 0 0.7 0.9 0.7 0.768221D 0 0 0 0.1 1 0
A B C Complex eigenvalues Eigenvector 3A 0.2 0 0.6 -0.05 0.597913 0B 0.8 0.3 0 -0.05 -0.597913 0C 0 0.7 0.4 1 0 0
Models in Biology 45
time to convergence is obviously connected to the probabilities in the matrix. The recurrence time of a state i
is now defined as the mean time at which the process returns to i. It can be shown that the recurrence times T
of any state i are inversely related to the stationary probabilities π.
(6.5)
The mean time to stay in any state is of course the inverse of the
probability not to leave the state. Hence in Fig. 6.3 the recurrence
time of state A is T = (0.38+0.51+0.77)/0.38) = 4.33 steps. The
question how long it will take to reach the stationary state is iden-
tical to the question what function describes the Fig. 6.3 and how
to calculate the parameter values of the function. With some
mathematics one can show that it is an exponential function of
the type
(6.6)
There are no simple solutions for the parameters.
A typical application of Markov chains in biology is succession.
For instance gravel pits have a distinct mosaic plant community
structure. Abandoned pits go through series of successional
stages. If we now map the plant distribution of the gravel pit im-
mediately after abandonment we get a matrix of initial states.
From other studies it is known with what frequency certain struc-
tural elements transform into others. Hence we have a transition
matrix. We can now describe the whole process of succession by
a Markov chain model. In this case we have a matrix of the initial
stage and the transition matrix. The model looks as follows
(6.7)
We consider six different plant community classes and have the
following transition matrix. Our
initial stage is given by Fig. 6.6
where the six communities are rep-
resented by different colours. After
t = 100 states (Fig. 6.7) our map
changed totally. Community types 1
and 2 dominate, 5 and 6 vanished.
After even 1000 states (Fig. 6.8) not
much had changed. However, very
slowly the frequency of community
3 raises. The proportion of commu-
ii
1T =π
btp ae −=
0= •ttX P XA B C D E F
1 4 2 1 1 1 32 3 1 1 3 2 23 3 3 1 1 3 14 3 3 2 4 2 25 1 1 3 2 2 36 2 3 2 4 3 37 1 2 3 1 3 48 1 2 3 1 3 4
Fig. 6.8
A B C D E F1 1 2 4 1 6 42 1 1 2 1 6 43 4 3 1 2 6 54 4 3 1 4 5 55 4 2 4 5 5 46 6 2 3 5 4 37 5 1 3 6 4 48 3 1 2 6 5 2
Fig. 6..6
A B C D E F1 3 2 3 3 4 22 2 1 2 4 3 33 2 2 3 1 3 14 3 4 1 2 2 35 2 1 1 3 2 16 2 3 2 1 2 37 4 1 1 2 1 38 2 3 2 1 3 2
Fig. 6.7
do while(jj.le.runs) do 100 i=1,arkol do 101 j=1,arnu ran1=ran(iseed) k=area(i,j) prob1=0 do 102 ii=1,spec prob1=prob(ii,k)+prob1 if(ran1.le.prob1)then area(i,j)=ii goto 101 endif 102 continue 101 continue 100 continue
Fig. 6.5, Photo Jan Meyer
1 2 3 4 5 61 0.12 0.20 0.21 0.29 0.18 0.062 0.26 0.02 0.31 0.03 0.31 0.033 0.05 0.05 0.08 0.12 0.28 0.284 0.09 0.29 0.03 0.26 0.02 0.165 0.26 0.22 0.27 0.21 0.08 0.226 0.21 0.22 0.09 0.09 0.13 0.25
Sum 1.00 1.00 1.00 1.00 1.00 1.00
46 Models in Biology
nity 4 remains stable but type 2, which dominated the intermediate stage of succession, decreases.
How to compute these pictures. Either you apply a commercial program that computes Monte Carlo
simulations and Markov chains or you write a program for your own. In our case I used a self written program
that iterates equation 6.2. For shorter series you can run a math program iteratively. Above a simple Fortran
solution is shown with which I computed the matrices on the left side.
Markov chains find application in probability theory. Assume for instance you have a virus with N
strains. Assume further that at each generation a strain mutates to another strain with probabilities ai→j. The
probability to stay is therefore 1-Σai→j. What is the probability that the virus is after k generations the same as
at the beginning. This can be modelled by a Markov chain with the following transition matrix
We get the desired probability from the matrix element p11 of Pk. Hence
The next table shows the respective Excel solution for a given transition matrix using the Matrix add in
for k =6. The requested probability is pii 0.23. Markov chains are therefore ideal tools for calculating probabili-
ties if we have multiple pathways to reach certain states. Particularly, they describe the probability to get in k
steps from state A to state B if the transition probabilities can be described using a transition matrix.
Random walk models A special example of Markov chains are random walks
Wee know already that random walks are defined by the general state equation
The state Nt is only defined by the previous state and a probability function of change. Typical examples
of such random walks are for instance animal movements. Let’s consider an animal A being at place x0. In a
next step it might turn to left with probability pl, turn to right with probability pr or walk straight on with prob-
ability ps Our random walk model looks at follows
i 1,1 1N
N1 1,i 1
1 a ap
a 1 a
≠
≠
⎛ ⎞−⎜ ⎟= ⎜ ⎟⎜ ⎟−⎝ ⎠
∑
∑
K
M O M
L
k k 1P U U−= • λ •
1t tN N ran+ = +
P A B C Eigenvalues EigenvectorsA 0.5 0.05 0.3 0.338197 0.814984 0.550947 0.368878B 0.3 0.8 0.1 0.561803 -0.450512 -0.797338 0.794506C 0.2 0.15 0.6 1 -0.364472 0.246391 0.482379
k = 5 Lk Inverse0.004424 0 0 0.878092 0.264583 -1.107265
0 0.055966 0 0.109323 -0.798204 1.2310890 0 1 0.607621 0.607621 0.607621
PN A B C ULk ULkU-1
A 0.230675 0.20048 0.258105 0.003606 0.030834 0.368878 0.230675 0.20048 0.258105B 0.47613 0.51785 0.43003 -0.001993 -0.044624 0.794506 0.47613 0.51785 0.43003C 0.293195 0.28167 0.311865 -0.001613 0.013789 0.482379 0.293195 0.28167 0.311865
Models in Biology 47
This is a recursive equation that describes a direc-
tional process. It’s two dimensional equivalent would
have the form
where the columns define forward or backward walk.
Recursive probability functions are also special cases of Markov chains. We can’t know, where the
animals ends his walk. But we might use a model of 5000 animals and try to give probabilities of the outcome.
Such a Monte Carlo simulation provides us with a frequency distribution of end points of the random walk.
Then we can tell that a typical animal ends his walk there or there. To model this we need the possible area into
our animal can walk, the number of possible states. This is indicated by the green area in Fig. 6.9. If this num-
ber is finite we speak of a bounded random walk. What is if the animal reaches the lower or upper boundary?
In the Figure the animal is reflected from the barrier.
1
l
n n s
r
px x p
p−
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
11 12
1 21 22
31 32
−
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
n n
p px x p p
p p
pl
pr
ps
Fig. 6.9
48 Models in Biology
7. The Weibull function and life history tables
Animals and plants have at each stage of their life history certain probabilities to die. These probabilities
can be combined in demographic or life history tables. A typical life table is shown in the first table. It is a
life table with discrete age categories. The first column gives the age t. The second column contains the num-
bers of individuals observed Nt. These are the individuals that survived to time t. At time t=0 we have the initial
population size at birth. Nt+1 - Nt =Dt gives the number of deaths in interval t. The mortality rate mt is the quo-
tient of deaths Dt and the original number Nt at interval t. The cumulative mortality rate Mt is the quotient of
the total numbers of deaths and the original population size N0.
(7.1)
lt = 1 - mt-1 is the proportion of individuals that survived to interval t. The cumulative proportion surviv-
ing st is of course 1 - mt. The mean number of individuals alive at each interval Lt is the arithmetic mean of Nt
and Nt+1.
(7.2)
To compute the further life expectancy Et from time t on we need the cumulative Lt. This is the total
number of years all the mean numbers of individuals will live. Tt is defined as
(7.3)
The mean life expectancy Et is then the quotient of Tt and Lt
(7.4)
t max
tt 1
t0
DM
N==
∑
t t 1t
N NL2
++=
t max
t ti t
T L=
= ∑
tt
t
TEL
=
Age Observed number of animals
Number dying
Mortality rate
Cumula-tive morta-
lity rate
Proportion surviving
Cumula-tive pro-portion
surviving
Mean number
alive
Cumula-tive Lt
Mean fur-ther life expec-tancy
t Nt Dt mt Mt lt st Lt Tt Et
0 1000 370 0.37 0.370 - - 1000.00 3028.00 3.03 1 630 210 0.33 0.580 0.63 0.630 815.00 2028.00 2.49 2 420 170 0.40 0.750 0.67 0.420 525.00 1213.00 2.31 3 250 140 0.56 0.890 0.60 0.250 335.00 688.00 2.05 4 110 50 0.45 0.940 0.44 0.110 180.00 353.00 1.96 5 60 26 0.43 0.966 0.55 0.060 85.00 173.00 2.03 6 34 19 0.56 0.985 0.57 0.034 47.00 88.00 1.86 7 15 10 0.67 0.995 0.44 0.015 24.50 41.00 1.65 8 5 2 0.40 0.997 0.33 0.005 10.00 16.00 1.60 9 3 2 0.67 0.999 0.60 0.003 4.00 6.00 1.50
10 1 1 1.00 1.000 0.33 0.001 2.00 2.00 1.00 11 0 - - 0.00 0.000 - - -
Models in Biology 49
The mean life expectancy of a six year old animal is therefore 1.86 years.
Next we have to deal with reproduction. Having discrete age classes we can assign each class a repro-
duction rate rt as the quotient of newborn individuals and total population size Nt. Hence
(7.5) This model does not contain deaths. To include death rates we extend the model to k equations contain-
ing population sizes
The population size of N0 is the sum of all reproduction processes at each age class. The mortalities are
given by eq. 7.5. The above equation can be expressed in matrix notation (the Leslie matrix)
or
(7.6) Hence N(2)=P•N(1)=P•P•N(0) an so on. We get
(7.7) From eq. 3.20 we get
where Λ is the matrix of eigenvalues of the transition matrix P.
To see whether the process is stationary we need
(7.8) In other words, the vector Nt is one of the eigenvectors of the transition matrix P having the eigenvalue λ
= 1.
One example from botany. Boucher and Mallona (Forest Ecol. Manage. 91(1997): 195-204) reported
survival rates of the lowland tropical rainforest tree Vochysia ferruginea after Hurrican devastation. Following
the population during 5 years they found that small adults produce in the mean 35.6 seedling and large adults
70.1 seedling. The probability for a seedling to stay a seedling in the next year was 0.209, the probability to
become a small sapling 0.01. All other seedling died. Using the respective data for small and large sapling and
r( t )tt t 1 t
t
r (t ) t0
Nr N N eN
dNr(t)dt N N eN
Δ+= → =
= → =
n
0 1 1 n n n ni 1
1 0 0
2 1 1
n n 1 n 1
N r N ...r N r N
N m NN m N...N m N
=
− −
= + =
==
=
∑
0 01 2 n 1 n
1 10
1
n 1 n 1
n 1n n
N (t 1) N (t)0 r r ... r rN (t 1) N (t)m 0 0 ... 0 0... ...0 m 0 ... 0 0N (t 1) N (t)... ... ... ... ... ...
0 0 0 ... m 0N (t 1) N (t)
−
− −
−
+⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟+⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟= •⎜ ⎟ ⎜ ⎟⎜ ⎟
+⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠ ⎝ ⎠
N(t 1) P N(t)+ = •
tt 0N P N= •
t 1t 0N U U N−= • • •Λ
t t tN P N 1N (P I) N 0= • = → − • =λ
50 Models in Biology
small and large adults they constructed the following Leslie matrix
The diagonal of this matrix give the probabilities of staying in the same class, the first row gives the
numbers of birth (propagules), the upper triangle the probabilities of regressing and the lower triangle the prob-
abilities of moving to another class. The largest eigenvector is > 1 that means the population was healthy
(why?). The age structure after 20 generations and the respective numbers of individuals starting from only 100
seedling comes from N20 = P20N0. Boucher and Mallona predicted a very fast recovery of the population after
destruction. Of course this projecting relies on the assumption that our matrix entries remain constant.
The net reproduction rate of a population is defined as
( 7 . 9 )
To compute R0 we need data of numbers of female offspring for each age class (so-called pivotal classes). This
is shown in the next table. R0 is now the sum of all lxbx
(7.10)
In our case the net reproductive rate is less than 1 and we infer that the population will decline.
From R0 we also get the mean generation length G. G is defined as
(7.11)
The example of the previous tables gives G = 29.9 years. The last values we need is the innate capacity
of increase. We know already the exponen-
tial growth model
( 7 . 1 2 )
r in this model gives the rate of increase.
Lotka gave an equation how to estimate r
from a life history table. He found that r
must satisfy the following condition
0Numbers of daughters in generation t+1RNumbers of daughters in generation t
=
t
0 i ii 1
R l b=
= ∑
n n
i i i ii 1 i 1
n0
i ii 1
l b i l b iG
Rl b
= =
=
= =∑ ∑
∑
rt0N N e=
Age Pivotal age
Observed number at pivotal age
Percent survi-ving
No of female
off-spring
Female offspring
per female ltbt
t t Nt lt Dt bt
0-9 4.5 950 0.95 0 0.000 0 10-19 14.5 905 0.905 50 0.055 0.05 20-29 24.5 870 0.87 410 0.471 0.41 30-39 34.5 740 0.74 300 0.405 0.3 40-49 44.5 710 0.71 100 0.141 0.1 50-59 54.5 640 0.64 5 0.008 0.005
R0 0.865
Seedling Small sapling Large sapling Small adult Large adult Complex Eigenvalues Starting densitiesSeedling 0.209 0.000 0.000 35.600 70.100 0.083 0.000 1000.000Small sapling 0.010 0.653 0.020 0.000 0.000 0.459 0.000 100.000Large sapling 0.000 0.170 0.407 0.000 0.000 0.650 0.335 0.000Small adult 0.000 0.000 0.570 0.731 0.000 0.650 -0.335 0.000Large adult 0.000 0.000 0.000 0.266 0.997 1.155 0.000 0.000
Final densitiesk=20 1.613 152.642 450.605 585.690 717.009 16877.634
0.032 3.068 9.061 11.780 14.425 339.2630.007 0.697 2.059 2.678 3.280 77.1230.010 0.939 2.770 3.600 4.408 103.8920.017 1.582 4.672 6.073 7.435 174.971
Models in Biology 51
( 7 . 1 3 )
Knowing R0 and G we can approximate r from
(7.14)
From our previous example we get
R = ln(0.865)/29.9 = -0.005 < 0. The population appears to decrease very slowly.
Our life tables also allow for the calculation of reproductive values in a population. The reproductive
value at age t is defined as the number of progenies plus the expected future number of progenies. It is
(7.15)
In the above example the reproductive value at age 25 is
0.9. The reproductive value at age 0 is of course identical
to the net reproductive rate R0.
Next we deal with is the Weibull distribution (after the
Swedish mathematician Waloddi Weibull, 1887-1979)
(7.16)
The Weibull distribution has the mean
and the variance
With Γ being the Gamma function described in the statstics lecture.
We get the cumulative density distribution from the integral
(7.17)
For β = 1 the Weibull distributions equals a simple expo-
nential function. For β = 3 the distributions approximates
(but not equals) a normal, for larger b the distribution be-
comes more and more left skewed (Fig. 7.1).
The Weibull distribution is particularly used in the analy-
sis of life expectancies and mortality rates. We simply
model the mortality rate m at time t using a general power
function model
nri
i ii 1
e l b 1−
=
=∑
0ln(R )rG
=
n ni i i i
t ti t 1 i tt t
l b l bV bl l= + =
= + =∑ ∑
1 xf ( , ) x eββ− −αα β = αβ
)11()( /1 +Γ= −
βα βxE
))11()12(())(( 2/22 +Γ−+Γ=− −
ββαμ βxE
xF( , ) 1 eβ−αα β = −
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2X
f(x)
β = 0.5
β = 1β = 2
β = 3
0
0.2
0.4
0.6
0.8
1
0 50 100 150t
F
b=1b=2b=3b=4
T = 100
Fig. 7.1
Fig. 7.2
52 Models in Biology
Using S(t) as the distribution of survival and modelling
this via an exponential model under the assumption that
the mortality rate is constant we get
For this usage the Weibull distribution is rewritten in a
two parametric form
( 7 . 1 8 )
where T denotes the characteristic life expectancy and t the age. f(β) gives then the probability that a given per-
son will die at age t. T is the age at which 63.2 % of the population already died. We get T from eq. 7.17 with α
= 1/T by setting t = T.
(7.19)
Having now data on age specific mortality rates f(β) we can estimate the characteristic life expectancy T
and the shape parameter β. The parameterized model then allows for the calculation of survival and mortality
rates and associated demographic variables at any given time t (Fig. 7.2). The Fig. shows the cumulative mor-
tality rates in dependence of time for T = 100 and different β using eq. 7.17.
Having data on mortality rates we can estimate the characteristic life time T from eq. 5.19. We use a
double log transformation
(7.20)
Using the cumulative mortality rates of the first tables we obtain b from the slope of a plot of ln[ln(1-F)]
against ln(t) (Fig. 7.3) We get a slope of 1.20, typical for many insects that have an exponential mortality - time
distribution. The intercept b is
This is the characteristic life expectancy. Interpolating the second column of the initial table give for 630 indi-
viduals to have died a very similar result around two years.
10m(t) m tβ−=
0m tS( t ) eβ−=
t1Ttf ( ) e
T T
β⎛ ⎞β − −⎜ ⎟⎝ ⎠β
β =
tTx 1F(1, ) 1 e 1 e 1 0.632
e
β
β
⎛ ⎞−⎜ ⎟⎜ ⎟− ⎝ ⎠β = − = − = − =
tln[ ln(1 F( )] ln ln(t) ln(T)T
β β β β⎛ ⎞− − = = −⎜ ⎟⎝ ⎠
0,891,2b (lnT) T e 2.09
−−
= − → = =β
y = 1.2009x - 0.8888
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
ln(t)
ln[-l
n(1-
F)]
Fig. 7.3
Models in Biology 53
8. Basic models in genetics
Our last lecture deals with simple models in genetics. Surely, genetics is one of the most mathematicized
parts of biology. This holds in particular for population genetics and evolutionary genetics. We will deal with
the logic behind some of the models. (Some examples of this lecture were inspired by the excellent Primer of
Population Biology by Edward O. Wilson and William H. Bossert, Stamford 1971)
One of the fundamental laws in genetics is the Hardy Weinberg law. It tells that without evolutionary
processes the frequency of alleles in a genome remains constant. To understand the
logic behind this law assume a gene with two alleles A and B. The frequencies of
A and B in the whole population are denoted with p and q. Hence p + q = 1. Now
assume crossing. The frequency of AA is the product pp, the frequency of BB is
the product qq. The frequency of AB is then pq + qp. The total frequencies of all
combinations are of course again 1. Hence pp + qq + 2pq = 1 or
(8.1)
This is the law first proposed in 1908 by the British mathematician Godfrey Harold Hardy (1877-1947)
and the German physician Wilhelm Weinberg (1867-1937).
Does the frequency of the alleles A and B
remain stable? Look at the following stan-
dard scheme. From this scheme we see
that the frequency of q in the next generation is q2 + qp. Hence
(8.2)
This is of course very simple and surely known from school. The biological interpretation is that with-
out selection, gene flow from one population to another or mutation events allele frequencies in a population
remain stable. No evolution occurs.
But evolution occurs! First we look at mutation events, where a gene or an allele A mutates to B. Muta-
tions are dose dependent, that means the number of mutation events M in a genome is proportional to the total
amount of the mutation inducing agent D, the dose. Hence
( 8 . 3 )
where k is a constant that describes the effectiveness of the agent to induce mutations.
The total amount of mutations of a certain gene locus is of course proportional to the total number of
that gene in the population N. M ∝ N. Hence we can define a new value, the mutation rate μ that describes this
proportionality.
(8.4)
The problem is now how to describe the rate of change of p due to mutation events. The rate of change
2 2 2( ) 2 1p q p pq q+ = + + =
2
2 2 2( )
( 2 ) ( )pq q q p q q
p pq q p q+ +
= =+ + +
M D M kD∝ → =
M kDN N
μ = =
AA AB BB SumAfter crossing p2 2pq q2 1Frequency of B 2pq / 2 q2 pq+q2
pp pqqp qqq
pqpBA
BA
Fig. 8.1
54 Models in Biology
is proportional to the frequency of the gene p. Hence
(8.5)
However, there are also back mutations from B to A The same law holds
(8.6)
Integration gives a fundamental rule for the change of gene frequencies in time under mutation pressure
(8.7)
Hence allele frequency decreases exponentially. We can also compute the equilibrium frequencies of p
and q due to mutation and back mutation rates. Because we are dealing with frequencies: p + q = 1. The total
rate of change in p is the sum of the change from p to q plus the change from q to p.
(8.8)
At equilibrium no further change should occur and we get
(8.9)
This will be the equilibrium frequency of A.
Next we look at gene flow. Assume a population has an allele A with frequency p. Due to migration the
next generation gets individuals from outside by immigration and looses individuals by emigration. What is the
new frequency of A if the frequency of A of the immigrating population is p*. Let i denote the immigration and
e the emigration rate. Both processes are again assumed to be proportional to actual density. The total number
of individuals before migration was N0. Ni individuals immigrated, Ne emigrated. Hence the new population
contains alleles A. The new frequency is
The new frequency of A can be determined solely from emigration and immigration rates and from the
frequency of A in the donor population. If the population size remains constant i equals e. The change of p
through time is then given by
(8.10)
Hence p = -i (p0 - p*) t. The change in allele frequency caused by gene drift is a linear process.
Next we consider selection. Assume a gene with two alleles A and B with frequencies p and q. If B is
under selection pressure the fraction of individuals having B is diminished by a factor s. s is termed the fitness.
dp pdt
μ= −
dq qdt
ν=
0
0
t
t
p p e
q q e
μ
ν
−=
=
dp p qdt
μ ν= − +
(1 )p q p p νμ ν νμ ν
= = − → =+
0 0 0 0* ( ) *e iN p N p N p N eN p iN p− + = − +
0 0 0
0 0 0
* (1 ) *1new
N p eN p iN p p e ippN eN iN e i
− + − += =
− + − +
0 0 0 0 0( *) ( *)newp dpp p p p i p p i p pt dt
Δ= − = − − − → =− −
Δ
Models in Biology 55
It is a value between 0 and 1. The
selection coefficient is now defined
as (1-s). With this coefficient we
have to multiply frequencies to get
the respective frequencies after selection. Now we have several possibilities to model this process of selection.
First assume that B is recessive and totally eliminated. The frequency of q after selection (in the next
generation) is
(8.11)
This looks like a recursive function. Because it doesn’t matter where we start we can generalise the last
equation and get
Can we simplify this equation to get the frequency of q after n generations? Look
(8.12)
An example: Assume a dog race should loose a recessive gene responsible for having curled hairs. This
gene has a frequency of 1 per 100 dogs. How long would it take to drop the frequency to a level 1 per 10000
dogs if all dogs with curled hairs are protected from breeding. Solving eq. 8.12 for n gives
More common than total elimination of recessive alleles is a partial elimination or partial selection
against this allele. Now our scheme
becomes
As before we compute the
frequency q of B after n generations
(8.13)
There is no general closed solution to this recursive equation. To simplify we obtain another strategy
and compute the change Δq = qn+1 - qn.
Now we consider only small changes in q and transform into a differential equation
0 0 01 2
0 0 0 02 1p q qq
p p q q= =
+ +
1
11n
nn
qqq−
−
=+
2
1 2 2 0
21 2 0
2
11 1 2 11
1
n
n n nn n
nn n
n
qq q q qq qqq q nq
q
−
− − −
−− −
−
+= = = → =
+ + +++
0
0
0.01 0.0001 9900 generations0.01*0.0001
n
n
q qn nq q
− −= → = =
20 0 0 0 0 1 1
1 2 2 20 0 1
(1 ) (1 ) (1 )1 1 1
n nn
n
p q s q q sq q sqq qsq sq sq
− −
−
+ − − −= = → =
− − −
2
1 2
(1 )1
n nn n
n
sq qq q qsq+
− −Δ = − =
−
AA AB BB SumBefore selection p0
2 2p0q0 q02 1
Selection coefficients 1 1 1-sSelection p0
2 2p0q0 0 p02+2p0q0
AA AB BB SumBefore selection p0
2 2p0q0 q02 1
Selection coefficients 1 1 1-sSelection p0
2 2p0q0 (1-s)q02 p0
2+2p0q0+(1-s)q02
56 Models in Biology
(8.14)
There is no simple solution to this equation. We
approximate a solution. s and q are both smaller
than 1. Hence sq2 << 1. We simplify 15.14 and get
(8.15)
Rearranging gives
There is an exact solution to this problem although complicated. It involves the ProductLog function a
solution of y = xex. However, numerical solutions are always possible. You need
(8.16)
Now assume that the allele B is selected against but on the other hand is produced by a constant muta-
tion rate μ. Does this process lead to an equilibrium frequency q of B. We model the change of B as before
An equilibrium means Δq = 0. Solving for q gives
This is the equilibrium frequency of B at constant s and μ.
Now we look at the important case where heterozygotes are superior. This is the well know heterosis
effect. By definition the fitness of the heterozygotes is 1. The change in q from one generation to the other
becomes
(8.17)
which has a quite simple solution for
q.
(8.18)
Now we treat the same problem from
2
2
(1 )1
dq sq qdt sq
− −=
−
2 (1 )dq sq qdt
= − −
2 (1 )dq sdn
q q= −
−
00
21 1 1ln( )
(1 )
nn qq
qdq snq qq q
−= + =
−∫
2
1 2(1 )(1 )
1osq qq p q q
sqμ μ→
− −Δ = − Δ = − −
−
2
2 (1 )1sq q
ssqμμ
μ= → =
+−
20 0 2 0
1 0 02 21 0 2 0
(1 ) (1 )1 (1 )q q s qq q q q
s q s q− + −
Δ = − = −− − −
1
1 2
sqs s
=+
AA AB BB SumBefore selection p0
2 2p0q0 q02 1
Selection coefficients 1-s1 1 1-s2
Selection (1-s1)p02 2p0q0 (1-s2)q0
2 1-s1p02-s2q0
2
Models in Biology 57
a matrix orientated perspective. The Excel
example beside shows the probability matri-
ces that we obtain if we cross the genotypes
AA, Aa and aa. Let P((A) = p be the prob-
ability of genotype A. Now we can con-
struct a matrix that contains the probabili-
ties to get each of the genotype in F1 from
each genotype in F0. For instance, the probabilities to get AA in F1 from AA in F0 is p: the probability to cross
with AA is p2 and this gives always AA. The probability to cross with Aa is 2pq and this give sin half the cases
AA, the probability to cross with aa in q2 and this gives never AA. The total probability to get AA in F1 from
AA in F0 is therefore p2 + 2pq/2 = p. Similar calculations lead to the Excel matrix above. Taking p = 0.6 and
q = 0.4 we obtain a probability matrix. The respective eigenvectors of the eigenvalue λ3 = 1 give the equilib-
rium frequencies. These are p = 0.6 and q = 0.4. We once again formulated the Hardy Weinberg law.
We can extend this result to a population of N di-
poloid individuals (2N genes). Consider random
genetic drift, that means a panmictic non-
overlapping population (random mating). The fre-
quency distribution of an allele A should follow a
Markov chain model (random walk) and therefore
approximate a binomial distribution. If we have i times A at the beginning of our drift the frequency of A is
i/2N. The probability to find j alleles in the next generation can be seen as a sampling of k genes out of the
original population with replacement. This is the classical Fischer-Wright model of genetic drift, the stan-
dard null model of population genetics. The probability to have j alleles in generation Fn under the condition
that there were i alleles in the previous generation Fn-1 is then
(8.19)
This probabilities can be expressed in a transition matrix where column probabilities add to 1. We see
that such a matrix has two absorbing states. In other words the process will end either in eliminating the allele
A or in a monodominance of A. The probability of loosing an allele A starting with k copies of A is then the
sum of all pi0. This sum has no trivial solution. Using diffusion theory from classical physicals it is possible to
show that the probability of extinction of an allele A with k copies is
(8.20)
Hence, a new mutation starting with 1 copy has a probability of pE = (1-1/2N) to go extinct and pS = 1-
(1-1/2N) = 1/2N to survive. The associated time to fix an allele in a population is approximately
(4.21)
In other words large effective population sizes prohibit the spread of new mutants in a population.
j 2N j
j
2N i ip 1j 2N 2N
−⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
E2N kp
2N−
=
FT 4N≈
P(A)=pP(a)=q0.6 0.4
AA Aa aa AA Aa aaAA p p/2 0 AA 0.6 0.3 0Aa q 1/2 p Aa 0.4 0.5 0.6aa 0 q/2 q aa 0 0.2 0.4
Eigenvalues Frequencies p q
0 0.408 0.802 0.58 0.36 0.6 0.40.5 -0.82 -0.27 0.773 0.48
1 0.408 -0.53 0.258 0.16Sum 1.61
Eigenvectors
Gneration n0 1 2 … 2N
0 1 p10 p20 … 01 0 p11 p21 … 02 0 p12 p22 … 0… … … … … …2N 0 p12N p22N … 1
Generation n-1
58 Models in Biology
Next we look at fitness. The frequencies of a genotype with respect to the genotype with the highest
frequency after selection is termed relative fitness. The sum of all relative fitness values after selection (the
sum column of our crossing schemes) is termed the average fitness of an individual. How does the average
fitness of an individual changes with respect to changes of the frequencies of A and B. We can include fitness
coefficients (1-si). To obtain new frequencies for p and q we multiply the three matrices with the respective
fitness values and get three new matrices that are shown below. Starting from initial frequencies we get then
new frequencies from the multiplication of the frequency vector with the transition matrix. After only five steps
we get fairly constant frequencies for AA = 0.04, Aa = 0.60, and aa = 0.36. Therefore p = 0.04+0.60/2 = 0.34
and q = 1-0.34= 0.66.
For simplicity we take our first scheme of eq. 8.11. The total elimination of the recessive allele. Let W
denote the average fitness. We need the derivative dW / dq, the change in average fitness with respect to
changes in the frequency of allele B.. Hence
But we are also interested in the change of W with respect to time, hence in dW / dt, For this we denote
dq / dt comes from eq. 8.11. dq / dt ≈ Δq = q1 –q0 ≡ q / (1+q) - q. We get for the change of average fit-
ness in time
This term is always positive and we see that eliminating B via selection leads to an increase in average
fitness.
2(1 ) 2(1 ) 2(1 ) 2 4 2dWW q q q q q qdq
= − + − → = − − + − = −
2dW dW dq dqqdt dq dt dt
= = −
322 ( )1 1
dW q qq qdt q q
= − − =+ +
Models in Biology 59
Literature (Coloured titles are available in the Institute or the library, red titles are of major importance)
Cornish Bowden A. 1999—Basic Mathematics for Biochemists—Oxford Univ. Press 2nd. Ed.
Ennos R. 1999—Statistical and Data Handling Skills in Biology—Longman.
Foster P. C. 1998—Easy Mathematics for Biologists— Taylor and Francis.
Jordan D. W., Smith P. 2002. Mathematical techniques. 3rd. Ed. Oxford Univ. Press
Grossman S., Tuner J. E. 1974—Mathematics for the Biological Sciences—Macmillan.
Martin J. 1972—Podstawy Matematyki i Statystyki—Warszawa.
Portenier C., Gromes W. 2003. Mathematik für Biologen und Humanbiologen. Script. Marburg
Scheiner S. M., Gurevitch J. (eds.) 2001—Design and Analysis of Ecological Experiments— Oxford Univ.
Press (2nd. Ed.).
Wilson E. O., Bossert W. H. 1971, A Primer of Population Biology. Sinauer (Stamford).
Murray J. D. 2003. Mathematical Biology. 3rd ed. Parts I and II. Springer New York.
Science Magazine special feature. 2004. Mathematics in biology. Science 303.
Napiórkowski K. 2001. Matematyka. http://info.fuw.edu.pl/~ajduk/FUW/matnkf/matematyka01_nkf.pdf
60 Models in Biology
Online archives and textbooks Online Mathematical textbooks (A large collection of textbooks) http://www.math.gatech.edu/~cain/textbooks/
onlinebooks.html
General mathematics (a collection of online lecture scripts and basic text on mathematics) http://
www.geocities.com/alex_stef/mylist.html
Mathematics online (a source of educational online texts) http://www.glencoe.com/sec/math/
Mathematics Virtual Library (Many links to interesting web pages and programs)
http://www.math.fsu.edu/Science/math.html
Math on the web (Search engine for all sorts of mathematics)
http://www.ams.org/mathweb/mi-mathinfo07.html
The Math Archive (Many links to interesting web pages and programs)
http://archives.math.utk.edu/
Eric Weisstein’s Mathematics ( a large online mathematics dictionary, with many examples) http://
mathworld.wolfram.com/
The Internet Mathematics library (a large collections of topics for pupils and students, math-beginners) http://
mathforum.org/library/
Mathematic resources (a large compilation of math internet pages)
http://www.clifton.k12.nj.us/cliftonhs/chsmedia/chsmath.html
Kolegium nauczyczielski. Materiały z wykładów. (Online scripts on various topics) http://info.fuw.edu.pl/
~ajduk/lect.html
Johannes Müller. 2003. Mathematical models in biology. Lecture term at TU Munich. http://www-
m12.ma.tum.de/lehre/model_2003/skript/skript.pdf
Population growth models (a nice collection of growth models) http://www.math.duke.edu/education/postcalc/
growth/contents.html.
Population growth models (A collection of growth model an animations) http://members.optusnet.com.au/
exponentialist/Growth_Models.htm
Competition models (for persons who are interested In a discussion of the Lotka Volterra models) http://
www.ub.rug.nl/eldoc/dis/fil/r.c.looijen/c11.pdf
The MacTutor history of mathematics (a very nice page on historical topics) http://www-history.mcs.st-
andrews.ac.uk/.
Excel Turorials (Many macros) http://www.herber.de/index.html?http://www.herber.de/forum/
archiv/104to108.htm.
Computational molecular Biology. (a very good side with examples how to use mathematics in molecular biol-
ogy). http://www.cs.bc.edu/~clote/ComputationalMolecularBiology/
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Mathematical software The Windows software collection (public domain and freeware)
http://archives.math.utk.edu/software/.msdos.directory.html (contains many very nice programs)
The mathematics virtual library (a collection of software pages) http://www.math.fsu.edu/Virtual/index.php?
f=21.
Guide to mathematical software (a search engine for math programs) http://gams.nist.gov//
Step by step derivatives (a very good program for computing derivatives) http://www.calc101.com/
webMathematica/derivatives.jsp#topdoit
Derivative calculator (a nice small but quite effective program for computing derivatives) http://cs.jsu.edu/
mcis/faculty/leathrum/Mathlets/derivcalc.html
JAVA Mathlets for Math Explorations (a nice collection of small math programs for everybody) http://
cs.jsu.edu/mcis/faculty/leathrum/Mathlets/
The integrator (a small but effective integration program)
http://www.integrals.com/index.en.cgi
The MathServ Calculus toolkit (a collection of Math applets for calculus computation)
http://www.math.vanderbilt.edu/~pscrooke/toolkit.html
Modelowanie reczwistości (a nice Polish page with a program collection and many further links) http://
www.wiw.pl/modelowanie/
Maple homepage. http://www.maplesoft.com/
Mathematica homepage (Wofram research) http://www.wri.com/
Mathworks homepage (Matlab) http://www.mathworks.com/
Mathtype (Office build in tool for mathematics writing) http://www.mathtype.com/en/products/mathtype/
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8. Important internet pages A very good elemtary math page for pupils and students: http://www.mathe-online.at/mathint.html
The best introduction to matrix algebra: http://numericalmethods.eng.usf.edu/matrixalgebrabook/downloadma/
matrixalgebra.pdf
Many links contains: http://archives.math.utk.edu/topics/linearAlgebra.html
Many good examples and a concise introduction At: http://people.hofstra.edu/faculty/Stefan_waner/RealWorld/
index.html
Matrix (a very good matrix algebra add in for excel) http://digilander.libero.it/foxes/index.htm
Markov chains and biology: http://www.statslab.cam.ac.uk/~james/Markov/