Demographic matrix models for structured populations

1. Demographic matrix models for structured populations Dominique ALLAINE

2. Structured populations 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

3. Structured populations Obviously, vital rates differ between individuals. These vital rates may differ according to age, to sex, to size category … Each individual is then described by a state at a given time. For example, an individual may be in the state « yearling female » at time t. Populations are made of individuals that are different. So, we can consider the structure of the population according to sex, to age, to size category, to status … The structure of a population at a given time is then characterised by a distribution function expliciting the number of individuals in each state

4. Structured-population models General characteristics Models based on the global size of a population does not take into account the differences between individuals. It is important to analyse the response of vital rates in structured populations to take into account differences between individuals Structured-population models aim at including the differences between individuals in vital rates. These models are applied to structured populations.

5. Structured-population models We will assume that all individuals in a same state experience the same environment and then respond in the same way. In other words, all individuals in a same state will have the same values of vital rates. Structured-population models are mathematical rules that allow to calculate the change over time of the distribution function of the number of individuals in each state. • Three types of mathematical approaches: • matrix models • delayed differential equation models • partial differential equation models

6. Matrix population models General characteristics The different states are discrete This means that individuals are classified into categories. For example, individuals are either males or females, are eitheir juveniles or yearlings or two-years old … When considering size, discrete categories have to be identified. We thus have to recognise categories and at a given time, each individual may be associated unambiguously to a category.

7. Matrix population models General characteristics The change in the population structure is in discrete time This means that the change of the distribution of the number of individuals in each state is given in discrete time. If we use, for example, an annual time scale, the number of individuals in each state is given for each year.

8. Matrix population models General characteristics First developped by P.H. Leslie during the 1940’s The number of individuals in each state at time t is given by a vector N(t) This vector is projected from time t to time t+1 by a population projection matrix M N(t+1) = M N(t) (n,1) (n,n) (n,1)

9. Matrix population models General characteristics • Different steps: • To determine the projection interval (time scale) • To determine the states of importance • To identify the vital rates necessary • To establish the life-cycle graph

10. Matrix population models Populations structured in age • The projection interval is usually the year in analyses on vertebrates • States of importance are often age, sex • Vital rates refer to: • proportion p of reproductive females per age class • fecundity f per age class • survival rate s per age and sex classes

11. 1 23 4 Matrix population models Populations structured in age Life cycle in the case of a populationstructured in age

12. Matrix population models Populations structured in age The matrix model corresponding to the previous life-cycle graph is:  Leslie matrix

13. R1 R2 E JAd Matrix population models Populations structured in stage Life cycle in the case of a populationstructured in stage

14. Matrix population models Populations structured in stage The matrix model corresponding to the previous life-cycle graph is:  Lefkovitch matrix

15. Matrix population models Constant linear models N(t+1) = M N(t) (n,1) (n,n) (n,1) In this case, the matrix M has constant coefficients Example: a two age-classes model

16. Matrix population models Constant linear models The Perron-Frobenius theorem • A non negative, irreductible, primitive matrix has 3 properties: • the first eigenvalue 1 is unique, real and positive • the right eigenvector w1 corresponding to 1 is strictly positive • the left eigenvector v1 corresponding to 1 is strictly positive

17. 1 23 4 Matrix population models Constant linear models A reductible matrix has some stages that make no contribution to some other stages Example:

18. 1 23 4 1 23 4 Matrix population models Constant linear models A matrix is primitive if the greatest common divisor of the lengths of the loops in the life-cycle graph is 1 Example: Primitive Imprimitive

19. Matrix population models Constant linear models Leslie matrices are often irreductible and primitive N(t+1) = M N(t) N(t+1) = M N(t) = Mt N(0)

20. Matrix population models Constant linear models • Let  an eigenvalue of the matrix M Let N(t) the right eigenvector of the matrix M associated to  Diagonalisation of the matrix M M = WW-1  M2 = WW-1 WW-1  M2 = W2W-1 Mt = WtW-1

21. Matrix population models Constant linear models Mt = WtW-1 Mt = WtV*

22. Matrix population models Constant linear models N(t) = Mt N(0) Remember: wi is a column vector and vi* is raw vector so their product is a matrix

23. Matrix population models Constant linear models It follows that:

24. Matrix population models Constant linear models Asymptotic results And consequently: The dynamic is driven by the dominant eigenvalue and associated eigenvectors

25. Matrix population models Constant linear models Asymptotic results It results that asymptotically: N(t+1) = M N(t) =  N(t)

26. Matrix population models Constant linear models Asymptotic results Asymptotically, the dominant eigenvalue  corresponds to the annual multiplication rate n(t+1) =  n(t)  = er where r is the population growth rate Asymptotically, the right eigenvector associated to the dominant eigenvalue  gives the stable state structure M W =  W

27. Matrix population models Constant linear models Asymptotic results Asymptotically, the left eigenvector associated to the dominant eigenvalue  gives the stable reproductive value, i.e. the contribution of each state to the population size VM =  V The annual multiplication rate, the stable state structure and the reproductive values depend on the values of vital rates but are independent of initial conditions (ergodicity)

28. Matrix population models Constant linear models Asymptotic results The damping ratio  expresses the rate of convergence to the stable population structure. In other words, the convergence will be more rapid when the dominant eigenvalue is large relative to the other eigenvalues.

29. Matrix population models Matrix population models Stochastic linear models N(t+1) = Mt N(t) (n,1) (n,n) (n,1) In this case, the matrix M has coefficients varying randomly with time Example: a two age-classes model

30. Matrix population models Stochastic linear models Growth rate The simple estimation of the multiplication rate, the stable state structure and the reproductive value calculated from eigenvalues and eigenvectors are no longer valid in stochastic linear models The population size at time t is: 

31. Matrix population models Stochastic linear models Growth rate It has been shown (Furstenberg & Kesten 1960) that: where Lns is the stochastic growth rate

32. Matrix population models Stochastic linear models Growth rate

33. Matrix population models Stochastic linear models Growth rate The analytical calculation of Ln s is not easy • The stochastic growth rate can be found by • Simulation • Approximation

34. - Ln ( n ) Ln ( n ) 1 å l = = l t 0 Ln Ln s i t t i Matrix population models Stochastic linear models Growth rate 1. Simulation The stochastic growth rate can be estimated from the average growth rate over a long simulation from the maximum likelihood estimator : An estimation of the stochastic growth rate from a simulation is:

35. Matrix population models Stochastic linear models Growth rate 1. Simulation Usually, the stochastic growth rate is calculated as the average of estimations obtained from several simulations

36. B1(s0,s0) P(1) B1(p1,p1) B1(s1,s1) B1(s2,s2) Matrix population models Stochastic linear models Growth rate 1. Simulation: environmental stochasticity

37. Matrix population models Stochastic linear models Growth rate 1. Simulation: demographic stochasticity

38. Number of young produced : Matrix population models Stochastic linear models Growth rate 1. Simulation: demographic stochasticity Number of reproductive females :

39. Matrix population models Stochastic linear models Growth rate 2. Approximation The stochastic growth rate can be analytically approximated when vital rates vary in a small way that is, their coefficients of variation are much less than 1 If we assume that all distributions are identical and independent, then, the stochastic growth rate is approximated by: where is the mean growth rate

40. Matrix population models Perturbation analysis From a biological point of view, it is important to know which factor has the greatest effect on  Perturbation analyses allow to predict the consequences of changes in the value of one (or more) vital rate on the value of  • Two concepts: • Sensitivity • Elasticity

41. Matrix population models Perturbation analysis 1. Sensitivity The sensitivity sij indicates how the value of  is impacted by a modification of the value of the parameter aij  The sensitivity is dependent on the metric of the parameter aij

42. Matrix population models Perturbation analysis 2. Elasticity The elasticity eij indicates the relative impact on  of a modification of the value of the parameter aij and The elasticity is independent on the metric of the parameter aij

43. Matrix population models Perturbation analysis 3. Prospective analyses The prospective analyses are done by calculating sensitivities or elasticities

44. Matrix population models Perturbation analysis 4. Retrospective analyses The objective of a retrospective analysis is to quantify the contribution of each vital rate to the variation in  The variability of  may be due to variations in vital rates in space (between populations) or in time (between years) for example The impact of a given component aij of the projection matrix on the variation of  depends on both the variation of the component and the sensitivity of  to this component

45. Matrix population models Perturbation analysis 4. Retrospective analyses A component aij will have a small contribution to the variation in  if this component does not change much or if  is not very sensitive to aij or if both are true. The contribution of aij to the variation in  involves the products of sij and the observed variation in aij • Two approaches are possible: • random effects • fixed effects

46. Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects We want to identify the contribution of vital rates to the variance in  during a time period for example (where time is considered as a random factor) Covariances between parameters are calculated directly from observed matrices Sensitivities are calculated from the mean matrix

47. Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects The contribution of the vital rate aij can be measured as: It is the sum of contributions including aij

48. Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects Tuljapurkar showed that: The contribution of the component aij can be measured as:

49. Matrix population models Perturbation analysis 4. Retrospective analyses a. Random effects These formula can also be rewritten as: The contribution of the component aij can be measured as:

50. Matrix population models Perturbation analysis 4. Retrospective analyses b. Fixed effects We want to identify the contribution of vital rates to the difference in multiplication rates between traitments. To simplify, consider two populations in different environments. These populations then vary according to the values of vital rates and, consequently in projection matrices. To these projection matrices A1 and A2 correspond two multiplication rates respectively equal to 1 and 2