Limit theorems for chain-binomial population models...Collaborators Fionnuala Buckley Department of Mathematics University of Queensland ∗Buckley, F.M. and Pollett, P.K. (2010) Limit

Limit theorems for chain-binomialpopulation models

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

Collaborators

Fionnuala BuckleyDepartment of MathematicsUniversity of Queensland

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

Collaborators

Ross McVinishDepartment of MathematicsUniversity of Queensland

∗McVinish, R. and Pollett, P.K. (2010) Limits of large metapopulationswith patch dependent extinction probabilities. Advances in Applied Prob-ability 42, 1172-1186.

Metapopulations

Metapopulations

Colonization

Metapopulations

Metapopulations

Local Extinction

Metapopulations

Metapopulations

Total Extinction

Metapopulations

Metapopulations

Colonization

from the mainland

Metapopulations

SPOM

A Stochastic Patch Occupancy Model (SPOM)

SPOM


Suppose that there are n patches.

SPOM



Let X(n)t = (X(n)1,t , . . . , X

(n)n,t ), where X

(n)i,t is a binary variable

indicating whether or not patch i is occupied.

SPOM



Let X(n)t = (X(n)1,t , . . . , X

(n)n,t ), where X



For each n, (X(n)t , t = 0, 1, . . . ) is assumed to be a Markovchain.

SPOM



Let X(n)t = (X(n)1,t , . . . , X

(n)n,t ), where X



For each n, (X(n)t , t = 0, 1, . . . ) is assumed to be a Markovchain.

Colonization and extinction happen in distinct, successivephases.

SPOM - Phase structure

For many species the propensity for colonization and localextinction is markedly different in different phases of theirlife cycle.


For many species the propensity for colonization and localextinction is markedly different in different phases of theirlife 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



t− 1 t t+ 1 t+ 2

t− 1 t t+ 1 t+ 2



t− 1 t t+ 1 t+ 2

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



Colonization: unoccupied patches become occupiedindependently with probability c(n−1

∑n

i=1X(n)i,t ), where

c : [0, 1] → [0, 1] is continuous, increasing and concave.

Examples of c(x)

c(x) = cx, where c ∈ (0, 1] is the maximum colonizationpotential.

Examples of c(x)


c(x) = c, where c ∈ (0, 1] is a fixed colonizationpotential—mainland colonization dominant.

Examples of c(x)



c(x) = c0 + cx, where c0 + c ∈ (0, 1] (mainland and islandcolonization).

Examples of c(x)



c(x) = c0 + cx, where c0 + c ∈ (0, 1] (mainland and islandcolonization).

c(x) = 1− exp(−xβ) (β > 0).




∑n

i=1X(n)i,t ), where





∑n

i=1X(n)i,t ), where


Extinction: occupied patch i remains occupiedindependently with probability Si (random).

SPOM

Thus, we have a Chain Bernoulli structure:

X(n)i,t+1

d= Bin

(

X(n)i,t + Bin

(

1−X(n)i,t , c(

1n

∑n

j=1X(n)j,t

)

)

, Si

)

SPOM


X(n)i,t+1

d= Bin

(

X(n)i,t +Bin

(

1−X(n)i,t , c(

1n

∑n

j=1X(n)j,t

)

)

, Si

)

SPOM


X(n)i,t+1

d= Bin

(

X(n)i,t + Bin

(

1−X(n)i,t , c(

1n

∑n

j=1X(n)j,t

)

)

, Si

)

SPOM

n = 30, Si ∼Beta(25.2, 19.8) (ESi = 0.56) and c(x) = 0.7x

0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0

c(x) = c(1130) = 0.7× 0.36̇ = 0.256̇

SPOM


0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0

SPOM


0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0

SPOM


0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0

0.60 0.56 0.63 0.62 0.52 0.61 0.68 0.49 0.49 0.49 0.500.41 0.59 0.63 0.60 0.61

SPOM


0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0

SPOM


0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0

c(x) = c(1030) = 0.7× 0.3̇ = 0.23̇

SPOM


0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0C 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0

SPOM


0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0C 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0

SPOM


0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0C 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0C 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0E 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0...C 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

SPOM - Homogeneous case

Compare this with the homogeneous case, where Si = s(non-random) is the same for each i, and we merely countthe number N (n)t of occupied patches at time t.

We have the following Chain Binomial structure:

N(n)t+1

d= Bin

(

N(n)t + Bin

(

n−N (n)t , c(

1nN

(n)t

)

)

, s)




N(n)t+1

d= Bin

(

N(n)t +Bin

(

n−N (n)t , c(

1nN

(n)t

)

)

, s)




N(n)t+1

d= Bin

(

N(n)t + Bin

(

n−N (n)t , c(

1nN

(n)t

)

)

, s)




N(n)t+1

d=Bin

(

N(n)t + Bin

(

n−N (n)t , c(

1nN

(n)t

)

)

, s)




N(n)t+1

d= Bin

(

N(n)t + Bin

(

n−N (n)t , c(

1nN

(n)t

)

)

, s)

A deterministic limit

Theorem [BP] If N (n)0 /np→ x0 (a constant), then

N(n)t /n

p→ xt, for all t ≥ 1,

with (xt) determined by xt+1 = f(xt), where

f(x) = s(x+ (1− x)c(x)).

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

Stability

xt+1 = f(xt), where f(x) = s(x+ (1− x)c(x)).

Stationarity : c(0) > 0. There is a unique fixed pointx∗ ∈ [0, 1]. It satisfies x∗ ∈ (0, 1) and is stable.Evanescence: c(0) = 0 and 1 + c ′(0) ≤ 1/s. Now 0 is theunique fixed point in [0, 1]. It is stable.

Quasi stationarity : c(0) = 0 and 1 + c ′(0) > 1/s. Thereare two fixed points in [0, 1]: 0 (unstable) and x∗ ∈ (0, 1)(stable).

[Notice that if c(0) = 0, we require c ′(0) > 0 for quasistationarity.]

CE Model - Evanescence

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100CE Model simulation (n = 100, s = 0.56, c(x) = cx with c =0.7)

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CE Model - Quasi stationarity

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A Gaussian limit

Theorem [BP] Further suppose that c(x) is twicecontinuously differentiable, and let

Z(n)t =

√n(N

(n)t /n− xt).

If Z(n)0d→ z0, then Z(n)• converges weakly to the Gaussian

Markov chain Z• defined by

Zt+1 = f′(xt)Zt + Et (Z0 = z0),

with (Et) independent and Et ∼ N(0, v(xt)), where

v(x) = s[

(1− s)x+ (1− x)c(x)(

1− sc(x))]

.


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100CE Model simulation (n =100, s =0.8, c(x) = cx with c =0.7)

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CE Model - Quasi-stationary distribution

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CE Model - Gaussian approximation

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Nx∗ = 64.2857


Returning to the general case, where patch survivalprobabilities are random and patch dependent , and wekeep track of which patches are occupied . . .

X(n)i,t+1

d= Bin

(

X(n)i,t + Bin

(

1−X(n)i,t , c(

1n

∑n

j=1X(n)j,t

)

)

, Si

)


Returning to the general case, where patch survivalprobabilities are random and patch dependent , and wekeep track of which patches are occupied . . .

X(n)i,t+1

d= Bin

(

X(n)i,t + Bin

(

1−X(n)i,t , c(

1n

∑n

j=1X(n)j,t

)

)

, Si

)

First, . . .

Notation: If σ is a probability measure on [0, 1) and let s̄kdenote its k-th moment, that is,

s̄k =∫ 1

0xkσ(dx).


Theorem Suppose there is a probability measure σ anddeterministic sequence {d(0, k)} such that

1n

∑n

i=1 Ski

p→ s̄k and 1n∑n

i=1 Ski X

(n)i,0

p→ d(0, k)

for all k = 0, 1, . . . , T . Then, there is a (deterministic)triangular array {d(t, k)} such that, for all t = 0, 1, . . . , T andk = 0, 1, . . . , T − t,

1n

∑n

i=1 Ski X

(n)i,t

p→ d(t, k),

where

d(t+ 1, k) = d(t, k + 1) + c (d(t, 0)) (s̄k+1 − d(t, k + 1)) .

A deterministic limit d(0,k)

0 1 2 3 T

t0

1

2

3

T

k


0 1 2 3 T

t0

1

2

3

T

k

A deterministic limit d(t,k)

0 1 2 3 T

t0

1

2

3

T

k

A deterministic limit d(t,0)

0 1 2 3 T

t0

1

2

3

T

k

Remarks

Typically, we are only interested in d(t, 0), being theasymptotic proportion of occupied patches at time t:

1n

∑n

i=1X(n)i,t

p→ d(t, 0).

Remarks: d(t,0)

0 1 2 3 T

t0

1

2

3

T

k

Remarks: d(t,k)

0 1 2 3 T

t0

1

2

3

T

k

Remarks: d(t,0)

0 1 2 3 T

t0

1

2

3

T

k

Remarks


1n

∑n

i=1X(n)i,t

p→ d(t, 0).

Remarks


1n

∑n

i=1X(n)i,t

p→ d(t, 0).

However, we may still interpret the ratio d(t, k)/d(t, 0)(k ≥ 1) as the k-th moment of the conditional distributionof the patch survival probability given that the patch isoccupied. (From these moments, the conditionaldistribution could then be reconstructed.)


Theorem Suppose there is a probability measure σ anddeterministic sequence {d(0, k)} such that

1n

∑n

i=1 Ski

p→ s̄k and 1n∑n

i=1 Ski X

(n)i,0

p→ d(0, k)

for all k = 0, 1, . . . , T . Then, there is a (deterministic)triangular array {d(t, k)} such that, for all t = 0, 1, . . . , T andk = 0, 1, . . . , T − t,

1n

∑n

i=1 Ski X

(n)i,t

p→ d(t, k),

where

d(t+ 1, k) = d(t, k + 1) + c (d(t, 0)) (s̄k+1 − d(t, k + 1)) .

Homogeneous case

When s̄k = s̄k1 for all k, that is the patch survivalprobabilities are the same, then it is possible to simplify

d(t+ 1, k) = d(t, k + 1) + c (d(t, 0)) (s̄k+1 − d(t, k + 1)) .

We can show by induction that d(t, k) = s̄k1xt, where

xt+1 = s̄1 (xt + (1− xt) c(xt)) .

Compare this with the earlier [BP] result....


Theorem [BP] If N (n)0 /np→ x0 (a constant), then

N(n)t /n

p→ xt, for all t ≥ 1,

with (xt) determined by xt+1 = f(xt), where

f(x) = s(x+ (1− x)c(x)).

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

Stability

Theorem Any fixed point d = (d(0), d(1), . . . ) is given by

d(k) =∫ 1

0c(ψ)xk+1

1−x+c(ψ)xσ(dx),

where ψ (= d(0)) solves

R(ψ) =∫ 1

0c(ψ)x

1−x+c(ψ)xσ(dx) = ψ. (1)

If c(0) > 0, there is a unique ψ > 0. If c(0) = 0 and

c ′(0)∫ 1

0x

1−xσ(dx) ≤ 1,

then ψ = 0 is the unique solution to (1). Otherwise, (1) hastwo solutions, one of which is ψ = 0.

Stability

Theorem If c(0) = 0 and

c ′(0)∫ 1

0x

1−xσ(dx) ≤ 1,

then d(k) ≡ 0 is a stable fixed point. Otherwise, thenon-zero solution to

R(ψ) =∫ 1

0c(ψ)x

1−x+c(ψ)xσ(dx) = ψ

provides the stable fixed point through

d(k) =∫ 1

0c(ψ)xk+1

1−x+c(ψ)xσ(dx).

CE Model (homogeneous) - Evanescence

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CE Model - Evanescence

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c′(0)

∫1

0

x

1−xσ(dx) =0.9383

0 0.5 10

2

4

6Beta(25.2,19.8)


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c′(0)

∫1

0

x

1−xσ(dx) =1.2572

0 0.5 10

1

2

3Beta(4.368,3.432)


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c′(0)

∫1

0

x

1−xσ(dx) = ∞

0 0.5 10

0.5

1

1.5

2Beta(1.176,0.924)


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c′(0)

∫1

0

x

1−xσ(dx) = ∞

0 0.5 10

2

4

6

8Beta(0.784,0.616)


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c′(0)

∫1

0

x

1−xσ(dx) = ∞

0 0.5 10

10

20

30Beta(0.3304,0.2596)


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c′(0)

∫1

0

x

1−xσ(dx) = ∞

0 0.5 10

2

4

6

8Beta(0.784,0.616)


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c′(0)

∫1

0

x

1−xσ(dx) =2.1

0 0.5 10

0.5

1

1.5

2Beta(3,2)

CollaboratorsCollaboratorsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsMetapopulationsSPOMSPOMSPOMSPOMSPOMSPOM - Phase structureSPOM - Phase structure

SPOM - Phase structureSPOM - Phase structureSPOM - Phase structureExamples of $�oldsymbol {c} �oldsymbol {(} �oldsymbol {x} �oldsymbol {)}$Examples of $�oldsymbol {c} �oldsymbol {(} �oldsymbol {x} �oldsymbol {)}$Examples of $�oldsymbol {c} �oldsymbol {(} �oldsymbol {x} �oldsymbol {)}$Examples of $�oldsymbol {c} �oldsymbol {(} �oldsymbol {x} �oldsymbol {)}$

SPOM - Phase structureSPOM - Phase structureSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOMSPOM - Homogeneous caseSPOM - Homogeneous caseSPOM - Homogeneous caseSPOM - Homogeneous caseSPOM - Homogeneous caseSPOM - Homogeneous caseSPOM - Homogeneous caseA deterministic limitStabilityCE Model - EvanescenceCE Model - Quasi stationarityA Gaussian limitCE Model - Quasi stationarityCE Model - Quasi stationarityCE Model - Quasi-stationary distributionCE Model - Gaussian approximationA deterministic limitA deterministic limit

A deterministic limitA deterministic limit $�oldsymbol {d}�oldsymbol {(}�oldsymbol {0}�oldsymbol {,}�oldsymbol {k}�oldsymbol {)}$A deterministic limit $�oldsymbol {d}�oldsymbol {(}�oldsymbol {1}�oldsymbol {,}�oldsymbol {k}�oldsymbol {)}$A deterministic limit $�oldsymbol {d}�oldsymbol {(}�oldsymbol {2}�oldsymbol {,}�oldsymbol {k}�oldsymbol {)}$A deterministic limit $�oldsymbol {d}�oldsymbol {(}�oldsymbol {3}�oldsymbol {,}�oldsymbol {k}�oldsymbol {)}$A deterministic limit $�oldsymbol {d}�oldsymbol {(}�oldsymbol {t}�oldsymbol {,}�oldsymbol {k}�oldsymbol {)}$A deterministic limit $�oldsymbol {d}�oldsymbol {(}�oldsymbol {t}�oldsymbol {,}�oldsymbol {0}�oldsymbol {)}$RemarksRemarks: $�oldsymbol {d}�oldsymbol {(}�oldsymbol {t}�oldsymbol {,}�oldsymbol {0}�oldsymbol {)}$Remarks: $�oldsymbol {d}�oldsymbol {(}�oldsymbol {t}�oldsymbol {,}�oldsymbol {k}�oldsymbol {)}$Remarks: $�oldsymbol {d}�oldsymbol {(}�oldsymbol {t}�oldsymbol {,}�oldsymbol {0}�oldsymbol {)}$RemarksRemarksA deterministic limitHomogeneous caseA deterministic limitStabilityStabilityCE Model (homogeneous)- EvanescenceCE Model - EvanescenceCE Model - Quasi stationarityCE Model - Quasi stationarityCE Model - Quasi stationarityCE Model - Quasi stationarityCE Model - Quasi stationarityCE Model - Quasi stationarity

Documents

Limit theorems for chain-binomial population models...Collaborators Fionnuala Buckley Department of Mathematics University of Queensland ∗Buckley, F.M. and Pollett, P.K. (2010) Limit