Metapopulations with infinitely many patches

Transcription

1 Metapopulations with infinitely many patches Phil. Pollett The University of Queensland UQ ACEMS Research Group Meeting 10th September 2018 Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 1 / 33

2 Metapopulations Glanville fritillary butterfly (Melitaea cinxia) in the Åland Islands in Autumn Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 2 / 33

3 N-patch SPOM A (homogeneous) stochastic patch occupancy model (SPOM) Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 3 / 33

4 N-patch SPOM A (homogeneous) stochastic patch occupancy model (SPOM) Suppose that there are N patches. Let n t {0, 1,..., N} be the number of occupied patches at time t, and assume that (n t, t = 0, 1,... ) is a discrete-time Markov chain. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 3 / 33

5 N-patch SPOM A (homogeneous) stochastic patch occupancy model (SPOM) Suppose that there are N patches. Let n t {0, 1,..., N} be the number of occupied patches at time t, and assume that (n t, t = 0, 1,... ) is a discrete-time Markov chain. Colonization and extinction happen in distinct, successive phases. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 3 / 33

6 N-patch SPOM A (homogeneous) stochastic patch occupancy model (SPOM) Suppose that there are N patches. Let n t {0, 1,..., N} be the number of occupied patches at time t, and assume that (n t, t = 0, 1,... ) is a discrete-time Markov chain. Colonization and extinction happen in distinct, successive phases. For many species the propensity for colonization and local extinction is markedly different in different phases of their life cycle. 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 Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 3 / 33

7 Phase structure Colonization and extinction happen in distinct, successive phases. t 1 t t + 1 t + 2 t 1 t t + 1 t + 2 Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 4 / 33

8 Phase structure Colonization and extinction happen in distinct, successive phases. t 1 t t + 1 t + 2 We will we assume that the population is observed after successive extinction phases (CE Model). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 4 / 33

9 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 5 / 33

10 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 5 / 33

11 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 5 / 33

12 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 6 / 33

13 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 7 / 33

14 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 8 / 33

15 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 9 / 33

16 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 10 / 33

17 N-patch SPOM Colonization and extinction happen in distinct, successive phases. Colonization: unoccupied patches become occupied independently with probability c(n t/n), where c : [0, 1] [0, 1] is continuous, increasing and concave, with c(0) = 0 and c (0) > 0. Extinction: occupied patches remain occupied independently with probability s. We thus have the following Chain Binomial structure 1 : ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 [Bin(m, p) is a binomial random variable with m trials and success probability p.] 1 Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-time metapopulation models. Probability Surveys 7, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 11 / 33

18 Evanescence: c (0) (1 s)/s 100 CE Model simulation (N = 100, n0 = 95, s = 0.56, c(x) = cx with c =0.7) nt t Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 12 / 33

19 Quasi stationarity: c (0) > (1 s)/s 100 CE Model simulation (N = 100, n0 = 5, s = 0.8, c(x) = cx with c =0.7) nt t Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 13 / 33

20 N patches Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 14 / 33

21 Infinitely many patches Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 15 / 33

22 Infinite-patch SPOM Prelude Recall the N-patch model: ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 16 / 33

23 Infinite-patch SPOM Prelude Recall the N-patch model: ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 Lemma If c has a continuous second derivative near 0, then, for fixed n, Bin(N n, c(n/n)) d Poi(mn), as N, where m = c (0). [Poi(λ) is a Poisson random variable with expectation λ.] Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 16 / 33

24 Infinite-patch SPOM Prelude Recall the N-patch model: ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 Lemma If c has a continuous second derivative near 0, then, for fixed n, Bin(N n, c(n/n)) d Poi(mn), as N, where m = c (0). [Interpretation: m is the mean per-capita number of colonizations.] [Poi(λ) is a Poisson random variable with expectation λ.] Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 16 / 33

25 Infinite-patch SPOM Prelude Recall the N-patch model: ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 Lemma If c has a continuous second derivative near 0, then, for fixed n, Bin(N n, c(n/n)) d Poi(mn), as N, where m = c (0). [Interpretation: m is the mean per-capita number of colonizations.] [Poi(λ) is a Poisson random variable with expectation λ.] This suggests the following infinite-patch model: n d t+1 = Bin ( n t + Poi ( ) ) mn t, s Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 16 / 33

26 Infinite-patch SPOM Prelude Recall the N-patch model: ( ( ) ) = d Bin n t + Bin N n t, c(n t/n), s n t+1 Lemma If c has a continuous second derivative near 0, then, for fixed n, Bin(N n, c(n/n)) d Poi(mn), as N, where m = c (0). [Interpretation: m is the mean per-capita number of colonizations.] [Poi(λ) is a Poisson random variable with expectation λ.] This suggests the following infinite-patch model: n t+1 d = Bin ( n t + Poi ( mn t ), s ) Claim The process (n t, t = 0, 1,... ) is a branching process (Galton-Watson-Bienaymé process) whose offspring distribution has pgf G(z) = (1 s(1 z))e ms(1 z). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 16 / 33

27 Infinite-patch SPOM Claim The process (n t, t = 0, 1,... ) is a branching process (Galton-Watson-Bienaymé process) whose offspring distribution has pgf G(z) = (1 s(1 z))e ms(1 z). (Think of the census times as marking the generations, the particles as being the occupied patches, and the offspring as being the occupied patches that they replace, notionally, in the succeeding generation.) Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 17 / 33

28 Infinite-patch SPOM Claim The process (n t, t = 0, 1,... ) is a branching process (Galton-Watson-Bienaymé process) whose offspring distribution has pgf G(z) = (1 s(1 z))e ms(1 z). (Think of the census times as marking the generations, the particles as being the occupied patches, and the offspring as being the occupied patches that they replace, notionally, in the succeeding generation.) The mean number of offspring is µ = (1 + m)s. So, for example, E(n t n 0) = n 0µ t. Theorem 1 Extinction occurs with probability 1 if and only if m (1 s)/s; otherwise extinction occurs with probability η n 0, where η is the unique fixed point of G in the interval (0, 1). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 17 / 33

29 Infinite-patch SPOM Claim The process (n t, t = 0, 1,... ) is a branching process (Galton-Watson-Bienaymé process) whose offspring distribution has pgf G(z) = (1 s(1 z))e ms(1 z). (Think of the census times as marking the generations, the particles as being the occupied patches, and the offspring as being the occupied patches that they replace, notionally, in the succeeding generation.) The mean number of offspring is µ = (1 + m)s. So, for example, E(n t n 0) = n 0µ t. Theorem 1 Extinction occurs with probability 1 if and only if m (1 s)/s; otherwise extinction occurs with probability η n 0, where η is the unique fixed point of G in the interval (0, 1). [Recall the earlier condition for evanescence: c (0) (1 s)/s] Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 17 / 33

30 Infinite-patch SPOM with regulation Assume the following structure: where m(n) 0. n t+1 d = Bin ( n t + Poi ( m(n t) ), s ) Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 18 / 33

31 Infinite-patch SPOM with regulation Assume the following structure: n t+1 d = Bin ( n t + Poi ( m(n t) ), s ) where m(n) 0. [A moment ago we had m(n) = mn.] Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 18 / 33

32 Infinite-patch SPOM with regulation Assume the following structure: n t+1 d = Bin ( n t + Poi ( m(n t) ), s ) where m(n) 0. [A moment ago we had m(n) = mn.] We will consider what happens when the initial number of occupied patches n 0 becomes large. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 18 / 33

33 Infinite-patch SPOM with regulation Assume the following structure: n t+1 d = Bin ( n t + Poi ( m(n t) ), s ) where m(n) 0. [A moment ago we had m(n) = mn.] We will consider what happens when the initial number of occupied patches n 0 becomes large. For some index N write m(n) = Nµ(n/N), where µ is a continuous function. We may take N to be simply n 0 or, more generally, following Klebaner 2, we may interpret N as being a threshold with the property that n 0/N x 0 as N. 2 Klebaner, F.C. (1993) Population-dependent branching processes with a threshold. Stochastic Process. Appl. 46, Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 18 / 33

34 Infinite-patch SPOM with regulation By choosing µ appropriately, we may allow for a degree of regulation in the colonization process. For example, µ(x) might be of the form µ(x) = rx(a x) (0 x a) (logistic growth); µ(x) = xe r(1 x) (x 0) (Ricker dynamics); µ(x) = λx/(1 + ax) b (x 0) (Hassell dynamics). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 19 / 33

35 Infinite-patch SPOM with regulation By choosing µ appropriately, we may allow for a degree of regulation in the colonization process. For example, µ(x) might be of the form µ(x) = rx(a x) (0 x a) (logistic growth); µ(x) = xe r(1 x) (x 0) (Ricker dynamics); µ(x) = λx/(1 + ax) b (x 0) (Hassell dynamics). µ(x) = mx (x 0) (branching). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 19 / 33

36 Infinite-patch SPOM with regulation By choosing µ appropriately, we may allow for a degree of regulation in the colonization process. For example, µ(x) might be of the form µ(x) = rx(a x) (0 x a) (logistic growth); µ(x) = xe r(1 x) (x 0) (Ricker dynamics); µ(x) = λx/(1 + ax) b (x 0) (Hassell dynamics). µ(x) = mx (x 0) (branching). We can establish a law of large numbers for Xt N at census t measured relative to the threshold. = n t/n, the number of occupied patches Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 19 / 33

37 Infinite-patch SPOM with regulation By choosing µ appropriately, we may allow for a degree of regulation in the colonization process. For example, µ(x) might be of the form µ(x) = rx(a x) (0 x a) (logistic growth); µ(x) = xe r(1 x) (x 0) (Ricker dynamics); µ(x) = λx/(1 + ax) b (x 0) (Hassell dynamics). µ(x) = mx (x 0) (branching). We can establish a law of large numbers for Xt N = n t/n, the number of occupied patches at census t measured relative to the threshold. Theorem 2 If X0 N p x 0 as N, then Xt N p x t for all t = 1, 2,..., where (x t) is determined by x t+1 = f (x t) (t = 0, 1,... ) with f (x) = s(x + µ(x)). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 19 / 33

38 Infinite-patch SPOM with regulation The proof uses the following very useful result. Lemma 3 Let U n, V n, and u be random variables, where U n and u are scalar. If E(U n V n) p u and Var (U n V n) p p 0 then U n u. 3 McVinish, R. and Pollett, P.K. (2012) The limiting behaviour of a mainland-island metapopulation. Journal of Mathematical Biology 64, Proof : We will use mathematical induction. Suppose Xt N p x t for some t {0, 1,... }. Since n d t+1 = Bin ( n t + Poi ( m(n ) t), s ), a simple calculation gives E(n t+1 n t) = s(n t + m(n t)). But, m(n) = Nµ(n/N). So, dividing by N gives E(Xt+1 N X t N ) = f (Xt N ), where f (x) = s(x + µ(x)). Since µ is continuous, so is f, and so E(X N t+1 X N t ) p f (x t) = x t+1. Another simple calculation yields Var (n t+1 n t) = s((1 s)n t + m(n t)), and so N Var (X N t+1 X N t ) = v(x N t ), where v(x) = s((1 s)x + µ(x)). Since v is continuous, v(xt N ) p v(x t), and hence Var (Xt+1 N X t N ) p 0. Using the technical lemma we arrive at Xt+1 N p x t+1, and the proof is complete. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 20 / 33

39 Infinite-patch SPOM with regulation x r Bifurcation diagram for the infinite-patch deterministic model with colonization following Ricker growth dynamics: x t+1 = 0.3 x t(1 + e r(1 x t ) ) (r ranges from 0 to 7.2). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 21 / 33

40 Infinite-patch SPOM with regulation 1 (a) 1 (b) (c) 3 (d) Simulation (blue circles) of the infinite-patch model with colonization following Ricker growth dynamics (x t+1 = 0.3 x t(1 + e r(1 x t ) )), together with the corresponding limiting deterministic trajectories (solid red). Here s = 0.3, N = 200, and (a) r = 0.84, (b) r = 1, (c) r = 4, (d) r = 5. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 22 / 33

41 Infinite-patch SPOM with regulation 2 (c) Simulation (blue circles) of the infinite-patch model with colonization following Ricker growth dynamics (x t+1 = 0.3 x t(1 + e r(1 x t ) )), together with the corresponding limiting deterministic trajectories (solid red). Here s = 0.3, N = 200, and r = 4. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 23 / 33

42 Infinite-patch SPOM with regulation 3 (d) Simulation (blue circles) of the infinite-patch model with colonization following Ricker growth dynamics (x t+1 = 0.3 x t(1 + e r(1 x t ) )), together with the corresponding limiting deterministic trajectories (solid red). Here s = 0.3, N = 200, and r = 5. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 24 / 33

43 Infinite-patch SPOM with regulation We can also get a handle on the fluctuations of (Xt N ) about (x t). Define Z N Zt N = N(Xt N x t) (t = 0, 1,... ). by Theorem 3 Suppose that µ is twice continuously differentiable with bounded second derivative, and suppose that Z0 N d z 0. Then, Z N converges weakly to the Gaussian Markov chain Z defined by Z d t+1 = s(1 + µ (x t))z t + E t, starting at (Z 0 =) z 0, with (E t) independent and E t N(0, v(x t)), where v(x) = s((1 s)x + µ(x)). The proof follows the programme laid out in the proof of Theorem 1 of Klebaner, F.C. and Nerman, O. (1994) Autoregressive approximation in branching processes with a threshold. Stochastic Process. Appl. 51, 1 7. But, note that (n t) is not a population-dependent branching processes with threshold; see note later. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 25 / 33

44 Infinite-patch SPOM with regulation 1 (a) 1 (b) (c) 3 (d) Same graphs as earlier, but now in (a), (b) and (c), the black dotted lines indicate ±2 standard deviations of the Gaussian approximation (in (c) every second point is proximate, thus indicating the extent of variation about each of the two limit cycle values). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 26 / 33

45 Infinite-patch SPOM with regulation 2 (c) Same graph as earlier, but now the black dotted lines indicate ±2 standard deviations of the Gaussian approximation (every second point is proximate, thus indicating the extent of variation about each of the two limit cycle values). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 27 / 33

46 Infinite-patch SPOM with regulation - stability Recall that f (x) = s(x + µ(x)). Notice that x will be a fixed point of f if and only if µ(x ) = ρ x, where ρ = (1 s)/s. Clearly 0 is a fixed point, but there might be others. If there is a unique positive fixed point x, it will be stable if µ (x ) < 1 and unstable if µ (x ) > 1 (need to consider higher derivatives when µ (x ) = 1). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 28 / 33

47 Infinite-patch SPOM with regulation - stability Recall that f (x) = s(x + µ(x)). Notice that x will be a fixed point of f if and only if µ(x ) = ρ x, where ρ = (1 s)/s. Clearly 0 is a fixed point, but there might be others. If there is a unique positive fixed point x, it will be stable if µ (x ) < 1 and unstable if µ (x ) > 1 (need to consider higher derivatives when µ (x ) = 1). Corollary 1 Suppose that f admits a unique positive stable fixed point x. Then, if X0 N p x, x t = x for all t and, assuming Z0 N z0, the limit process Z is an AR-1 process of the form Z d t+1 = s(1 + µ (x ))Z t + E t, starting at (Z 0 =)z 0, with iid errors E t N(0, (1 s 2 )x ). Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 28 / 33

48 Infinite-patch SPOM with regulation - stable equilibrium 1 (b) Simulation (blue circles) of the infinite-patch model with colonization following Ricker growth dynamics (x t+1 = 0.3 x t(1 + e r(1 x t ) )), together with the corresponding limiting deterministic trajectories (solid red). The black dotted lines indicate ±2 standard deviations of the Gaussian approximation. Here s = 0.3, N = 200, and r = 1, and x (stable) Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 29 / 33

49 Infinite-patch SPOM with regulation - stability Corollary 2 Suppose that f admits a stable limit cycle x0, x1,..., xd 1 with X0 N x0. Then, x nd+j = xj (n 0, j = 0,..., d 1) and, assuming Z0 N z0, the limit process Z has the following representation: (Yn, n 0), where Yn = (Z nd, Z nd+1,..., Z (n+1)d 1 ) d with Z 0 = z 0, is a d-variate AR-1 process of the form Yn+1 = AYn + E n, with iid errors E n N(0, Σ d ); A is the d d matrix 0 0 a a 2 A = , 0 0 a d where a j = s j j 1 i=0 (1 + µ (xi )), Σ d = (σ ij ) is the d d symmetric matrix with entries σ ij = a i a j i 1 k=0 v(x k )/a 2 k+1 (1 i j d), where v(x) = s((1 s)x + µ(x)), and the random entries, (Z 1,..., Z d 1 ), of Y0 have a Gaussian N(az 0, Σ d 1 ) distribution, where a = (a 1,..., a d 1 ). Furthermore, (Yn) has a Gaussian N(0, V ) stationary distribution, where V = (v ij ) has entries v ij = σ ij /(1 a 2 d). p Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 30 / 33

50 Infinite-patch SPOM with regulation - limit cycle (d = 2) 2 (c) Simulation (blue circles) of the infinite-patch model with colonization following Ricker growth dynamics (x t+1 = 0.3 x t(1 + e r(1 x t ) )), together with the corresponding limiting deterministic trajectories (solid red). The black dotted lines indicate ±2 standard deviations of the Gaussian approximation. Here s = 0.3, N = 200, and r = 4, and, x , x Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 31 / 33

51 Note Recall that n d t+1 = Bin ( n t + Poi(m(n t)), s ). Whilst (n t) does not exhibit the branching property (required for it to be a population-dependent branching processes with threshold), we can say the following. Theorem n t+1 Proof : d = Bin(n t, s) + Poi ( sm(n t) ) (independent RVs). ( E(z n t+1 n t) = E E (z ) ) n t+1 nt Poi(m(nt)), n t ) = E ((1 s + sz) nt +Poi(m(nt )) nt ) = (1 s + sz) nt E ((1 s + sz) Poi(m(nt )) nt = (1 s(1 z)) nt sm(nt )(1 z) e Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 32 / 33

52 Future directions An inhomogeneous SPOM keeps track of which patches are occupied: X N t = (X1,t N, X 2,t N,... ), where X i,t N is a binary variable indicating whether or not patch i is occupied at time t. (Again we consider a sequence of models indexed by a threshold N.) Assume that (X N t, t = 0, 1,... ) is a (countable-state) Markov chain with ( d = Bin (X Ni,t + Bin 1 Xi,t N, c( ) ) ) X N t, s i, X N i,t+1 a Chain Bernoulli structure. Approach: Following McVinish, R. and Pollett, P.K. (2010) Limits of large metapopulations with patch dependent extinction probabilities. Adv. Appl. Probab. 42, use point processes (St N = {s i : Xi,t N = 1}) and probability generating functionals [ ] G S N t (ξ) = E s i S N ξ(s i ), and hope that (St N ) converges weakly to a point process S t t, with G St+1 (ξ) = G St (H (ξ)) for suitable H. Phil. Pollett (The University of Queensland) Infinite-patch metapopulations 33 / 33