Matrix Population Models

1. Matrix Population Models • Life tables • Intro to matrix multiplication • Examples of age and stage structured models • Elasticity analysis

2. Life Tables and Matrices:Accounting demographic parameters Killer Whale Lefkovich matrix from Brault, S. and H. Caswell (1993) Sardine life table(Sardinops sagax) from Murphy (1967)

3. Matrix multiplication Scalar Multiplication - each element in a matrix is multiplied by a constant.

4. Matrix multiplication Multiply rows times columns. You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix.

5. Matrix multiplication Multiply rows times columns. You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix. Dimensions: 3 x 2 2 x 3 They must match. The dimensions of your answer.

6. 2 x 2 2 x 2 *Answer should be dimension ? 0(4) + (-1)(-2) 0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5)

7. Matrix Multiplication (Population Model): x = Answer should be dimension ?

8. Matrix Multiplication (Population Model): x =

9. Introduction to Matrix Models • Vital rates describe the development of individuals through their life cycle (Caswell 1989) • Vital rates are : birth, growth, development, reproductive, mortality rates • The response of these rates to the environment determines: • population dynamics in ecological time • the evolution of life histories in evolutionary time

10. The general form of an age-structured Leslie Matrix models “Projection Matrix”:

11. The general form of an age-structured Leslie Matrix models “Projection Matrix”:

12. Age Class 4 Age based matrix population model • Fecundity fx • Survivorship sx Age Class 3 Age Class 1 Age Class 2

13. The general form of Lefkovitch Matrix modelStage – structured Projection Matrix

14. The general form of Lefkovitch Matrix modelStage – structured Projection Matrix

15. Size Class 4 Stage-based matrix population model • Fecundity fx • Growth gx and Survivorship sx Size Class 3 Size Class 1 Size Class 2

16. Some of the utility of matrix population models • Population projection – deterministic and stochastic • Elasticity Analysis • Conservation • Management • Meta population dynamics

17. Caswell • Types of model variability

18. Population projectionExample: What is the population at t1? Juvenile Adult t0 fx NJuvenile NAdult px

19. x = fx Juvenile Adult px

20. Projecting this sample matrix indefinitely will result in the finite population growth rate: λ t0 x =

21. Look at the y-axis λ is on the natural log scale…

22. Stable age distribution

23. Stable age distribution. Expected in a static environment…

24. Population trajectories with process and observation error

25. Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations λ = 2.00 A 25% decrease in Age 1 survivorship results in a 12% decrease in population growth. λ = 1.76

26. Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations λ = 2.00 A 25% increase in Age 2 fecundity results in a 9% increase in population growth. λ = 2.18

27. Brault and Caswell

28. Elasticity • A type of “perturbation” analysis • The elasticityeijindicates the relative impact on  of a modification of the value of the parameteraij • Scaled, therefore The elasticityisindependent on the metric of the parameteraij and

29. Stage-specific survival and reproduction • G, P, F

30. Stage-specific survival and reproduction • G, P, F

31. Stage-specific survival and reproduction • Initialize matrix A