Simple & Multiple Regression 1: Simple Regression - Prediction models 1.
Simple population models
description
Transcript of Simple population models
Simple population models
BirthsDeaths
Population increase
Population increase = Births – deaths
t
Equilibrium
tttttttttt NdbNdNbNN )1(1
N: population sizeb: birthrated: deathrate
ttt
t
t
t
t
tt
t
tt
t
tt
t
tt
dbN
deathsN
birthsNdeathsbirths
NNr
Ndeathsd
Nbirthsb
ttt RNNrN )1(1
The net reproduction rate R = (1+bt-dt)
If the population is age structured and contains k age classes we get
k
ikkkk NbNbNbNbN
122110 ...
The numbers of surviving individuals from class i to class j are given by
211
122
011
)1(...
)1()1(
kkk NdN
NdNNdN
Leslie matrix
Assume you have a population of organisms that is age structured.Let fX denote the fecundity (rate of reproduction) at age class x.
Let sx denote the fraction of individuals that survives to the next age class x+1 (survival rates).Let nx denote the number of individuals at age class x
We can denote this assumptions in a matrix model called the Leslie model. We have w-1 age classes, w is the maximum age of an individual.
L is a square matrix.
1
2
1
0
...
wn
nnn
tN
000000...............0...0000...0000...000
...
2
2
1
0
13210
w
w
s
ss
sfffff
L
tt LNN 1
Numbers per age class at time t=1 are the dot product of the Leslie matrix with the abundance vector N at time t
01 NLN tt
k
ikkkk NbNbNbNbN
12211 ...
211
122
011
)1(...
)1()1(
kkk NdN
NdNNdN
ttn
nnn
s
ss
sfffff
n
nnn
1
2
1
0
2
2
1
0
13210
11
2
1
0
...
...
000000...............0...0000...0000...000
...
...
...
ww
w
w
vThe sum of all fecundities gives
the number of newborns
vn0s0 gives the number of
individuals in the first age class
Nw-1sw-2 gives the number of individuals in the last classv
The Leslie model is a linear approach.It assumes stable fecundity and mortality rates
The effect pof the initial age composition disappears over timeAge composition approaches an equilibrium although the whole
population might go extinct.Population growth or decline is often exponential
An example
Age class N0 L1 1000 0 0.5 1.2 1.5 1.1 0.2 0.0052 2000 0.4 0 0 0 0 0 03 2500 0 0.8 0 0 0 0 04 1000 0 0 0.5 0 0 0 05 500 0 0 0 0.3 0 0 06 100 0 0 0 0 0.1 0 07 10 0 0 0 0 0 0.004 0
Generation0 1 2 3 4 5 6 7 8 9 10 11 12
1000 6070.05 4335.002 3216.511 3709.4 3822.356 3338.88 3195.559 3199.811 3037.552 2873.77 2783.134 2681.0592000 400 2428.02 1734.001 1286.604 1483.76 1528.942 1335.552 1278.224 1279.924 1215.021 1149.508 1113.2542500 1600 320 1942.416 1387.201 1029.284 1187.008 1223.154 1068.442 1022.579 1023.939 972.0165 919.60631000 1250 800 160 971.208 693.6003 514.6418 593.504 611.5769 534.2208 511.2894 511.9697 486.0083
500 300 375 240 48 291.3624 208.0801 154.3925 178.0512 183.4731 160.2662 153.3868 153.5909100 50 30 37.5 24 4.8 29.13624 20.80801 15.43925 17.80512 18.34731 16.02662 15.33868
10 0.4 0.2 0.12 0.15 0.096 0.0192 0.116545 0.083232 0.061757 0.07122 0.073389 0.064106
At the long run the population dies out.Reproduction rates
are too low to counterbalance the high mortality rates
0.01
0.1
1
10
100
1000
10000
0 5 10 15 20 25
Abun
danc
e
Time
12345
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7
Important properties:1. Eventually all age classes
grow or shrink at the same rate
2. Initial growth depends on the age structure
3. Early reproduction contributes more to population growth than late reproduction
1
2
1
0
...
wn
nnn
tN
tt LNN 1
01 NLN tt
000000...............0...0000...0000...000
...
2
2
1
0
13210
w
w
s
ss
sfffff
L
0.01
0.1
1
10
100
1000
10000
0 5 10 15 20 25
Abun
danc
e
Time
12345
6
7
Does the Leslie approach predict a stationary point where population abundances doesn’t change any more?
ttt NLNN 1
We’re looking for a vector that doesn’t change direction when multiplied with the Leslie matrix.
This vector is the eigenvector U of the matrix.Eigenvectors are only defined for square matrices.
ULU
0dtdN
0][0
UILULU
I: identity matrix
ttt rNNNN 1
Exponential population growth
rteNNrNdtdN
0
ttt RNNrN )1(1
0
2000
4000
6000
8000
10000
0 5 10 15 20
Popu
latio
n si
ze
Time
r=0.1r=-0.1 The exponential growth model predicts continuous increase or decrease in population size.
What is if there is an upper boundary of population size?
0
2
4
6
8
10
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14
0 10 20 30 40 50
Time [h]
Vol
ume
Saccharomyces cerevisiae
0
5
10
15
20
25
0 5 10 15 20
Popu
latio
n gr
owth
Time
Maximum growth
)()1(2
KNKrN
KNrNN
KrrN
dtdN
We assume that population growth is a simple quadratic function with a maximum growth at an
intermediate level of population size
The Pearl-Verhulst model of population growth
The logistic growth equation
2 0 2 2( )0
( )1 1 1 ( / 1) a t t a t a t
K K KN te Ce K N e
0.2612.74( )
1 9.32 tN te
Second order differential equation
00.20.40.60.8
11.2
0 5 10 15 20 25
Popu
latio
n si
ze
Time
)10(5.011)(
tetN
)10(5.011)(
tetN
Logistic population increase and decrease
K is the carrying capacity (maximum population size)
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
)()1(2
KNKrN
KNrNN
KrrNN t
tt
tttt
Time lags
)()1( )()(
)()(
2)()( K
NKrN
KN
rNNKrrNN t
tt
tttt
The time lag model assumes that population growth might dependet not on th eprevious but on some even
earlier population states.
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1000
0 20 40 60 80 100 120
Time
N(t)
r=0.2; = 0;K =500
A
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=2.099; = 0;K =500
B
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1000
0 20 40 60 80 100 120
Time
N(t)
r=1; = 1;K =500
C
Low growth rates generates a typical logistic growth
High growth rates can generate increasing population cycles
Intermediate growth rates give damped oscillations
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=2.95; = 0;K =500
G
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=2.7; = 0;K =500
F
0100200300400500600700800900
1000
0 20 40 60 80 100 120
Time
N(t)
r=3.05; = 0;K =500
H
High growth rates give irregular but stable oscillations
Certain high growth rates produce pseudochaos
Too high growth rates lead to extinction
A simple deterministic model is able to produce very different time series and even pseudochaos
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10
0 0.5 1 1.5 2 2.5 3
N(t)
dN/d
t
N1 N2
m = rK / 4
Critical harvesting rate
mKNKrN
dtdN )(
Constant harvesting
Constant harvesting rate m
mKNKrN )(0
Where is the stationary point whwere fish population becomes stable
-4m + rK > 0
Alfred James Lotka (1880-1949)
Vito Volterra (1860-1940)
Life tables
AgeObserved number of animals
Number dying
Mortality rate
Cumula-tive
mortality rate
Propor-tion survi-
ving
Cumula-tive
proportion surviving
Mean number
alive
Cumula-tive Lt
Mean further 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 - - -
Demographic or life history tables
Cumulative mortality rate Mt
t max
tt 1
t0
DM
N
lt = 1 - mt-1 is the proportion of individuals that survived to interval t
Cumulative proportion surviving st is 1 - mt
t t 1t
N NL2
Mean number of individuals alive at each interval
AgeObserved number of animals
Number dying
Mortality rate
Cumula-tive
mortalityrate
Propor-tion survi-
ving
Cumula-tive
proportion surviving
Mean number
alive
Cumula-tive Lt
Mean further 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 - - -
t max
t ti t
T L
tt
t
TEL
Mean life expectancy at age t
AgeObserved number of animals
Number dying
Mortality rate
Cumula-tive
mortalityrate
Propor-tion survi-
ving
Cumula-tive
proportion surviving
Mean number
alive
Cumula-tive Lt
Mean further 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 - - -
0Numbers of daughters in generation t+1RNumbers of daughters in generation t
Net reproduction rate of a population
t
0 i ii 1
R l b
Mean generation length is the mean period elapsing betwee the birth of prents and the birth
of offspringn n
i i i ii 1 i 1
n0
i ii 1
l b i l b iG
Rl b
G = 30.2 years
Age Pivotal age (class mean)
Observed number at pivotal age
Fraction surviving
No. of female offspring
Female offspring per
femaleR
t Nt lt Dt bt ltbt ltbt0 1000
0-9 4.5 950 0.95 0 0 0 010-19 14.5 905 0.905 50 0.055249 0.05 0.72520-29 24.5 870 0.87 410 0.471264 0.41 10.04530-39 35.5 740 0.74 300 0.405405 0.3 10.6540-49 44.5 710 0.71 100 0.140845 0.1 4.4550-59 54.5 640 0.64 5 0.007813 0.005 0.272560-69 64.5 530 0.53 0 0 0 070-79 74.5 410 0.41 0 0 0 080-89 84.5 210 0.21 0 0 0 090-99 94.5 50 0.05 0 0 0 0
Sum 0.865 26.1425Generation time 30.22254
1 xf ( , ) x e
xF( , ) 1 e
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2X
f(x)
= 0.5
= 1 = 2
= 3
The Weibull distribution is particularly used in the analysis of life expectancies and mortality rates
t1Ttf ( ) e
T T
The two parametric form
The characteristic life expectancy T is the age at which 63.2% of the population already died.
For t = T we get
0
0.2
0.4
0.6
0.8
1
0 50 100 150t
Fb=1b=2b=3b=4
T = 100
1 xf ( , ) x e
Tt
eF 1)(
B: shape paramterT: timeT: characteristic life time
632.0111)(
eeF T
T
How to estimate the parameter and the characteristic life expectancy T from life history tables?
tln[ ln(1 F( )] ln ln(t) ln(T)T
We obtain b from the slope of a plot of ln[ln(1-F)] against ln(t)
---0.0000.00--011
1.002.002.000.0010.331.0001.001110
1.506.004.000.0030.600.9990.67239
1.6016.0010.000.0050.330.9970.40258
1.6541.0024.500.0150.440.9950.6710157
1.8688.0047.000.0340.570.9850.5619346
2.03173.0085.000.0600.550.9660.4326605
1.96353.00180.000.1100.440.9400.45501104
2.05688.00335.000.2500.600.8900.561402503
2.311213.00525.000.4200.670.7500.401704202
2.492028.00815.000.6300.630.5800.332106301
3.033028.001000.00--0.3700.3737010000
EtTtLtstltMtmtDtNtt
Meanfurther life
expec-tancy
Cumula-tive Lt
Meannumber
alive
Cumula-tive
proportionsurviving
Propor-tion survi-
ving
Cumula-tive
mortalityrate
Mortalityrate
Numberdying
Observednumber ofanimals
Age
---0.0000.00--011
1.002.002.000.0010.331.0001.001110
1.506.004.000.0030.600.9990.67239
1.6016.0010.000.0050.330.9970.40258
1.6541.0024.500.0150.440.9950.6710157
1.8688.0047.000.0340.570.9850.5619346
2.03173.0085.000.0600.550.9660.4326605
1.96353.00180.000.1100.440.9400.45501104
2.05688.00335.000.2500.600.8900.561402503
2.311213.00525.000.4200.670.7500.401704202
2.492028.00815.000.6300.630.5800.332106301
3.033028.001000.00--0.3700.3737010000
EtTtLtstltMtmtDtNtt
Meanfurther life
expec-tancy
Cumula-tive Lt
Meannumber
alive
Cumula-tive
proportionsurviving
Propor-tion survi-
ving
Cumula-tive
mortalityrate
Mortalityrate
Numberdying
Observednumber ofanimals
Age 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)]
1.20.89b (ln T) T e 3.85
Tt
eF 1)(