LIMITING CONDITIONAL DISTRIBUTIONS FOR STOCHASTIC METAPOPULATION MODELS
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
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 .
(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.
Let (now arbitrary) and suppose that our interaction matrix depends on in the following way:
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.
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.
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.