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 timeuandextinction has not yet occurred. How can we incorporate this information?

The conditional state distribution

Evaluate the state probabilities at timeuconditioned 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 patchjwillnotbe colonized by migration from patchi, and is the probability that (in the absence of immigration) patchiwill 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.