Quasistationarity in populations that are subject to large-scale mortality or emigration
by
Phil Pollett
Department of Mathematics
The University of Queensland
WHAT ARE WE MODELLING?
Populations which are subject to crashes. Dramatic losses can occur due to
Quasistationary behaviour. Such populations can survive for long periods before extinction occurs and can settle down to an apparently stationary regime.
Our goal. We seek to model this behaviour in order to properly manage these populations: to predict persistence times and to estimate population size.
Our model. The birth-death and catastrophe process predicts eventual extinction, but the time till extinction can be very long. The stationarity exhibited by these populations over any reasonable time scale can be explained using a quasistationary distribution.
THE MODEL
We use a continuous-time Markov process
, where X(t) is the
population size at time t, with transition
rates
given by
with the other transition rates equal to 0. Here, , a>0,
for at least one i in
, and
.
Interpretation.
For ,
is the instantaneous rate at which the population
size changes from j to k,
is the per capita rate of change
and, given a change occurs, a is the probability that this results in
a birth and
is the probability that this results in a catastrophe
of size i (corresponding to the death or emigration of individuals).
SOME PROPERTIES
The state space. Clearly 0 is an absorbing state (corresponding to population extinction) and C is an irreducible class.
Extinction probabilities.
If is the probability of extinction
starting with i individuals, then
for all
if and only if D (the expected increment size), given by
is less than 0 (the subcritical case) or equal to 0 (the critical case).
In the supercritical case (D>0), the extinction probabilities can be expressed in terms of the probability generating function (pgf)
We find that
where b(s)=f(s)-s.
QUASISTATIONARY DISTRIBUTIONS
In order to describe the long-term behaviour of the process, we shall use two types of quasistationary distribution (QSD), called Type I and Type II, corresponding to the limits:
where . Thus, we seek the limiting probability that the
population size is j, given that extinction has not occurred, or (in
the second case) will not occur in the distant future, but that
eventually it will occur; we have conditioned on eventual extinction
to deal with the supercritical case, where this event has probability
less than 1.
THE EXISTENCE OF QSDS
Consider the two eigenvector equations
where and C is the irreducible class.
In order that both QSDs exist, it is necessary that these
equations have strictly positive solutions for some , these being
the positive left and right eigenvectors of
(the transition-rate
matrix restricted to C) corresponding to a strictly negative
eigenvalue
.
Let be the maximum value of
for which
positive eigenvectors exist (
is known to be finite), and denote
the corresponding eigenvectors by
and
.
THE EXISTENCE OF QSDS
Proposition 5.1 of [1] can be restated for our purposes as follows (all sums are
over k in C):
Proposition 1. Suppose that Q is regular.
GEOMETRIC CATASTROPHES
We first examine the important special case ,
, where b>0, 0<q<1 and a + b = 1. Thus, given a jump
occurs, it is a birth with probability a or a catastrophe with
probability b, and the size of the catastrophe is determined by a
geometric distribution.
We need to solve the
right and left eigenvector equations. These are, respectively,
for ,
and, for ,
with the understanding that .
GEOMETRIC CATASTROPHES
We find that D = a - b/(1 - q), and that the maximum value of
for which there exist strictly positive left and right eigenvectors is
where r=a/(b+qa).
We deduce immediately that no QSD exists in the critical case (D=0). However, in both the supercritical (D>0) and subcritical (D<0) cases, both the Type I and the Type II QSDs exist.
GEOMETRIC CATASTROPHES
Supercritical case. When D>0 we find that
(note that r>1 since D>0)
and
where ; note that
.
Subcritical case. When D<0 we find that , where as above r=a/(b+qa), but now r<1 since
D<0. We also find that
where ; note that
.
THE GENERAL CASE
Recall that b(s)=f(s)-s, where f is the pgf given by , and D=-b'(1-). Recall also that, in the
supercritical case, the absorption probabilities have generating
function
( for all
in the subcritical case). We will use the
fact that b(s)=0 has a unique solution
on [0,1], and that
or
according as
or D<0.
Lemma 1. .
Theorem 1. In the subcritical case both types of
QSD exist. The Type I QSD is given by
(nice!), and the Type II QSD has
pgf
, where
and, for ,
.
THE GENERAL CASE
The supercritical case is more delicate. We require two extra conditions:
Condition (A). The catastrophe-size distribution has
finite second moment, that is,
(equivalently
).
Condition (B). The function b can be written
where L is slowly varying, that is,
for large t.
Theorem 2. In the supercritical case both types of
QSD exist under (A) and (B).
The Type I QSD is given by
, and the Type II QSD has
pgf
, where
and, for s<1, .