Mathematical Biology Aim : To understand exponential growth Objectives: 1) Understand derivation...
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Transcript of Mathematical Biology Aim : To understand exponential growth Objectives: 1) Understand derivation...
Mathematical Biology
Aim : To understand exponential growth
Objectives: 1) Understand derivation of the model 2) Introduce relative and absolute rates
3) Solve a simple differential equation (separable variables)
4) Obtain expressions for doubling time 5) Introduce radio-carbon dating 6) Summarise the properties of the exponential distribution
Lecture 2: Growth without limit
Steps of model making
1) Collect data2) Identify main processes3) Write a “word” model for these main processes4) Express the model as mathematical formulae5) Solve the model 6) Interpret properties of solution in biological terms7) Make testable predictions8) Test, and see that model is not perfect9) Back to step 1)
USA Population
0
50
100
150
1750 1800 1850 1900 1950
Popu
lati o
n x1
06
Year
Collared Dove
0
5
10
15
20
1956 1958 1960 1962 1964 1966
Popu
lati o
n x1
03
Year
0
100
200
300
400
0 1 2 3 4
Tur b
idi ty
Time (h)
Exponential Growth of E. coli
Solving differential equations
Procedure:
1) Classify the equation (for now ignore this)
2) Find general solution (includes arbitrary constant)
3) Find particular solution (constant fixed to a value)
4) Rearrange the solution if necessary (i.e. Y=…)
5) Check the solution- using the differential equation + init. cond.- using dimensional analysis
Dimensional Analysis
Allows us to interpret parameters in our equations…andto check that our maths has all worked out correctlyThis just convention, but I shall use square brackets for dimensions, and introduce the following generic classes L to represent some sort of length (cm, feet, miles, etc.) T to represent a time (seconds, years, days, etc.) M to represent a mass (grams, kilos, etc.)Rules… 1) if you have A = B then must have [A] = [B] 2) if you have A + B then must have [A] = [B] (= [A+B]) 3) [AB] = [A][B] and [A/B] = [A]/[B] 4) if you have exp(A), sin(A) etc., then [A] = 1 5) [dY/dt] = [Y]/[t] = [Y] T-1
0
5
10
15
0 2 4 6 8 10
Con
c en t
r at io
n
Time (min)
Maths all works fine for exponential decay :e.g. drug concentration in blood
Rate of metabolism is proportional to concentration…just take < 0 to reflect decreasing concentration
Radioactive decay
0 expY Y t
dY Ydt
(again < 0)
12
ln(2)T
Radio Carbon Dating
Summary: the exponential function
0tY Y e
0
CumulativeY
tY0
Rate
t
ddYt
0Y
Linearized
t
lnY
lnY0
0
t
Y0
Y
0
ddYt
0Y
t t
lnYlnY0
0