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 thatis 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:
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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.