Demographic PVAs

Simulating Demographic Stochasticity and Density

Dependence

Demographic stochasticity

• Simulated by performing so-called Monte Carlo simulations: the fate of each individual in a certain class and a certain year is decided by a set of independent random choices, all of which are based on the same set of mean vital rates

• However this can greatly slow a program

Variability caused by demographic stocahsticity in binomial vital rates

0 5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Number of individuals

Var

ianc

e in

out

com

e

Techniques to how to perform Monte Carlo simulations

• For vital rates that are probabilities:

Pick a uniform random number and compare its value to the probabilities of different fates an individual might experience

How to use a uniform random number to decide between fates in a Monte Carlo simulation

a34

a34+a44

1

Die

a34+a44+a54

a34+a44+a54+a64

Survive and shrink to class 3

Survive and stay to class 4

Survive and grow to class 5

Survive and grow to class 6

0

Adding demographic stochasticity to reproduction

• Determine if the individual lives

• Use a Poisson or another discrete distributions to obtain the individual fertility

Number of flowers

0

5

10

15

20

25

30

35

40

0 1 2 3 5

Number of flowers

Fre

qu

ency

Observed

Predicted

0

5

10

15

20

25

30

35

40

0 1 2 3 5

Number of flowers

Fre

qu

ency

Observed

Predicted

P (y=r) = (e –μ μ r) / r!

Population 1; 2000

Population 2; 2000Sampling from Poisson distributions to estimate flower production

Avg. # Flowers

2000 2001 2002

pop1 1.45 0.75 1.26

pop2 1.39 0 1.63

pop3 3.41 1.89 4.10

pop4 3.22 3.62 5.00

pop5 5.71 1.73 4.86

pop6 2.69 0.69 1.65

Number of seeds per flower

number of seeds per flower

138.0

124.0

110.0

96.0

82.0

68.0

54.0

40.0

26.0

12.0

8

6

4

2

0

Std. Dev = 28.97

Mean = 63.2

N = 30.00

Population 1; 2000

Sampling from normal distributions to estimate seed production

Avg. # Seeds

2000 2001 2002

pop1 63 63 53

pop2 56 62 42

pop3 54 43 43

pop4 55 49 38

pop5 63 54 41

pop6 68 50 33

Including density dependence

• Two factors make it more difficult to account for density dependence in demographic PVAs

1. We will rarely have as many years of vital rate estimates from a demographic study

2. There are many more variable that are potentially density dependent

Three questions we must consider:

• Which vital rates are density-dependent?

• How do those rates change with density?

• Which classes contribute to the density that each vital rate “feels”?

Two more limited approaches to building density-dependent projection matrix models

1. Assume that there is a maximum number or density of individuals in one or more classes, or of the population as a whole , that the available resources can support, and construct a simulation model that prevents the population vector from exceeding this limit.

Two more limited approaches to building density-dependent projection matrix models

2. Choose one or at most a few vital rates that, on the basis some evidence are suspected to be strongly density dependent

Placing a limit on the size of one or more classes

Caps or ceilings on population density are most often used to introduce density dependence when the focal species is territorial

The Iberian Lynx

Gaona et al. (1998)lynx population in Doñana National Park post

breeding census birth-pulse population

Cubs Juveniles Floaters Breeders

Cubs 0 0 0 b x c x p x s4

Juveniles s1 0 0 0

Floaters 0 s2 (1-g) s3 (1-g) 0

Breeders 0 s2 (g) s3 (g) s4

b = probability that a territory-holding female will breed in a given yearc = number of cubs produced by females that do breed

p = proportion of cubs that are female

• Density dependence acts on g, which represent the probability that a surviving juvenile or floater will acquire a territory next year

• Just before the birth pulse that precedes census t+1, there will be s4n4(t) breeding females still in possession of a territory and K- s4n4(t) vacant territories available.

Gaona et al. (1998)lynx population in Doñana National Park post

breeding census birth-pulse population

Gaona et al. (1998)lynx population in Doñana National Park post

breeding census birth-pulse population

• g= [K-s4n4(t)]/[s2n2(t)+ s3n3(t)]

Density dependent functions

tEstEs exp000

The Ricker function

Density dependent functions

tE

stEs

1

000

Beverton-Holt model

Density dependent functions

)(tnfbtE iii

I II III IV V

I 0 0 b3f3s0E(t) b4f4s0E(t) b5f5s0E(t)

II s1 0 0 0 0

III 0 s2 0 0 0

IV 0 0 s2 (1-b3) 0 0

V 0 0 0 s3 (1-b4) 0