Count based PVA

Incorporating Density Dependence, Demographic Stochasticity, Correlated Environments,

Catastrophes and Bonanzas

Assumptions of the diffusion appoximation

• Population growth • Is unaffected by population density • Its only source of variability is environmental

stochasticity• No trends in its mean and the variance • Its values are not correlated in successive years• Moderate variability• No observation error

But..

• Incorporating these effects into PVA models require:

• more and better data

• more mathematically complex models

Negative density dependence

• The simplest way to incorporate negative density dependence is introduce a population ceiling to the density-independent population growth model

Nt+1=λtNt ;if Nt < K

K ;if Nt > K

The ceiling model

Program algo2 (prepared by Matt; 10,50,.55,.45,60)

Mean time to extinction

cdeec

T cdck 212

1 22

KcKc

T c log212

1 2

Where c=μ/σ2, d=log(Nc/Nx), and k=log(K/Nx)

If Nc=K and Nx =1 then:

Extinction risk predicted by the Ceiling Model

μ= 0.1

μ= 0.001

μ= -0.1σ2= μ

σ2= 2μ

σ2= 4μ

σ2= 8μ

Program tbarpedro

The theta logistic model

• A gradually changing growth rate

tt

tt K

NrNN 1exp1

0

20

40

60

80

100

120

0 10 20 30 40 50 60

Tiempo

Ta

ma

ño

de

po

bla

cio

nK = 100

r = 0.2

Theta:

4

1

0.3

tt

tt K

NrNN 1exp1

The theta logistic model

0

1

2

3

4

5

6

0 20 40 60 80 100 120

Population size

log(lam

bda)

.

Θ = 0.3

Θ = 1

Θ = 4

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60

Tiempo

Ta

ma

ño

de

po

bla

cio

nK = 100

r = 0.8

Theta:

4

1

0.3

tt

tt K

NrNN 1exp1

The Bay checkerspot butterfly

Euphydryas editha bayensis

0

1000

2000

3000

4000

5000

6000

7000

8000

1955 1960 1965 1970 1975 1980 1985 1990

Year

Estim

ate

d n

um

ber

of fe

male

s .

front

Harrison et al., 1991

JRC population

The negative association remains after removing the outlier in the right

-3

-2

-1

0

1

2

3

0 1000 2000 3000 4000 5000 6000 7000 8000

Tamaño de la poblacion

ln(la

mbd

a)

back

Density Dependent model

• Find the best model: Fit three models to the data using nonlinear least-squares regression of log(Nt+1/Nt) against Nt

Models to be tested:Density independent model: log(Nt+1/Nt)=r

The Ricker model log(Nt+1/Nt)=r(1-Nt/K)

The theta logistic model log(Nt+1/Nt)=r[1-(Nt/K)Θ]

Estimate the parameters of each model

Model Least-squares parameter estimates

r K Θ Residual Variance

Density independent

0.001673 1.3999

Ricker 0.3488 846.02 1.0722

Theta logistic 0.9941 551.38 0.4566 1.0165

Model maximum likelihood of a model

assuming normally distributed deviations is

• ln(Lmax) = -(q/2)[ln(2Vr) +1)

• Vr = residual variance

• q= Sample number

Maximum log likelihood

• The probability of obtaining the observed data given a particular set of parameter values for a particular model

• Information criterion statistics combine the maximum log likelihood for a model with the number of parameters it include to provide a measure of “support”

“Support” is higher for:

• models with higher likelihoods, and • models with fewer parameters

More complex models are penalized because more parameters will always lead to a better fit to the data, but at the cost of less precision in the estimate of each parameter and incorporation of spurious patterns from the data into future populations

Akaike Information Criteria

• To identify the best model:• AICc = -2 ln(Lmax ) + (2pq)/ (q-p-1)

p = Number of estimated parameters (including the residual variance)

q= sampling number

Akaike weights

Wi =exp[-0.5(AICc,i-AICc,best)]

exp[-0.5(AICc,i-AICbest)]

Compute the maximum log likelihood and Akaike weights for each model

Model Number of parameters

Including Vr

Log Lmax AIC Akaike weights

Density independent

2 -41.266 87.054 .07

Ricker 3 -37.799 82.689 .62

Theta logistic 4 -37.105 84.115 .31

Simulate the model to predict population viability

σ2 =qVr

q-1

Program theta_logistic

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Years into the future

Cum

ula

tive p

robabili

ty o

f quasi

-ext

inct

ion

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Years into the future

Cum

ula

tive p

robabili

ty o

f quasi

-ext

inct

ion

Program extprobpedro

Simulate the model to predict population viability

Program extprobpedro Program theta_denindeppedro

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Years into the future

Cum

ula

tive p

robabili

ty o

f quasi

-ext

inct

ion

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Years into the future

Cum

ula

tive p

robabili

ty o

f quasi

-ext

inct

ion

Allee effects

• We can simply set the quasi-extinction threshold at or above the population size at which Alee effects become important

• Explicitly include Alee effects in the population model

Nt+1 =Nt

2

A+Nt

е r-βNt

The parameters

0

2

4

6

8

1 10 100 1000

N

O

The potential offspring

Value at A

maximum

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000

N

'-b

etaN

.

Fraction of potential reproduction that is actually achieved

A discrete-time model with Alee effects generated by mate-finding

problems

00.5

11.5

22.5

33.5

1 10 100 1000

N

"r"

A discrete-time model with Alee effects generated by mate-

finding problems

0

10

20

30

40

50

60

0 5 10 15 20

Time, t

Po

pu

lati

on

siz

e, N

t

.

10

15

25

Combined effects of Demographic

Environmental stochasticity

t

m

iitt mCC

t

/1

tm

itit

td CC

mtV

1

2

1

1)(

t

dt

t

t

N

tVNf

N

Nt

)()()(

2

12

Combined effects of Demographic and

Environmental stochasticity

5 10 15 2010

-2

10-1

100

101

102

Time (years)

Pop

ulat

ion

dens

ity

5 10 15 2010

-2

10-1

100

101

102

Time (years)P

opul

atio

n de

nsity

r=0.1,K=15, Θ=1, b=.1 r=0.1,K=15, Θ=1, b=1.5

Correlation of deviations

y = -0.1877x - 0.0135

R2 = 0.0354

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

-3.000

-2.000

-1.000

0.000 1.000 2.000 3.000

Deviation in year t

Devia

tio

n in

year

t+1 .

Environmental correlation

• When the environmental effects on the population growth rate are correlated, the “effective” environmental variance in the log population growth rate is (Foley 1994):

[(1+ρ)/(1-ρ)]σ2

[(1+ρ)/(1-ρ)]σ2

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1

rho

vari

an

ce

Variance without correlation

Generate the correlated environmental variation

• Є= ρ Єt-1 +√σ2√ (1-ρ2)zt]

ρ = correlation coefficient

zt= random number drawn from a normal distribution with mean 0 and variance 1

Єt-1= is the sum of a term due to correlation with the previous environment deviation and a new random term, scaled by a factor to assure that the long string of Є is σ2

Extinction risk and correlation

0

5

10

15

0 5 10 15

0

5

10

15

0 5 10 15

Nt+

1

NtNt

r=0.8 r=1.4

)/1(exp1 KNrNN ttT

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Autocorrelation in environmental effects

Pro

babi

lity

of h

ittin

g qu

asi-e

xtin

ctio

n th

resh

old

Nx

at o

r be

fore

tm

ax

r=0.8

r=1.4

Catastrophes and Bonanzas

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Probabilty of extreme value

Pro

babi

lity

of h

ittin

g qu

asi-e

xtin

ctio

n th

resh

old

Nx

at o

r be

fore

tm

ax