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

**-**- disease (eg a new virus)
**-**- food shortages (eg overbrowsing)
**-**-
significant changes in climate.

**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.

**(i)**If converges, and either converges or is bounded, then the Type II QSD exists and defines a proper probability distribution over*C*, given by**(ii)**If in addition converges, then the Type I QSD exists and defines a proper probability distribution over*C*, given by

**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, .