Limit theorems for discrete-time metapopulation models

Transcription

1 MASCOS Insiu de Mahémaiques de Toulouse, June Page 1 Limi heorems for discree-ime meapopulaion models Phil Polle Deparmen of Mahemaics The Universiy of Queensland hp:// pkp AUSTRALIAN RESEARCH COUNCIL Cenre of Excellence for Mahemaics and Saisics of Complex Sysems

2 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 2

3 MASCOS Insiu de Mahémaiques de Toulouse, June Page 3 Meapopulaions Colonizaion

4 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 4

5 MASCOS Insiu de Mahémaiques de Toulouse, June Page 5 Meapopulaions Local Exincion

6 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 6

7 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 7

8 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 8

9 MASCOS Insiu de Mahémaiques de Toulouse, June Page 9 Meapopulaions Toal Exincion

10 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 10

11 Mainland-island configuraion MASCOS Insiu de Mahémaiques de Toulouse, June Page 11

12 MASCOS Insiu de Mahémaiques de Toulouse, June Page 12 Mainland-island configuraion Colonizaion from he mainland

13 Meapopulaions MASCOS Insiu de Mahémaiques de Toulouse, June Page 13

14 MASCOS Insiu de Mahémaiques de Toulouse, June Page 14 Pach-occupancy models We record he number n of occupied paches a each ime. A ypical approach is o suppose ha (n, 0) is Markovian.

15 MASCOS Insiu de Mahémaiques de Toulouse, June Page 14 Pach-occupancy models We record he number n of occupied paches a each ime. A ypical approach is o suppose ha (n, 0) is Markovian. Suppose ha here are N paches. Each occupied pach becomes empy a rae e (he local exincion rae), colonizaion of empy paches occurs a rae c/n for each suiable pair (c is he colonizaion rae) and immigraion from he mainland occurs ha rae v (he immigraion rae).

16 MASCOS Insiu de Mahémaiques de Toulouse, June Page 15 A coninuous-ime sochasic model The sae space of he Markov chain (n, 0) is S = {0, 1,...,N} and he ransiions are: n n + 1 a rae n n 1 a rae en (ν + c ) N n (N n)

17 MASCOS Insiu de Mahémaiques de Toulouse, June Page 15 A coninuous-ime sochasic model The sae space of he Markov chain (n, 0) is S = {0, 1,...,N} and he ransiions are: n n + 1 a rae n n 1 a rae en (ν + c ) N n (N n) This an example of Feller s sochasic logisic (SL) model, sudied in deail by J.V. Ross. Ross, J.V. (2006) Sochasic models for mainland-island meapopulaions in saic and dynamic landscapes. Bullein of Mahemaical Biology 68, Feller, W. (1939) Die grundlagen der volerraschen heorie des kampfes ums dasein in wahrscheinlichkeiseoreischer behandlung. Aca Bioheoreica 5,

18 MASCOS Insiu de Mahémaiques de Toulouse, June Page 16 Accouning for life cycle Many species have life cycles (ofen annual) ha consis of disinc phases, and he propensiy for colonizaion and local exincion is differen in each phase.

19 MASCOS Insiu de Mahémaiques de Toulouse, June Page 16 Accouning for life cycle Many species have life cycles (ofen annual) ha consis of disinc phases, and he propensiy for colonizaion and local exincion is differen in each phase. Examples: The Vernal pool fairy shrimp (Branchineca lynchi) and he California linderiella (Linderiella occidenalis), boh lised under he Endangered Species Ac (USA) The Jasper Ridge populaion of Bay checkerspo buerfly (Euphydryas ediha bayensis), now exinc

20 MASCOS Insiu de Mahémaiques de Toulouse, June Page 17 Colonizaion and exincion phases For he buerfly, colonizaion is resriced o he adul phase and here is a greaer propensiy for local exincion in he non-adul phases.

21 MASCOS Insiu de Mahémaiques de Toulouse, June Page 17 Colonizaion and exincion phases For he buerfly, colonizaion is resriced o he adul phase and here is a greaer propensiy for local exincion in he non-adul phases. We will assume ha ha colonizaion (C) and exincion (E) occur in separae disinc phases.

22 MASCOS Insiu de Mahémaiques de Toulouse, June Page 17 Colonizaion and exincion phases For he buerfly, colonizaion is resriced o he adul phase and here is a greaer propensiy for local exincion in he non-adul phases. We will assume ha ha colonizaion (C) and exincion (E) occur in separae disinc phases. There are several ways o model his: A quasi-birh-deah process wih wo phases A non-homogeneous coninuous-ime Markov chain (cycle beween wo ses of ransiion raes) A discree-ime Markov chain

23 MASCOS Insiu de Mahémaiques de Toulouse, June Page 18 Colonizaion and exincion phases For he buerfly, colonizaion is resriced o he adul phase and here is a greaer propensiy for local exincion in he non-adul phases. We will assume ha ha colonizaion (C) and exincion (E) occur in separae disinc phases. There are several ways o model his: A quasi-birh-deah process wih wo phases A non-homogeneous coninuous-ime Markov chain (cycle beween wo ses of ransiion raes) A discree-ime Markov chain

24 MASCOS Insiu de Mahémaiques de Toulouse, June Page 19 A discree-ime Markovian model Recall ha here are N paches and ha n is he number of occupied paches a ime. We suppose ha (n, = 0, 1,...) is a discree-ime Markov chain aking values in S = {0, 1,...,N} wih a 1-sep ransiion marix P = (p ij ) consruced as follows.

25 MASCOS Insiu de Mahémaiques de Toulouse, June Page 19 A discree-ime Markovian model Recall ha here are N paches and ha n is he number of occupied paches a ime. We suppose ha (n, = 0, 1,...) is a discree-ime Markov chain aking values in S = {0, 1,...,N} wih a 1-sep ransiion marix P = (p ij ) consruced as follows. The exincion and colonizaion phases are governed by heir own ransiion marices, E = (e ij ) and C = (c ij ). We le P = EC if he census is aken afer he colonizaion phase or P = CE if he census is aken afer he exincion phase.

26 MASCOS Insiu de Mahémaiques de Toulouse, June Page 20 EC versus CE P = EC { P = CE {

27 MASCOS Insiu de Mahémaiques de Toulouse, June Page 21 Assumpions The number of exincions when here are i paches 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 colonizaions when here are i paches occupied follows a Bin(N i,c i ) law: c i,i+k = (c ij = 0 if j < i.) ( ) N i c k i (1 c i ) N i k (k = 0, 1,...,N i). k

28 MASCOS Insiu de Mahémaiques de Toulouse, June Page 22 Chain-binomial srucure Thus, we have he following chain-binomial srucure: n +1 = ñ + Bin(N ñ,cñ ) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Bin(N n,c n ). (CE)

29 MASCOS Insiu de Mahémaiques de Toulouse, June Page 22 Chain-binomial srucure Thus, we have he following chain-binomial srucure: n +1 = ñ + Bin(N ñ,cñ ) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Bin(N n,c n ). (CE) For he CE model (only) i is easy o show ha n +1 has he same disribuion as he sum of wo independen binomial random variables: n +1 D = Bin(n, 1 e) + Bin(N n, (1 e)c n ).

30 MASCOS Insiu de Mahémaiques de Toulouse, June Page 22 Chain-binomial srucure Thus, we have he following chain-binomial srucure: n +1 = ñ + Bin(N ñ,cñ ) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Bin(N n,c n ). (CE) For he CE model (only) i is easy o show ha n +1 has he same disribuion as he sum of wo independen binomial random variables: n +1 D = Bin(n, 1 e) + Bin(N n, (1 e)c n ). So, (1 e)c i is he effecive colonisaion probabiliy when here are i occupied paches.

31 MASCOS Insiu de Mahémaiques de Toulouse, June Page 23 Examples of c i c i = (i/n)c, where c (0, 1] is he maximum colonizaion poenial. (This enails c 0j = δ 0j, so ha 0 is an absorbing sae and {1,...,N} is a communicaing class.)

32 MASCOS Insiu de Mahémaiques de Toulouse, June Page 23 Examples of c i c i = (i/n)c, where c (0, 1] is he maximum colonizaion poenial. (This enails c 0j = δ 0j, so ha 0 is an absorbing sae and {1,...,N} is a communicaing class.) c i = c, where c (0, 1] is a fixed colonizaion poenial mainland colonizaion dominan. (Now {0, 1,...,N} is irreducible.)

33 MASCOS Insiu de Mahémaiques de Toulouse, June Page 23 Examples of c i c i = (i/n)c, where c (0, 1] is he maximum colonizaion poenial. (This enails c 0j = δ 0j, so ha 0 is an absorbing sae and {1,...,N} is a communicaing class.) c i = c, where c (0, 1] is a fixed colonizaion poenial mainland colonizaion dominan. (Now {0, 1,...,N} is irreducible.) Oher possibiliies include c i = c 0 (1 (1 c 1 /c 0 ) i ), c i = 1 exp( iβ/n) and c i = c 0 + (i/n)c, where c 0 + c (0, 1] (mainland and island colonizaion).

34 MASCOS Insiu de Mahémaiques de Toulouse, June Page 24 The proporion of occupied paches Henceforh we shall be concerned wih X (N) he proporion of occupied paches a ime. = n /N,

35 MASCOS Insiu de Mahémaiques de Toulouse, June Page 25 Simulaion: EC Model wih c i = c 1 Mainland-Island simulaion P = EC (N=100, x 0 =0.05, e =0.01, c =0.05) X (N)

36 MASCOS Insiu de Mahémaiques de Toulouse, June Page 26 The proporion of occupied paches Henceforh we shall be concerned wih X (N) he proporion of occupied paches a ime. = n /N,

37 MASCOS Insiu de Mahémaiques de Toulouse, June Page 26 The proporion of occupied paches Henceforh we shall be concerned wih X (N) he proporion of occupied paches a ime. = n /N, In he mainland-island case c i = c, he disribuion of n can be evaluaed explicily, and we have esablished large-n deerminisic and Gaussian approximaions for (X (N) ). Buckley, F.M. and Polle, P.K. (2009) Analyical mehods for a sochasic mainlandisland meapopulaion model. Ecological Modelling. In press (acceped 24/02/10).

38 MASCOS Insiu de Mahémaiques de Toulouse, June Page 27 Mainland-Island c i = c (Summary) Le p = 1 e(1 c) q = c (EC model) p = 1 e q = (1 e)c. (CE model) and define sequences (p ) and (q ) by q = q (1 a ) and p = q + a ( 0), where a = p q = (1 e)(1 c) (he same for boh EC and CE) and q = q/(1 a).

39 MASCOS Insiu de Mahémaiques de Toulouse, June Page 27 Mainland-Island c i = c (Summary) Le p = 1 e(1 c) q = c (EC model) p = 1 e q = (1 e)c. (CE model) and define sequences (p ) and (q ) by q = q (1 a ) and p = q + a ( 0), where a = p q = (1 e)(1 c) (he same for boh EC and CE) and q = q/(1 a). Then, n D = Bin(n0,p ) + Bin(N n 0,q ) (independen binomial random variables).

40 MASCOS Insiu de Mahémaiques de Toulouse, June Page 28 Mainland-Island c i = c (Summary) Le p = 1 e(1 c) q = c (EC model) p = 1 e q = (1 e)c. (CE model) and define sequences (p ) and (q ) by q = q (1 a ) and p = q + a ( 0), where a = p q = (1 e)(1 c) (he same for boh EC and CE) and q = q/(1 a). Then, n D = Bin(n0,p ) + Bin(N n 0,q ) ( D Bin(N,q ) ) (independen binomial random variables).

41 MASCOS Insiu de Mahémaiques de Toulouse, June Page 29 Mainland-Island c i = c (Summary) Le X (N) = n /N be he proporion occupied a ime. If X (N) 0 P x 0, as N, hen X (N) x = x 0 p + (1 x 0 )q. P x, where

42 MASCOS Insiu de Mahémaiques de Toulouse, June Page 30 Simulaion: EC Model wih c i = c 1 Mainland-Island simulaion P = EC (N=100, x 0 =0.05, e =0.01, c =0.05) X (N)

43 MASCOS Insiu de Mahémaiques de Toulouse, June Page 31 Simulaion: EC Model (Deerminisic pah) 1 Mainland-Island simulaion P = EC (N=100, x 0 =0.05, e =0.01, c =0.05) X (N) Deerminisic pah

44 MASCOS Insiu de Mahémaiques de Toulouse, June Page 32 Mainland-Island c i = c (Summary) Le X (N) = n /N be he proporion occupied a ime. If X (N) 0 P x 0, as N, hen X (N) x = x 0 p + (1 x 0 )q. P x, where

45 MASCOS Insiu de Mahémaiques de Toulouse, June Page 32 Mainland-Island c i = c (Summary) Le X (N) = n /N be he proporion occupied a ime. If X (N) 0 P x 0, as N, hen X (N) x = x 0 p + (1 x 0 )q. P x, where Now pu Z (N) := N(X (N) x ).

46 MASCOS Insiu de Mahémaiques de Toulouse, June Page 32 Mainland-Island c i = c (Summary) Le X (N) = n /N be he proporion occupied a ime. If X (N) 0 P x 0, as N, hen X (N) x = x 0 p + (1 x 0 )q. P x, where Now pu Z (N) Z (N) D := N(X (N) N(a z 0,V ), where x ). Then, if Z (N) 0 D z 0, V = x 0 p (1 p ) + (1 x 0 )q (1 q ).

47 MASCOS Insiu de Mahémaiques de Toulouse, June Page 33 Simulaion: EC Model (Gaussian approx.) 1 Mainland-Island simulaion P = EC (N=100, x 0 =0.05, e =0.01, c =0.05) X (N) Deerminisic pah ± wo sandard deviaions

48 MASCOS Insiu de Mahémaiques de Toulouse, June Page 34 Gaussian approximaions Can we esablish deerminisic and Gaussian approximaions for he basic N-pach models (where he disribuion of n is no known explicily)?

49 MASCOS Insiu de Mahémaiques de Toulouse, June Page 35 Simulaion: EC Model wih c i = (i/n)c 100 Meapopulaion simulaion P = EC (N=100, n 0 =95, e =0.3, c =0.8) n

50 MASCOS Insiu de Mahémaiques de Toulouse, June Page 36 Sim. & qsd: EC Model wih c i = (i/n)c 100 Meapopulaion simulaion P = EC (N=100, n 0 =95, e =0.3, c =0.8) n

51 MASCOS Insiu de Mahémaiques de Toulouse, June Page 37 Gaussian approximaions Can we esablish deerminisic and Gaussian approximaions for he basic N-pach models (where he disribuion of n is no known explicily)?

52 MASCOS Insiu de Mahémaiques de Toulouse, June Page 37 Gaussian approximaions Can we esablish deerminisic and Gaussian approximaions for he basic N-pach models (where he disribuion of n is no known explicily)? Is here a general heory of convergence for discree-ime Markov chains ha share he salien feaures of he pach-occupancy models presened here?

53 MASCOS Insiu de Mahémaiques de Toulouse, June Page 38 General srucure: densiy dependence We have a sequence of Markov chains (n (N) ) indexed by N, ogeher wih funcions (f ) such ha E(n (N) +1 n(n) ) = Nf (n (N) /N).

54 MASCOS Insiu de Mahémaiques de Toulouse, June Page 39 General srucure: densiy dependence We have a sequence of Markov chains (n (N) ) indexed by N, ogeher wih funcions (f ) such ha E(n (N) +1 n(n) ) = Nf (n (N) /N). We hen define (X (N) ) by X (N) = n (N) /N.

55 MASCOS Insiu de Mahémaiques de Toulouse, June Page 40 General srucure: densiy dependence We have a sequence of Markov chains (n (N) ) indexed by N, ogeher wih funcions (f ) such ha E(X (N) +1 X(N) ) = f (X (N) ).

56 MASCOS Insiu de Mahémaiques de Toulouse, June Page 41 General srucure: densiy dependence We have a sequence of Markov chains (n (N) ) indexed by N, ogeher wih funcions (f ) such ha E(n (N) +1 n(n) ) = Nf (n (N) /N). We hen define (X (N) D ) by X (N) = n (N) /N. We hope ha ) FDD (x ), where (x ) if X (N) 0 x 0 as N, hen (X (N) saisfies x +1 = f (x ) (he limiing deerminisic model).

57 MASCOS Insiu de Mahémaiques de Toulouse, June Page 42 General srucure: densiy dependence Nex we suppose ha here are funcions (s ) such ha Var(n (N) +1 n(n) ) = Ns(n (N) /N).

58 MASCOS Insiu de Mahémaiques de Toulouse, June Page 43 General srucure: densiy dependence Nex we suppose ha here are funcions (s ) such ha N Var(X (N) +1 X(N) ) = s(x (N) ).

59 MASCOS Insiu de Mahémaiques de Toulouse, June Page 44 General srucure: densiy dependence Nex we suppose ha here are funcions (s ) such ha Var(n (N) +1 n(n) ) = Ns (n (N) /N). We hen define (Z (N) ) by Z (N) = N(X (N) x ).

60 MASCOS Insiu de Mahémaiques de Toulouse, June Page 45 General srucure: densiy dependence Nex we suppose ha here are funcions (s ) such ha Var(Z (N) +1 X(N) ) = s (X (N) ). We hen define (Z (N) ) by Z (N) = N(X (N) x ).

61 MASCOS Insiu de Mahémaiques de Toulouse, June Page 46 General srucure: densiy dependence Nex we suppose ha here are funcions (s ) such ha Var(n (N) +1 n(n) ) = Ns (n (N) /N). We hen define (Z (N) ) by Z (N) hope ha if N(X (N) = N(X (N) x ). We ) FDD (Z ), 0 x 0 ) D z 0, hen (Z (N) where (Z ) is a Gaussian Markov chain wih Z 0 = z 0.

62 MASCOS Insiu de Mahémaiques de Toulouse, June Page 47 General srucure: densiy dependence Wha will be he form of his chain?

63 MASCOS Insiu de Mahémaiques de Toulouse, June Page 47 General srucure: densiy dependence Wha will be he form of his chain? Consider he ime-homogeneous case, f = f and s = s.

64 MASCOS Insiu de Mahémaiques de Toulouse, June Page 47 General srucure: densiy dependence Wha will be he form of his chain? Consider he ime-homogeneous case, f = f and s = s. Formally, by Taylor s heorem, f(x (N) ) f(x ) = (X (N) x )f (x ) + and so, since E(X (N) +1 X(N) ) = f(x (N) ) and x +1 = f(x ), E(Z (N) +1 ) = N (E(X (N) +1 ) f(x )) = f (x ) E(Z (N) ) +, suggesing ha E(Z +1 ) = a E(Z ), where a = f (x ).

65 MASCOS Insiu de Mahémaiques de Toulouse, June Page 48 General srucure: densiy dependence We have Var(X (N) +1 ) = Var(E(X(N) So, since N Var(X (N) +1 X(N) Var(Z (N) +1 ) = N Var(X(N) +1 X(N) )) + E(Var(X (N) +1 X(N) )). ) = s(x (N) ), +1 ) = N Var(f(X(N) )) + E(s(X (N) )) a 2 N Var(X (N) ) + E(s(X (N) )) (where a = f (x )) = a 2 Var(Z (N) ) + E(s(X (N) )), suggesing ha Var(Z +1 ) = a 2 Var(Z ) + s(x ).

66 MASCOS Insiu de Mahémaiques de Toulouse, June Page 48 General srucure: densiy dependence We have Var(X (N) +1 ) = Var(E(X(N) So, since N Var(X (N) +1 X(N) Var(Z (N) +1 ) = N Var(X(N) +1 X(N) )) + E(Var(X (N) +1 X(N) )). ) = s(x (N) ), +1 ) = N Var(f(X(N) )) + E(s(X (N) )) a 2 N Var(X (N) ) + E(s(X (N) )) (where a = f (x )) = a 2 Var(Z (N) ) + E(s(X (N) )), suggesing ha Var(Z +1 ) = a 2 Var(Z ) + s(x ). And, since (Z ) will be Markovian,...

67 MASCOS Insiu de Mahémaiques de Toulouse, June Page 49 General srucure: densiy dependence And, since (Z ) will be Markovian, we migh hope ha Z +1 = a Z + E (Z 0 = z 0 ), where a = f (x ) and E ( = 0, 1,...) are independen Gaussian random variables wih E N(0,s(x )).

68 MASCOS Insiu de Mahémaiques de Toulouse, June Page 49 General srucure: densiy dependence And, since (Z ) will be Markovian, we migh hope ha Z +1 = a Z + E (Z 0 = z 0 ), where a = f (x ) and E ( = 0, 1,...) are independen Gaussian random variables wih E N(0,s(x )). If x eq is a fixed poin of f, and N(X (N) 0 x eq ) z 0, hen we migh hope ha (Z (N) (Z ), where (Z ) is he AR-1 process defined by Z +1 = az + E, Z 0 = z 0, where a = f (x eq ) and E ( = 0, 1,...) are iid Gaussian N(0,s(x eq )) random variables. ) FDD

69 MASCOS Insiu de Mahémaiques de Toulouse, June Page 50 Convergence of Markov chains We can adap resuls of Alan Karr for our purpose. Karr, A.F. (1975) Weak convergence of a sequence of Markov chains. Probabiliy Theory and Relaed Fields 33, He considered a sequence of ime-homogeneous Markov chains (X (n) ) on a general sae space (Ω, F) = (E, E) N wih ransiion kernels (K n (x,a), x E,A E) and iniial disribuions (π n (A),A E). He proved ha if (i) π n π and (ii) x n x in E implies K n (x n, ) K(x, ), hen he corresponding probabiliy measures (P π n n ) on (Ω, F) also converge: P π n n P π.

70 MASCOS Insiu de Mahémaiques de Toulouse, June Page 51 N-pach models: convergence Theorem For he N-pach models wih c i = (i/n)c, if D x 0 as N, hen X (N) 0 (X (N) 1,X (N) 2,...,X (N) n ) D (x 1,x 2,...,x n ), for any finie sequence of imes 1, 2,..., n, where (x ) is defined by he recursion x +1 = f(x ) wih f(x) = (1 e)(1 + c c(1 e)x)x f(x) = (1 e)(1 + c cx)x (EC model) (CE model)

71 MASCOS Insiu de Mahémaiques de Toulouse, June Page 52 N-pach models: convergence Theorem If, addiionally, N(X (N) 0 x 0 ) D z 0, hen (Z (N) ) FDD (Z ), where (Z ) is he Gaussian Markov chain defined by Z +1 = f (x )Z + E (Z 0 = z 0 ), where E ( = 0, 1,...) are independen Gaussian random variables wih E N(0,s(x )) and s(x) = (1 e)[c(1 (1 e)x)(1 c(1 e)x) + e(1 + c 2c(1 e)x) 2 ]x (EC model) s(x) = (1 e)[e + c(1 x)(1 c(1 e)x)]x (CE model)

72 MASCOS Insiu de Mahémaiques de Toulouse, June Page 53 Simulaion: EC Model 1 Meapopulaion simulaion P = EC (N=100, x 0 =0.95, e =0.4, c =0.8) X (N)

73 MASCOS Insiu de Mahémaiques de Toulouse, June Page 54 Simulaion: EC Model (Deerminisic pah) 1 Meapopulaion simulaion P = EC (N=100, x 0 =0.95, e =0.4, c =0.8) Deerminisic pah X (N)

74 MASCOS Insiu de Mahémaiques de Toulouse, June Page 55 Simulaion: EC Model (Gaussian approx.) 1 Meapopulaion simulaion P = EC (N=100, x 0 =0.95, e =0.4, c =0.8) Deerminisic pah ± wo sandard deviaions X (N)

75 MASCOS Insiu de Mahémaiques de Toulouse, June Page 56 N-pach models: convergence In boh cases (EC and CE) he deerminisic model has wo equilibria, x = 0 and x = x, given by x = 1 1 e x = 1 ( 1 e c(1 e) e c(1 e) ) (EC model) (CE model)

76 MASCOS Insiu de Mahémaiques de Toulouse, June Page 56 N-pach models: convergence In boh cases (EC and CE) he deerminisic model has wo equilibria, x = 0 and x = x, given by x = 1 1 e x = 1 ( 1 e c(1 e) e c(1 e) ) (EC model) (CE model) Indeed, we may wrie f(x) = x (1 + r (1 x/x )), r = c(1 e) e for boh models (he form of he discree-ime logisic model), and we obain he condiion c > e/(1 e) for x o be posiive and hen sable.

77 MASCOS Insiu de Mahémaiques de Toulouse, June Page 57 N-pach models: convergence Corollary If c > e/(1 e), so ha x given above is sable, and N(X (N) 0 x ) D z 0, hen (Z (N) where (Z ) is he AR-1 process defined by ) FDD (Z ), Z +1 = (1 + e c(1 e))z + E (Z 0 = z 0 ), where E ( = 0, 1,...) are independen Gaussian N(0,σ 2 ) random variables wih σ 2 = (1 e)[c(1 (1 e)x )(1 c(1 e)x ) + e(1 + c 2c(1 e)x ) 2 ]x (EC model) σ 2 = (1 e)[e + c(1 x )(1 c(1 e)x )]x (CE model)

78 MASCOS Insiu de Mahémaiques de Toulouse, June Page 58 Simulaion: EC Model 1 Meapopulaion simulaion P = EC (N=100, x 0 =0.95, e =0.3, c =0.8) X (N) x =

79 MASCOS Insiu de Mahémaiques de Toulouse, June Page 59 Simulaion: EC Model (AR-1 approx.) 1 Meapopulaion simulaion P = EC (N=100, x 0 =0.95, e =0.3, c =0.8) X (N) x =

80 MASCOS Insiu de Mahémaiques de Toulouse, June Page 60 AR-1 Simulaion: EC Model 1 AR-1 simulaion P = EC (N=100, x 0 = , e =0.3, c =0.8) X (N) x =

81 MASCOS Insiu de Mahémaiques de Toulouse, June Page 61 Recen developmens Buckley, F.M. and Polle, P.K. (2010) Limi heorems for discree-ime meapopulaion models. Probabiliy Surveys 7,

82 MASCOS Insiu de Mahémaiques de Toulouse, June Page 61 Recen developmens Buckley, F.M. and Polle, P.K. (2010) Limi heorems for discree-ime meapopulaion models. Probabiliy Surveys 7, A general heory of convergence for sequences of ime-inhomogeneous densiy-dependen Markov chains.

83 MASCOS Insiu de Mahémaiques de Toulouse, June Page 61 Recen developmens Buckley, F.M. and Polle, P.K. (2010) Limi heorems for discree-ime meapopulaion models. Probabiliy Surveys 7, A general heory of convergence for sequences of ime-inhomogeneous densiy-dependen Markov chains. Analysis of he scheme n +1 = ñ + Bin(N ñ,c(ñ /N)) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Bin(N n,c(n /N)), (CE) where c is coninuous, increasing and concave, wih c(0) 0 and c(x) 1.

84 MASCOS Insiu de Mahémaiques de Toulouse, June Page 62 Recen developmens Sabiliy analysis of he limiing deerminisic model: (i) Saionariy: c(0) > 0. (ii) Evanescence: c(0) = 0 and c (0) e/(1 e). (iii) Quasi saionariy: c(0) = 0 and c (0) > e/(1 e).

85 MASCOS Insiu de Mahémaiques de Toulouse, June Page 62 Recen developmens Sabiliy analysis of he limiing deerminisic model: (i) Saionariy: c(0) > 0. (ii) Evanescence: c(0) = 0 and c (0) e/(1 e). (iii) Quasi saionariy: c(0) = 0 and c (0) > e/(1 e). Infinie-pach models. If c(0) = 0 and c(x) has a coninuous second derivaive near 0, hen Bin(N n,c(n/n)) D Poi(mn) as N, where m = c (0).

86 MASCOS Insiu de Mahémaiques de Toulouse, June Page 62 Recen developmens Sabiliy analysis of he limiing deerminisic model: (i) Saionariy: c(0) > 0. (ii) Evanescence: c(0) = 0 and c (0) e/(1 e). (iii) Quasi saionariy: c(0) = 0 and c (0) > e/(1 e). Infinie-pach models. If c(0) = 0 and c(x) has a coninuous second derivaive near 0, hen Bin(N n,c(n/n)) D Poi(mn) as N, where m = c (0). This leads o he scheme n +1 = ñ + Poi(mñ ) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Poi(mn ), (CE)

87 MASCOS Insiu de Mahémaiques de Toulouse, June Page 63 Recen developmens...which urns ou o be a (Galon-Wason) branching process.

88 MASCOS Insiu de Mahémaiques de Toulouse, June Page 63 Recen developmens...which urns ou o be a (Galon-Wason) branching process. Analysis of he more general scheme n +1 = ñ + Poi(m(ñ )) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Poi(m(n )), (CE) assuming m(n) = n 0 µ(n/n 0 ).

89 MASCOS Insiu de Mahémaiques de Toulouse, June Page 63 Recen developmens...which urns ou o be a (Galon-Wason) branching process. Analysis of he more general scheme n +1 = ñ + Poi(m(ñ )) ñ = n Bin(n,e) (EC) n +1 = ñ Bin(ñ,e) ñ = n + Poi(m(n )), (CE) assuming m(n) = n 0 µ(n/n 0 ). In he limi as n 0 X (N) := n /n 0 has a deerminisic approximaion ha can exhibi he full range of dynamic behaviour (including chaos).

90 MASCOS Insiu de Mahémaiques de Toulouse, June Page 64 Ricker dynamics: µ(x) = x exp(r(1-x)) 1 (a) 1 (b) x x (c) 3 (d) x 1 x