Limit theorems for discrete-time metapopulation models

Transcription

1 MASCOS AustMS Meeting, October Page 1 Limit theorems for discrete-time metapopulation models Phil Pollett Department of Mathematics The University of Queensland pkp AUSTRALIAN RESEARCH COUNCIL Centre of Excellence for Mathematics and Statistics of Complex Systems

2 MASCOS AustMS Meeting, October Page 2 Collaborator Fionnuala Buckley (MASCOS) Department of Mathematics The University of Queensland

3 Metapopulations MASCOS AustMS Meeting, October Page 3

4 MASCOS AustMS Meeting, October Page 4 Metapopulations Colonization

5 Metapopulations MASCOS AustMS Meeting, October Page 5

6 MASCOS AustMS Meeting, October Page 6 Metapopulations Local Extinction

7 Metapopulations MASCOS AustMS Meeting, October Page 7

8 Metapopulations MASCOS AustMS Meeting, October Page 8

9 Metapopulations MASCOS AustMS Meeting, October Page 9

10 MASCOS AustMS Meeting, October Page 10 Metapopulations Total Extinction

11 Metapopulations MASCOS AustMS Meeting, October Page 11

12 Mainland-island configuration MASCOS AustMS Meeting, October Page 12

13 MASCOS AustMS Meeting, October Page 13 Mainland-island configuration Colonization from the mainland

14 MASCOS AustMS Meeting, October Page 14 Patch-occupancy models We record the number n t of occupied patches at each time t. A typical approach is to suppose that (n t, t 0) is Markovian.

15 MASCOS AustMS Meeting, October Page 14 Patch-occupancy models We record the number n t of occupied patches at each time t. A typical approach is to suppose that (n t, t 0) is Markovian. Suppose that there are J patches. Each occupied patch becomes empty at rate e (the local extinction rate), colonization of empty patches occurs at rate c/j for each suitable pair (c is the colonization rate) and immigration from the mainland occurs that rate v (the immigration rate).

16 MASCOS AustMS Meeting, October Page 15 A continuous-time stochastic model The state space of the Markov chain (n t, t 0) is S = {0, 1,...,J} and the transitions are: n n + 1 at rate n n 1 at rate en (ν + c ) J n (J n)

17 MASCOS AustMS Meeting, October Page 15 A continuous-time stochastic model The state space of the Markov chain (n t, t 0) is S = {0, 1,...,J} and the transitions are: n n + 1 at rate n n 1 at rate en (ν + c ) J n (J n) This an example of Feller s stochastic logistic (SL) model, studied in detail by J.V. Ross. Ross, J.V. (2006) Stochastic models for mainland-island metapopulations in static and dynamic landscapes. Bulletin of Mathematical Biology 68, Feller, W. (1939) Die grundlagen der volterraschen theorie des kampfes ums dasein in wahrscheinlichkeitsteoretischer behandlung. Acta Biotheoretica 5,

18 MASCOS AustMS Meeting, October Page 16 Accounting for life cycle Many species have life cycles (often annual) that consist of distinct phases, and the propensity for colonization and local extinction is different in each phase.

19 MASCOS AustMS Meeting, October Page 16 Accounting for life cycle Many species have life cycles (often annual) that consist of distinct phases, and the propensity for colonization and local extinction is different in each phase. Examples: The Vernal pool fairy shrimp (Branchinecta lynchi) and the California linderiella (Linderiella occidentalis), both listed under the Endangered Species Act (USA) The Jasper Ridge population of Bay checkerspot butterfly (Euphydryas editha bayensis), now extinct

20 MASCOS AustMS Meeting, October Page 17 Colonization and extinction phases For the butterfly, colonization is restricted to the adult phase and there is a greater propensity for local extinction in the non-adult phases.

21 MASCOS AustMS Meeting, October Page 17 Colonization and extinction phases For the butterfly, colonization is restricted to the adult phase and there is a greater propensity for local extinction in the non-adult phases. We will assume that that colonization (C) and extinction (E) occur in separate distinct phases.

22 MASCOS AustMS Meeting, October Page 17 Colonization and extinction phases For the butterfly, colonization is restricted to the adult phase and there is a greater propensity for local extinction in the non-adult phases. We will assume that that colonization (C) and extinction (E) occur in separate distinct phases. There are several ways to model this: A quasi-birth-death process with two phases A non-homogeneous continuous-time Markov chain (cycle between two sets of transition rates) A discrete-time Markov chain

23 MASCOS AustMS Meeting, October Page 18 Colonization and extinction phases For the butterfly, colonization is restricted to the adult phase and there is a greater propensity for local extinction in the non-adult phases. We will assume that that colonization (C) and extinction (E) occur in separate distinct phases. There are several ways to model this: A quasi-birth-death process with two phases A non-homogeneous continuous-time Markov chain (cycle between two sets of transition rates) A discrete-time Markov chain

24 MASCOS AustMS Meeting, October Page 19 A discrete-time Markovian model Recall that there are J patches and that n t is the number of occupied patches at time t. We suppose that (n t, t = 0, 1,...) is a discrete-time Markov chain taking values in S = {0, 1,...,J} with a 1-step transition matrix P = (p ij ) constructed as follows.

25 MASCOS AustMS Meeting, October Page 19 A discrete-time Markovian model Recall that there are J patches and that n t is the number of occupied patches at time t. We suppose that (n t, t = 0, 1,...) is a discrete-time Markov chain taking values in S = {0, 1,...,J} with a 1-step transition matrix P = (p ij ) constructed as follows. The extinction and colonization phases are governed by their own transition matrices, E = (e ij ) and C = (c ij ). We let P = EC if the census is taken after the colonization phase or P = CE if the census is taken after the extinction phase.

26 MASCOS AustMS Meeting, October Page 20 EC versus CE P = EC { t 1 t t + 1 t + 2 P = CE { t 1 t t + 1 t + 2

27 MASCOS AustMS Meeting, October Page 21 Assumptions The number of extinctions when there are i patches occupied follows a Bin(i,e) law (0 < e < 1): e i,i k = ( ) i e k (1 e) i k k (k = 0, 1,...,i). (e ij = 0 if j > i.) The number of colonizations when there are i patches occupied follows a Bin(J i,c i ) law: c i,i+k = (c ij = 0 if j < i.) ( ) J i c k i (1 c i ) J i k (k = 0, 1,...,J i). k

28 MASCOS AustMS Meeting, October Page 22 Examples of c i c i = (i/j)c, where c (0, 1] is the maximum colonization potential. (This entails c 0j = δ 0j, so that 0 is an absorbing state and {1,...,J} is a communicating class.)

29 MASCOS AustMS Meeting, October Page 22 Examples of c i c i = (i/j)c, where c (0, 1] is the maximum colonization potential. (This entails c 0j = δ 0j, so that 0 is an absorbing state and {1,...,J} is a communicating class.) c i = c, where c (0, 1] is a fixed colonization potential mainland colonization dominant. (Now {0, 1,...,J} is irreducible.)

30 MASCOS AustMS Meeting, October Page 22 Examples of c i c i = (i/j)c, where c (0, 1] is the maximum colonization potential. (This entails c 0j = δ 0j, so that 0 is an absorbing state and {1,...,J} is a communicating class.) c i = c, where c (0, 1] is a fixed colonization potential mainland colonization dominant. (Now {0, 1,...,J} is irreducible.) Other possibilities include c i = c(1 (1 c 1 /c) i ), c i = 1 exp( iβ/j) and c i = d + (i/j)c, where c + d (0, 1] (mainland and island colonization).

31 MASCOS AustMS Meeting, October Page 23 The proportion of occupied patches Henceforth we shall be concerned with X t (J) the proportion of occupied patches at time t. = n t /J,

32 MASCOS AustMS Meeting, October Page 24 Simulation: P = EC with c i = c 1 Mainland-Island simulation P = EC (J =100, x 0 =0.05, e =0.01, c =0.05) X (J) t t

33 MASCOS AustMS Meeting, October Page 25 The proportion of occupied patches Henceforth we shall be concerned with X t (J) the proportion of occupied patches at time t. = n t /J,

34 MASCOS AustMS Meeting, October Page 25 The proportion of occupied patches Henceforth we shall be concerned with X t (J) the proportion of occupied patches at time t. = n t /J, In the case c i = c, where the distribution of n t can be evaluated explicitly, we have established large-j deterministic and Gaussian approximations for (X t (J) ). Buckley, F.M. and Pollett, P.K. (2009) Analytical methods for a stochastic mainlandisland metapopulation model. In (Eds. Anderssen, R.S., Braddock, R.D. and Newham, L.T.H.) Proceedings of the 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation, Modelling and Simulation Society of Australia and New Zealand and International Association for Mathematics and Computers in Simulation, July 2009, pp

35 MASCOS AustMS Meeting, October Page 26 Simulation: P = EC with c i = c 1 Mainland-Island simulation P = EC (J =100, x 0 =0.05, e =0.01, c =0.05) X (J) t t

36 MASCOS AustMS Meeting, October Page 27 Simulation: P = EC (Deterministic path) 1 Mainland-Island simulation P = EC (J =100, x 0 =0.05, e =0.01, c =0.05) X (J) t Deterministic path t

37 MASCOS AustMS Meeting, October Page 28 Simulation: P = EC (Gaussian approx.) 1 Mainland-Island simulation P = EC (J =100, x 0 =0.05, e =0.01, c =0.05) X (J) t Deterministic path ± two standard deviations t

38 MASCOS AustMS Meeting, October Page 29 Gaussian approximations Can we establish deterministic and Gaussian approximations for the basic J-patch models (where the distribution of n t is not known explicitly)?

39 MASCOS AustMS Meeting, October Page 29 Gaussian approximations Can we establish deterministic and Gaussian approximations for the basic J-patch models (where the distribution of n t is not known explicitly)? Is there a general theory of convergence for discrete-time Markov chains that share the salient features of the patch-occupancy models presented here?

40 MASCOS AustMS Meeting, October Page 30 Simulation: P = EC with c i = (i/j)c 100 Metapopulation simulation P = EC (J =100, n 0 =95, e =0.3, c =0.8) nt t

41 MASCOS AustMS Meeting, October Page 31 General structure: density dependence We have a sequence of Markov chains (n (J) t ) indexed by J, together with a function f such that E(n (J) t+1 n(j) t ) = Jf(n (J) t /J), or, more generally, a sequence of functions (f (J) ) such that E(n (J) t+1 n(j) t ) = Jf (J) (n (J) t /J) and f (J) converges uniformly to f.

42 MASCOS AustMS Meeting, October Page 32 General structure: density dependence We have a sequence of Markov chains (n (J) t ) indexed by J, together with a function f such that E(n (J) t+1 n(j) t ) = Jf(n (J) t /J), or, more generally, a sequence of functions (f (J) ) such that E(n (J) t+1 n(j) t ) = Jf (J) (n (J) t /J) and f (J) converges uniformly to f. We then define (X t (J) ) by X t (J) = n (J) t /J.

43 MASCOS AustMS Meeting, October Page 33 General structure: density dependence We have a sequence of Markov chains (n (J) t ) indexed by J, together with a function f such that E(X (J) t+1 X(J) t ) = f(x (J) t ), or, more generally, a sequence of functions (f (J) ) such that E(X (J) t+1 X(J) t ) = f (J) (X (J) t ) and f (J) converges uniformly to f.

44 MASCOS AustMS Meeting, October Page 34 General structure: density dependence We have a sequence of Markov chains (n (J) t ) indexed by J, together with a function f such that E(n (J) t+1 n(j) t ) = Jf(n (J) t /J), or, more generally, a sequence of functions (f (J) ) such that E(n (J) t+1 n(j) t ) = Jf (J) (n (J) t /J) and f (J) converges uniformly to f. We then define (X t (J) ) by X t (J) = n t (J) /J. We hope that if X (J) 0 x 0 as J, then (X t (J) ) FDD (x t ), where (x t ) satisfies x t+1 = f(x t ) (the limiting deterministic model).

45 MASCOS AustMS Meeting, October Page 35 General structure: density dependence Next we suppose that there is a function s such that Var(n (J) t+1 n(j) t ) = Js(n (J) t /J), or, more generally, a sequence of functions (s (J) ) such that Var(n (J) t+1 n(j) t ) = Js (J) (n t (J) /J) and s (J) converges uniformly to s.

46 MASCOS AustMS Meeting, October Page 36 General structure: density dependence Next we suppose that there is a function s such that J Var(X (J) t t+1 X(J) ) = s(x (J) t ), or, more generally, a sequence of functions (s (J) ) such that J Var(X (J) t t+1 X(J) and s (J) converges uniformly to s. ) = s (J) (X (J) t )

47 MASCOS AustMS Meeting, October Page 37 General structure: density dependence Next we suppose that there is a function s such that Var(n (J) t+1 n(j) t ) = Js(n (J) t /J), or, more generally, a sequence of functions (s (J) ) such that Var(n (J) t+1 n(j) t ) = Js (J) (n t (J) /J) and s (J) converges uniformly to s. We then define (Z t (J) ) by Z t (J) = J(X (J) t x t ).

48 MASCOS AustMS Meeting, October Page 38 General structure: density dependence Next we suppose that there is a function s such that Var(Z (J) t t+1 X(J) ) = s(x (J) t ), or, more generally, a sequence of functions (s (J) ) such that Var(Z (J) t t+1 X(J) and s (J) converges uniformly to s. ) = s (J) (X (J) t ) We then define (Z t (J) ) by Z t (J) = J(X (J) t x t ).

49 MASCOS AustMS Meeting, October Page 39 General structure: density dependence Next we suppose that there is a function s such that Var(n (J) t+1 n(j) t ) = Js(n (J) t /J), or, more generally, a sequence of functions (s (J) ) such that Var(n (J) t+1 n(j) t ) = Js (J) (n t (J) /J) and s (J) converges uniformly to s. We then define (Z t (J) ) by Z t (J) that if J(X (J) = J(X (J) t x t ). We hope 0 x 0 ) z 0, then (Z (J) (Z t ), where (Z t ) is a Gaussian Markov chain with Z 0 = z 0. t ) FDD

50 MASCOS AustMS Meeting, October Page 40 General structure: density dependence What will be the form of this chain?

51 MASCOS AustMS Meeting, October Page 40 General structure: density dependence What will be the form of this chain? Consider the simplest case, f (J) = f and s (J) = s.

52 MASCOS AustMS Meeting, October Page 40 General structure: density dependence What will be the form of this chain? Consider the simplest case, f (J) = f and s (J) = s. Formally, by Taylor s theorem, f(x t (J) ) f(x t ) = (X t (J) x t )f (x t ) + and so, since E(X (J) t t+1 X(J) ) = f(x (J) t ) and x t+1 = f(x t ), E(Z (J) t+1 ) = J (E(X (J) t+1 ) f(x t)) = f (x t ) E(Z (J) t ) +, suggesting that E(Z t+1 ) = a t E(Z t ), where a t = f (x t ).

53 MASCOS AustMS Meeting, October Page 41 General structure: density dependence We have Var(X (J) t+1 ) = Var(E(X(J) So, since J Var(X (J) t+1 X(J) Var(Z (J) t+1 ) = J Var(X(J) t t t+1 X(J) ) = s(x (J) t ), t+1 ) = J Var(f(X(J) t )) + E(Var(X (J) t+1 X(J) t )). )) + E(s(X (J) t )) a 2 tj Var(X t (J) ) + E(s(X t (J) )) (where a t = f (x t )) = a 2 t Var(Z (J) t ) + E(s(X (J) t )), suggesting that Var(Z t+1 ) = a 2 t Var(Z t ) + s(x t ).

54 MASCOS AustMS Meeting, October Page 41 General structure: density dependence We have Var(X (J) t+1 ) = Var(E(X(J) So, since J Var(X (J) t+1 X(J) Var(Z (J) t+1 ) = J Var(X(J) t t t+1 X(J) ) = s(x (J) t ), t+1 ) = J Var(f(X(J) t )) + E(Var(X (J) t+1 X(J) t )). )) + E(s(X (J) t )) a 2 tj Var(X t (J) ) + E(s(X t (J) )) (where a t = f (x t )) = a 2 t Var(Z (J) t ) + E(s(X (J) t )), suggesting that Var(Z t+1 ) = a 2 t Var(Z t ) + s(x t ). And, since (Z t ) will be Markovian,...

55 MASCOS AustMS Meeting, October Page 42 General structure: density dependence And, since (Z t ) will be Markovian, we might hope that Z t+1 = a t Z t + E t (Z 0 = z 0 ), where a t = f (x t ) and E t (t = 0, 1,...) are independent Gaussian random variables with E t N(0,s(x t )).

56 MASCOS AustMS Meeting, October Page 42 General structure: density dependence And, since (Z t ) will be Markovian, we might hope that Z t+1 = a t Z t + E t (Z 0 = z 0 ), where a t = f (x t ) and E t (t = 0, 1,...) are independent Gaussian random variables with E t N(0,s(x t )). If x eq is a fixed point of f, and J(X (J) 0 x eq ) z 0, then we might hope that (Z t (J) ) FDD (Z t ), where (Z t ) is the AR-1 process defined by Z t+1 = az t + E t, Z 0 = z 0, where a = f (x eq ) and E t (t = 0, 1,...) are iid Gaussian N(0,s(x eq )) random variables.

57 MASCOS AustMS Meeting, October Page 43 Convergence of Markov chains We can adapt results of Alan Karr for our purpose. Karr, A.F. (1975) Weak convergence of a sequence of Markov chains. Probability Theory and Related Fields 33, He considered a sequence of time-homogeneous Markov chains (X (n) t ) on a general state space (Ω, F) = (E, E) N with transition kernels (K n (x,a), x E,A E) and initial distributions (π n (A),A E). He proved that if (i) π n π and (ii) x n x in E implies K n (x n, ) K(x, ), then the corresponding probability measures (P π n n ) on (Ω, F) also converge: P π n n P π.

58 MASCOS AustMS Meeting, October Page 44 J-patch models: convergence Theorem For the J-patch models with c i = (i/j)c, if X (J) 0 x 0 as J, then (X (J) t 1,X (J) t 2,...,X (J) t n ) P (x t1,x t2,...,x tn ), for any finite sequence of times t 1, t 2,..., t n, where (x t ) is defined by the recursion x t+1 = f(x t ) with EC-model: f(x) = (1 e)(1 + c c(1 e)x)x CE-model: f(x) = (1 e)(1 + c cx)x

59 MASCOS AustMS Meeting, October Page 45 J-patch models: convergence Theorem If, additionally, J(X (J) 0 x 0 ) z 0, then (Z t (J) ) FDD (Z t ), where (Z t ) is the Gaussian Markov chain defined by Z t+1 = f (x t )Z t + E t (Z 0 = z 0 ), where E t (t = 0, 1,...) are independent Gaussian random variables with E t N(0,s(x t )) and EC-model: s(x) = (1 e)[c(1 (1 e)x)(1 c(1 e)x) +e(1 + c 2c(1 e)x) 2 ]x CE-model: s(x) = (1 e)[e + c(1 x)(1 c(1 e)x)]x

60 MASCOS AustMS Meeting, October Page 46 Simulation: P = EC 1 Metapopulation simulation P = EC (J =100, x 0 =0.95, e =0.4, c =0.8) X (J) t t

61 MASCOS AustMS Meeting, October Page 47 Simulation: P = EC (Deterministic path) 1 Metapopulation simulation P = EC (J =100, x 0 =0.95, e =0.4, c =0.8) Deterministic path X (J) t t

62 MASCOS AustMS Meeting, October Page 48 Simulation: P = EC (Gaussian approx.) 1 Metapopulation simulation P = EC (J =100, x 0 =0.95, e =0.4, c =0.8) Deterministic path ± two standard deviations X (J) t t

63 MASCOS AustMS Meeting, October Page 49 J-patch models: convergence In both cases (EC and CE) the deterministic model has two equilibria, x = 0 and x = x, given by EC-model: x = 1 1 e CE-model: x = 1 ( 1 e c(1 e) e c(1 e) )

64 MASCOS AustMS Meeting, October Page 49 J-patch models: convergence In both cases (EC and CE) the deterministic model has two equilibria, x = 0 and x = x, given by EC-model: x = 1 1 e CE-model: x = 1 ( 1 e c(1 e) e c(1 e) Indeed, we may write f(x) = x (1 + r (1 x/x )), r = c(1 e) e for both models (the form of the discrete-time logistic model), and we obtain the condition c > e/(1 e) for x to be positive and then stable. )

65 MASCOS AustMS Meeting, October Page 50 J-patch models: convergence Corollary If c > e/(1 e), so that x given above is stable, and J(X (J) 0 x ) z 0, then (Z t (J) ) FDD (Z t ), where (Z t ) is the AR-1 process defined by Z t+1 = (1 + e c(1 e))z t + E t (Z 0 = z 0 ), where E t (t = 0, 1,...) are independent Gaussian N(0,σ 2 ) random variables with EC-model: σ 2 = (1 e)[c(1 (1 e)x )(1 c(1 e)x ) +e(1 + c 2c(1 e)x ) 2 ]x CE-model: σ 2 = (1 e)[e + c(1 x )(1 c(1 e)x )]x

66 MASCOS AustMS Meeting, October Page 51 Simulation: P = EC 1 Metapopulation simulation P = EC (J =100, x 0 =0.95, e =0.3, c =0.8) X (J) t x = t

67 MASCOS AustMS Meeting, October Page 52 Simulation: P = EC (AR-1 approx.) 1 Metapopulation simulation P = EC (J =100, x 0 =0.95, e =0.3, c =0.8) X (J) t x = t

68 MASCOS AustMS Meeting, October Page 53 AR-1 Simulation: P = EC 1 AR-1 simulation P = EC (J =100, x 0 = , e =0.3, c =0.8) X (J) t x = t