1. Demographic PVAs Simulating Demographic Stochasticity and Density Dependence

2. 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

3. Variability caused by demographic stocahsticity in binomial vital rates

4. 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

5. How to use a uniform random number to decide between fates in a Monte Carlo simulation a 34 a 34 +a 44 1 Die a 34 +a 44 +a 54 a 34 +a 44 +a 54 +a 64 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

6. Adding demographic stochasticity to reproduction Determine if the individual lives Use a Poisson or another discrete distributions to obtain the individual fertility

7. Number of flowers P (y=r) = (e –μ μ r ) / r! Population 1; 2000 Population 2; 2000 Sampling from Poisson distributions to estimate flower production Avg. # Flowers 200020012002 pop11.450.751.26 pop21.3901.63 pop33.411.894.10 pop43.223.625.00 pop55.711.734.86 pop62.690.691.65

8. Number of seeds per flower Population 1; 2000 Sampling from normal distributions to estimate seed production Avg. # Seeds 200020012002 pop163 53 pop2566242 pop35443 pop4554938 pop5635441 pop6685033

9. 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

10. 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”?

11. 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.

12. 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

13. 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

14. The Iberian Lynx

15. Gaona et al. (1998) lynx population in Doñana National Park post breeding census birth-pulse population CubsJuvenilesFloatersBreeders Cubs000b x c x p x s 4 Juveniless1s1 000 Floaters0s 2 (1-g)s 3 (1-g)0 Breeders0s 2 (g)s 3 (g)s4s4 b = probability that a territory-holding female will breed in a given year c = number of cubs produced by females that do breed p = proportion of cubs that are female

16. 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 s 4 n 4 (t) breeding females still in possession of a territory and K- s 4 n 4 (t) vacant territories available. Gaona et al. (1998) lynx population in Doñana National Park post breeding census birth-pulse population

17. g= [K-s 4 n 4 (t)]/[s 2 n 2 (t)+ s 3 n 3 (t)]

19. Density dependent functions The Ricker function

20. Density dependent functions Beverton-Holt model

21. Density dependent functions

22. IIIIIIIVV I00b 3 f 3 s 0 E(t)b 4 f 4 s 0 E(t)b 5 f 5 s 0 E(t) IIs1s1 0000 III0s2s2 000 IV00s 2 (1-b 3 )00 V000s 3 (1-b 4 )0