Demographic PVA’s

1. Demographic PVA’s Assessing Population Growth and Viability

2. Structured populations in a deterministic environment • Deterministic projection models, when we do not have (or use) estimates of variation

3. 1456 123 9 vector: a representation of the population structure • It is a column of numbers that indicates the densities of individuals in each class in the population at one point in time n(t) =

4. a11(t) a21(t) a31(t) a12(t) a22(t) a32(t) a13(t) a23(t) a33(t) Each entry aij(t) in a projection matrix A(t) gives the number of individuals in class i at census (t+1) produced on average by a single individual in class j at census (t) A(t) =

5. a11(t) a21(t) a31(t) a12(t) a22(t) a32(t) a13(t) a23(t) a33(t) n1(t+1) n2(t+1) n3(t+1) n1(t) n2(t) n3(t) If we know the densities at census tn(t), we can project the densities at the next census n(t+1) =

6. a13(t) n3(t+1) a23(t) n3(t+1) a33(t) n3(t+1) a12(t) n2(t+1)+ a22(t) n2(t+1)+ a32(t) n2(t+1)+ a11(t) n1(t+1)+ a21(t) n1(t+1)+ a31(t) n1(t+1)+ n1(t+1) n2(t+1) n3(t+1) If we know the densities at census tn(t), we can project the densities at the next census n(t+1) =

7. In a constant environment… • A(t)=A • Population convergence

8. 0.1189 0.1047 0.7764 w= The stable distribution (w) is The unique vector containing the ultimate proportions of the population in each class given the constant projection matrix A The ultimate or long term growth rate (λ) is λ1=0.6389

9. 1.0000 1.0973 1.1139 v= The reproductive values (v) is The relative contribution to future population growth an individual currently in a particular class is expected to make Reproductive values take into account the number of offspring an individual might produce in each of the classes it passes through the future, the likelihood of the individual reaching those classes, the time required to do so, and the population growth rate

10. Eigenvalue sensitivities • The ultimate rate of population growth in a constant environment, λ1, depends on the magnitudes of all the elements in A, so changing any of them will change λ1. • Sij Sensitivity: is a useful measure of how much changes in a particular matrix element will change λ1

11. Sij Sensitivity: • It is the partial derivative of λ1 with respect to aij • It measures the change in λ1 that would result from a small change in aij , keeping all other elements of the matrix A fixed at their present values

12. 0.1082 0.1187 0.1205 0.0953 0.1046 0.1062 0.7067 0.7755 0.7872 S= Sensitivities

13. Sensitivity Slope=0.787

14. How to include stochastic environmental effects on matrix models? • Matrix selection

15. Choose 1: or or Year 1 matrix 1 Year 1 matrix 3 or or Choose 1: or or Choose 1: Year 2 matrix 1 Year 0 matrix 3 Year 2 matrix 2 Year 0 matrix 1 Year 2 matrix 3 If environmental conditions are aperiodic and uncorrelated, and moreover the probability of choosing a particular matrix does not change over time then environmental conditions are said to be: “independently and identically distributed” or “iid” Year 0 Matrix 2 Yar 1 matrix 2 Modeling using matrix selection

16. Mountain golden heather Number of realizations Population size at t = 50

17. Year 3 matrix 3.1 Year 3 matrix 3.2 Modeling samples from matrices by time since fire. In this (simplified) example, the fire return interval is 3 years: Year 0 fire matrix Use this: Year 1 matrix 1.1 Year 1 matrix 1.2 or or Year 1 matrix 1.3 Choose 1: Other years or Use this: Beyond interpolation, input pooled matrices reset fire

18. Estimating the Stochastic log Growth rate λs • By simulation Log[N(t+1)/N(t)] over all pairs of adjacent years

19. Estimating the Stochastic log Growth rate λs • Tuljapurkar’s approximation (an analytical solution)

20. Calculating Quasi-Extinction probability • Box 7.5