LIMITING CONDITIONAL DISTRIBUTIONS FOR STOCHASTIC METAPOPULATION MODELS

by

Phil Pollett
Department of Mathematics
The University of Queensland

CONDITIONAL STATE DISTRIBUTION

X(t) - state of the population at time t

We suppose that is a continuous time Markov chain with a discrete state space , where 0 is the state corresponding to extinction and comprises the remaining states.

- state probabilities

We observe the population at an arbitrary time u and extinction has not yet occurred. How can we incorporate this information?

The conditional state distribution

Evaluate the state probabilities at time u conditioned on non-extinction:

A STOCHASTIC MODEL FOR METAPOPULATIONS

There are n geographical regions (patches): Let , where is 1 or 0 according as patch i is occupied or not at time t ( ). Note that the state space is .

- Interaction matrix:

, for , is the probability that patch j will not be colonized by migration from patch i, and is the probability that (in the absence of immigration) patch i will become extinct.

Assume (Gyllenberg and Silvestrov) that

where is the distance between patches i and j ( and ), is the area of patch i and a( ) measures how badly individuals are at migrating.

THE TRANSITION PROBABILITIES

Assume that the various colonization processes and local extinction processes are independent. Define , where , by

to be the probability that patch j will become extinct at the next epoch given a present configuration x.

The transition matrix

Note that, since , , state (corresponding to the extinction of all patches) is an absorbing state for the chain:

A 5-PATCH METAPOPULATION

Patches 2, 3, 4 & 5 are equally spaced (a distance 0.1 apart), while Patch 1 is 10 times that distance away from the others ( ). All patches have the same area.

PERSISTENCE OF METAPOPULATIONS: QUASI-STATIONARITY

Simulation of the 5-patch model with a=7. The number of occupied patches is plotted against time (up to extinction at t=728).

Do the conditional state probabilities account for the observed behaviour?

Comparison of the observed frequencies with the conditional state distribution at t=1,2,5,10. The brown bar is the proportion of time for which i patches were occupied ( ) during the period of the simulation. The blue bar is the distribution of the number of occupied patches evaluated using .

LIMITING CONDITIONAL DISTRIBUTIONS

The observed trend has a simple theoretical explanation. Since is a finite set, the limit

exists and defines a proper distribution , called a limiting conditional distribution, and m is the left eigenvector of  (P restricted to  ) corresponding to the eigenvalue, , with maximal modulus (Darroch and Seneta (1965)). Note that the expected time till absorption, , is approximately .

We can be precise about the rate of convergence by examining the eigenvalue, , of with second-largest modulus. It might not be real, and it has multiplicity (for simplicity, suppose ). It can be shown that

where .

THE CONVERGENCE OF TO

Comparison between the conditional state distribution (blue) and the limiting conditional distribution (brown) of the number of occupied patches for the 5-patch metapopulation model with a=7.

For the 5-patch metapopulation model with a=7, we find that , (real with multiplicity 1), and .

PSEUDO-STATIONARY DISTRIBUTIONS

(The method of Gyllenberg and Silvestrov)

Rationale: If we had assumed that Patch 1 (say) had a zero local extinction probability ( ), that patch would behave as a mainland. State 0 would no longer be accessible from all states: would decompose into two classes, and , consisting of those states in which have and respectively; either the process would start in (mainland inhabited) and remain there, or, start in (mainland uninhabited) and eventually enter either or the absorbing state.

We identify a ``quasi-mainland'', namely a single patch i with small (say Patch 1), and consider a sequence of processes indexed by , treating as a perturbation.

PERTURBATION THEORY

Let (now arbitrary) and suppose that our interaction matrix depends on in the following way:

where

the latter assumed to be non-negative and finite, and, that satisfies .

Then, in an obvious notation,

where is the transition matrix corresponding to and is the transition matrix corresponding to Q.

THE G&S LIMITING REGIME

Let and in such a way that , where . Since the expected lifetime of the quasi-mainland is of order , one is able to study the process on different time scales (s=0, , and ).

G&S showed that the limit

exists and is given by a mixture of the limiting probabilities for the (ergodic) chain generated by Q and the degenerate distribution which assigns all its mass to state 0, the mixing probability being , where is a positive constant which is specified in terms of .

A COMPARISON USING THE 5-PATCH MODEL

Comparison between the limiting conditional distribution (blue), the simulated proportions (green) and the pseudo-stationary distribution (brown).

The disparity is marked: for this example, the two ways of analysing the model lead to quite different predictions.

THE EFFECT OF VARYING s

The effect of varying s on the pseudo-stationary distribution (blue). The brown bar represents the simulated proportions of occupied patches.

Observe that the disparity becomes worse as the time-scale parameter s increases.

RECONCILIATION

Denote the state probabilities corresponding to by , and denote the corresponding conditonal probabilities by . Gosselin (1997) proved that

which he compared with Theorem 6.2 of G&S:

Thus, in the important case s=0, the limiting conditonal distribution and the pseudo-stationary agree when is small.

The problem with the 5-patch model is that - not sufficiently small.

REMARKS

Quasi-stationarity is a property of the model and not the means of analysing it. The 5-patch model exhibits quasi-stationarity, demonstrated emphatically using simulation, yet is not small. The pseudo-stationary distribution does not capture this behaviour. On the other hand, the conditional state distribution m(t) does: afterall, it is the most information our model can provide at any time t given that we know extinction has not occured by time t. In cases when the convergence of m(t) to the limiting conditional distribution m is rapid, this distribution can be used instead.