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

6

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

12

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.

0100200300400500600700800900

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

0100200300400500600700800900

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

-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

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)(