The Role of Ceilings in Population Models

1
The Role of Ceilings in Population Models

• Ben Cairns
• Department of Mathematics
• Supervisor Phil Pollett
• Assoc. Supervisor Hugh Possingham
• bjc_at_maths.uq.edu.au, http//www.maths.uq.edu.au/
  bjc

number of suitable nesting sites. In many
circumstances, however, soft-limiting dynamics
may be preferable to a hard limit. Even
unbounded models (those which do not impose a
hard upper limit on the population) can be
guaranteed to remain finite, and otherwise
faithfully represent the dynamics of the
population. If the goal is to approximate a
population with a finite model, however, ceilings
play an important role. Approximating unbounded
populations One form of unbounded population
model is the birth, death and catastrophe
process, a continuous-time Markov chain model for
the size of a population. Birth, death and
catastrophe processes represent the dynamics of a
population as rates at which the population makes
(Markovian) transitions from one size to another.
Figure 2 illustrates a birth, death and
catastrophe process in which, for a population of
i individuals, births occur at a rate Bi, deaths
occur at a rate Di, and catastrophe drops in
population from size i to size j occur at rates
Cij. The model has a ceiling if the boundary, N,
is finite. Birth death and
catastrophe processes are typically represented
by a transition rate matrix, Q, which uniquely
determines the behaviour of the population model.
The elements, qij, of this matrix are given by
When a population is not subject to a hard
upper bound, it may be difficult to analyse its
behaviour. Expected times to extinction (or
simply extinction times) are found by solving
systems of linear equations, and in the case of
unbounded models these systems of equations are
infinite. If, in Figure 2, N is made finite, then
the model is subject to an upper bound, which
allows the approximation of the true extinction
times by values that are much easier to obtain
the solution, T, to the finite system of
equations where M is the restriction of Q to
states i, j 2 1, , N.
This approach assumes that the ceiling N is
reflecting. When reflecting, if the population
reaches N, then it will remain there until it
drops to a lower value. We may instead take N to
be absorbing such that the model allows the
population to jump from N up into some absorbing
state A. This latter form of ceiling has an
important advantage in approximating unbounded
populations (Figure 1). The expected time to
either extinction or to absorption at A is always
less than or equal to the extinction time for the
unbounded model, and so provides a conservative
estimate of this quantity (which may be
calculated according to the program laid out in
9.2 of 2). Figure 3 illustrates that
conversely, where the ceiling is reflecting, very
bad approximate extinction times may be obtained
for poor choices of N.
Introduction Population ceilings are features of
many population models, in which they play
important roles as representatives of the
physical limits on the size of the population.
Here we present an overview of the use of
population ceilings in the mathematical modelling
of populations. We will argue that fixed
ceilings to populations are often misplaced in
cases where the value of the ceiling does not
have a clear physical interpretation beyond that
of a maximum population size. A population
ceiling may still be useful, however, if the aim
is to approximate an unbounded population by one
that is bounded, but such a value must be chosen
with care. The use of population ceilings In
many cases, hard limits on the size of a
population are imposed by its environment. For
example, in a classical metapopulation the number
of suitable patches is also the ceiling for the
number of occupied patches (the size of the
metapopulation). Other natural limits such as
minimum home ranges or the availability of
nesting sites may limit a population. In 1,
for example, the population of breeding pairs of
the California Spotted Owl, Strix occidentalis
occidentalis, is limited by a ceiling that
represents the total