inﬁnite-patch metapopulation models: branching ... · we have the following chain binomial...

Infinite-patch metapopulation models:branching, convergence and chaos

Phil Pollett

Department of MathematicsThe University of Queensland

http://www.maths.uq.edu.au/˜pkp

AUSTRALIAN RESEARCH COUNCILCentre of Excellence for Mathematicsand Statistics of Complex Systems

MASCOS MASCOS Annual Conference, October 2010 - Page 1

Collaborator

Fionnuala BuckleyMASCOS PhD ScholarUniversity of Queensland

∗Buckley, F.M. and Pollett, P.K. (2010) Limit theorems for discrete-timemetapopulation models. Probability Surveys 7, 53-83.


Metapopulations


Metapopulations

Colonization


Metapopulations


Metapopulations

Local Extinction


Metapopulations


Metapopulations

Total Extinction


Metapopulations


SPOM

A Stochastic Patch Occupancy Model (SPOM)


SPOM


Suppose that there are N patches.


SPOM



Let nt ∈ {0, 1, . . . , N} be the number occupied at time t.


SPOM




Assume (nt, t = 0, 1, . . . ) to be Markov chain.


SPOM





Colonization and extinction happen in distinct,successive phases.


SPOM - Phase structure


t − 1 t t + 1 t + 2

t − 1 t t + 1 t + 2


SPOM - Phase structure


t − 1 t t + 1 t + 2

We will we assume that the population is observedafter successive extinction phases (CE Model).


SPOM






We will assume that the population is observed aftersuccessive extinction phases (CE Model).


SPOM


Colonization: unoccupied patches become occupiedindependently with probability c(nt/N), wherec : [0, 1] → [0, 1] is continuous, increasing and concave,and c ′(0) > 0.


SPOM


Colonization: unoccupied patches become occupiedindependently with probability c(nt/N), wherec : [0, 1] → [0, 1] is continuous, increasing and concave,and c ′(0) > 0.

Extinction: occupied patches remain occupiedindependently with probability s.


N -patch SPOM

We have the following Chain Binomial structure:

nt+1D= Bin

(

nt + Bin(

N − nt, c(nt/N))

, s)


N -patch SPOM


nt+1D= Bin

(

nt + Bin(

N − nt, c(nt/N))

, s)

Notation: Bin(m, p) is a binomial random variable withm trials and success probability p.


N -patch SPOM


nt+1D= Bin

(

nt + Bin(

N − nt, c(nt/N))

, s)


N -patch SPOM


nt+1D= Bin

(

nt + Bin(

N−nt, c(nt/N))

, s)


N -patch SPOM


nt+1D= Bin

(

nt + Bin(

N − nt, c(nt/N))

, s)


CE Model

0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

80

90

100CE Model simulation (N =100, n0 =95, s =0.56, c(x) = cx with c =0.7)

t

nt


CE Model

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

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90


t

nt


CE Model - Evanescence

0 5 10 15 20 25 30 35 40 45 500

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90


t

nt


CE Model - Quasi stationarity

0 10 20 30 40 50 60 70 80 90 1000

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90


t

nt


CE Model - Evanescence

0 5 10 15 20 25 30 35 40 45 500

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90


t

nt


CE Model - Quasi stationarity

0 10 20 30 40 50 60 70 80 90 1000

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90


t

nt


CE Model c ′(0) < (1 − s)/s

0 5 10 15 20 25 30 35 40 45 500

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90


t

nt


CE Model c ′(0) > (1 − s)/s

0 10 20 30 40 50 60 70 80 90 1000

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t

nt


Metapopulations


Infinite-patch SPOM

Prelude If c(0) = 0 and c has a continuous secondderivative near 0, then, for fixed n,

Bin(N − n, c(n/N))D→ Poi(mn), as N → ∞,

where m = c ′(0).


Infinite-patch SPOM

We have the following structure:

nt+1D= Bin

(

nt + Poi(

mnt)

, s)


N -patch SPOM


nt+1D= Bin

(

nt + Bin(

N − nt, c(nt/N))

, s)

Bin(N − n, c(n/N))D→ Poi(mn) (as N → ∞)


Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)


Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)

Claim The process (nt, t = 0, 1, . . . ) is a branchingprocess (Galton-Watson process) whose offspringdistribution has pgf G(z) = (1 − s + sz)e−ms(1−z).


Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)


(We think of the census times as marking the‘generations’, the ‘particles’ being the occupiedpatches, and the ‘offspring’ being the occupiedpatches that they notionally replace in the succeedinggeneration.)


Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)



Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)


The mean number of offspring is µ = (1 + m)s.


Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)


The mean number of offspring is µ = (1 + m)s.

So, for example, E(nt|n0) = n0µt (t ≥ 1).


Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)



Infinite-patch SPOM


nt+1D= Bin

(

nt + Poi(

mnt)

, s)


Theorem Extinction occurs with probability 1 if andonly if m ≤ (1 − s)/s; otherwise total extinction occurswith probability ηn0, where η is the unique fixed point ofG on the interval (0, 1).


CE Model c ′(0) < (1 − s)/s (η = 1)

0 5 10 15 20 25 30 35 40 45 500

10

20

30

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90


t

nt


CE Model c ′(0) > (1 − s)/s (ηn0 = 0.0020837)

0 10 20 30 40 50 60 70 80 90 1000

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t

nt


Infinite-patch SPOM with regulation

Assume the following structure:

nt+1D= Bin

(

nt + Poi(

m(nt))

, s)

where m(n) ≥ 0.




nt+1D= Bin

(

nt + Poi(

m(nt))

, s)

where m(n) ≥ 0. A moment ago we had m(n) = mn.




nt+1D= Bin

(

nt + Poi(

m(nt))

, s)

where m(n) ≥ 0.



nt+1D= Bin

(

nt + Poi(

m(nt))

, s)



nt+1D= Bin

(

nt + Poi(

m(nt))

, s)

We will consider what happens when the initialnumber of occupied patches n0 becomes large.



nt+1D= Bin

(

nt + Poi(

m(nt))

, s)


For some index N write m(n) = Nµ(n/N), and assumeµ is continuous with bounded first derivative.



nt+1D= Bin

(

nt + Poi(

m(nt))

, s)


For some index N write m(n) = Nµ(n/N), and assumeµ is continuous with bounded first derivative.

We may take N to be simply n0 or, more generally,following Klebaner∗, we may interpret N as being a‘threshold’ with the property that n0/N → x0 as N → ∞.

∗Klebaner (1993) Population-dependent branching processes with a threshold.Stochastic Process. Appl. 46, 115–127.



By choosing µ appropriately, we may allow for adegree of regulation in the colonisation process.




For example, µ(x) might be of the form

• µ(x) = rx(a − x) (0 ≤ x ≤ a) (logistic growth);

• µ(x) = xer(1−x) (x ≥ 0) (Ricker dynamics);

• µ(x) = λx/(1 + ax)b (x ≥ 0) (Hassell dynamics).







• µ(x) = λx/(1 + ax)b (x ≥ 0) (Hassell dynamics);

• µ(x) = mx (x ≥ 0) (branching).







• µ(x) = λx/(1 + ax)b (x ≥ 0) (Hassell dynamics);

• µ(x) = mx (x ≥ 0) (branching).

We can establish a law of large numbers forX (N)

t= nt/N , the number of occupied patches at

census t measured relative to the threshold.


Infinite-patch SPOM - Convergence

Theorem For the infinite-patch CE model withparameters s and µ(x), let X (N)

t= nt/N be the number

of occupied patches at census t relative to thethreshold N .

Suppose that µ is continuous with bounded firstderivative.

If X (N)02→ x0 as N → ∞, then X (N)t

2→ xt for all t ≥ 1,

where (xt) is determined by xt+1 = f(xt) (t ≥ 0), wheref(x) = s(x + µ(x)).


Infinite-patch SPOM - Ricker dynamics

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

r

x

Bifurcation diagram for the infinite-patch deterministic CE model with Rickergrowth dynamics: xn+1 = 0.3 xn(1 + er(1−xn)) (r ranges from 0 to 7.2).


Infinite-patch SPOM - Ricker dynamics

0 20 40 600

0.2

0.4

0.6

0.8

1(a)

t

xt

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1(b)

t

xt

0 20 40 60 80 1000

0.5

1

1.5

2(c)

t

xt

0 50 100 150 2000

0.5

1

1.5

2

2.5

3(d)

t

xt

Simulation (open circles) of the infinite-patch CE model with Ricker growth dy-namics, together with the corresponding limiting deterministic trajectories (solidcircles). Here s = 0.3, N = 200 and (a) r = 0.84, (b) r = 1 (c) r = 4, (d) r = 5.


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inﬁnite-patch metapopulation models: branching ... · we have the following chain binomial...

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