#### model formulation

To develop a model of density-dependent catastrophes we expanded upon a birth and death process model developed by MacArthur & Wilson (1967) and later modified to include catastrophes by Mangel & Tier (1993). Taylor & Karlin (1998) present an introduction to these methods. We assume a minimum size at which the population is extinct (the critical population size) and known values for the maximum population size, carrying capacity, and birth and death rates. Then the mean time to drop to the critical population size from any population size *x*, assuming the population changes by at most one individual per time step, is:

*T*(*x*) = 1 + *B*(*x*)*T*(*x* + 1) + (1 − *B*(*x*) − *D*(*x*))*T*(*x*) + *D*(*x*)*T*(*x* − 1).(eqn 1)

Here *B*(*x*) is the instantaneous birth rate at population size *x*, *D*(*x*) is the instantaneous death rate at size *x*. This equation can be written simultaneously for all population sizes as a vector of *T*(*x*)'s and a matrix, called the infinitesimal generator, containing the *B*(*x*)'s and *D*(*x*)'s (Taylor & Karlin 1998). If **T** is the vector of mean persistence times for populations from the minimum size to the carrying capacity and **M** is the matrix of population size specific birth and death rates, Mangel & Tier (1993) show that

This is an analytical method for calculating mean persistence times that can accommodate very complex dynamics, including density-dependence (Mangel & Tier 1993). In most cases the matrix inversion will have to be done numerically, and there are many software packages available that can perform this operation. We used True BASIC™ (http://www.truebasic.com), a modern structured programming language which is interpreted into C, running on a Macintosh G3 computer.

Mangel & Tier (1993) extend this basic model to include a two part process for catastrophes, composed of the rate of occurrence for catastrophes at a given population size (size is interchangeable with density because the model assumes a constant habitat area) *C*(*x*), and the intensity of the catastrophes. Given that a catastrophe occurs, we define intensity, *Q*(*y* |* x*), as:

*Q*(*y* |* x*) = Pr{decrease is *y* individuals | current population size is *x*} for *y* > 1 (eqn 3)

where *Q*(*y* |* x*) must sum to one over all possible catastrophe sizes:

- (eqn 4)

We depart from Mangel & Tier (1993) in ignoring catastrophes with no deaths, *Q*(0 |* x*), and ones that are analogous to the individual death term, *Q*(1 |* x*). This modification yields a matrix of population size specific rates similar to eqn 2, with the addition of catastrophe probabilities in the subdiagonal:

- ( eqn 5)

Where *C*(*x*) is the catastrophe rate at size *x*, *x*_{c} is the population size below which extinction occurs, and

*R*(*x*) = *B*(*x*) + *D*(*x*) + *C*(*x*)(eqn 6)

is the rate of change in population size at *x*. This matrix can be inverted as above in eqn 2 to yield a vector of mean persistence times for all population sizes.

In order to investigate the effect of density-dependence in catastrophe probability, we chose a function for the catastrophe rate that would be flexible but would maintain a sigmoidal shape. We defined the catastrophe rate as:

- (eqn 7)

Here the rate of density-independent catastrophes is ω_{0}. The rate of density-dependent catastrophes is , where *x* is the current population size, *x*_{th} is the size at which the rate is , and Γ is a parameter allowing modification of the strength of the density-dependence (Fig. 1). We chose this particular functional form for three reasons: (i) it agrees with the conclusion that disease outbreaks in wildlife populations generally occur when the population exceeds a threshold value (Dobson & Hudson 1995); (ii) it matches the observed pattern in the empirical data from crabeater seals that we use; and (iii) it is a relatively simple form that requires few parameters and thus is straightforward to analyse. In discussing density-dependent catastrophes we will use ω_{1}, the maximum rate, to refer to a particular catastrophe rate. The actual values for *C*(*x*) will be much less for most population sizes, with its value reaching ω_{1} only near the maximum population size.

In the paper we will generally discuss catastrophe frequency as a probability instead of a rate for clarity, therefore we show the method for conversion from rates. This is also a necessary step for simulations, as although the rates can be represented directly as probabilities (*B*(*x*)Δ*t* + *o*(*x*)Δ*t*), these are not bounded by 1 when Δ*t* is very close to 0 (Hilborn & Mangel 1997; p. 69). A change in population size occurs with probability:

- (eqn 8)

The probability of a particular type of change, e.g. a birth, is the product of the probability of a change (eqn 8) times the relative probability of the type of change of interest (a birth):

- (eqn 9)

Analogously, death and catastrophe rates can be converted to probabilities by replacing the birth rate in the numerator of the last term in eqn 9 with appropriate rate.

We modelled the intensity of catastrophes as either a uniform or binomially distributed decrease in population size (Fig. 2). For the uniform distribution, if a catastrophe occurs, all population sizes two or more less than the current size are equally likely outcomes. Although we use this as a density-independent case, the population size will have some effect on the size of the catastrophe. On average, the population after a catastrophe will be approximately half of the original population. We use the uniform distribution as our density-independent case for three reasons: (i) if the size of a catastrophe were wholly density-independent, i.e. a set value, it would either devastate the populations at small population sizes or be trivial at large sizes; (ii) it is a common distribution used in many population viability models; and (iii) in comparison with the binomial model the effect of uniformly distributed catastrophes is much less dependent on density. For the binomial model we assume all individuals are equally likely to die in a catastrophe with probability 1 − *p*. The most likely reduction in the population size is (1 − *p*) (*x* − 2), where *x* is the pre-catastrophe population size (Fig. 2). Resulting populations will be concentrated near *px* rather than distributed across the range of values; thus population reductions due to binomial catastrophes are more sensitive to the pre-catastrophe population size. Although it would be possible to make the intensity of catastrophes more strictly density-dependent by allowing the survival probability to depend on density, we chose to avoid this complication to maintain comparability with the empirical example we use.

It is important to note that our model does not include environmental stochasticity, otherwise known as process error (Hilborn & Mangel 1997). To incorporate process error in a model one needs to deal with the variation in demographic rates over time, and ideally incorporate not only their variance, but also the covariance among rates. This is a complex, but solvable problem for demographic simulation models (e.g. Doak *et al*. 1994). However, incorporating process error into the type of birth-death process model we use is difficult. The primary problem is that it would be necessary to take the expectation of the inverse of the matrix of birth, death, and catastrophe rates, **M**, across the joint distribution of the demographic rates:

**T** = E_{ɛ}[−**1****M**^{−1}(*b*(ɛ),*d*(ɛ), *c*(ɛ))](eqn 10)

Where ɛ denotes the process error in the birth (*b*), death (*d*) and catastrophe rates (*c*). It is important to be clear, introducing process error does not alter the fundamental underlying process – the population still changes by only single births or deaths and catastrophes – the values in the inverse matrix are now just an expectation across the variation in each rate.

An alternative approach for calculating equation 10 would be to discretize the joint distribution of the birth and death rates, calculate **M** for each possible joint realization of the rates, invert each **M**, and then take the expectation across these inverted matrices. We performed this discretized analysis assuming a symmetric beta distribution for additive process error in either the birth rate or the death rate, holding the other rate constant, to evaluate the effects of environmental stochasticity on our model results. We varied each rate in steps of 5% from −100% to +100%, yielding 20 possible realizations for either the birth or death rate. We explored the effects of this additive process error as the variance in error distribution increased up to a maximum of 0·4. To illustrate the range of this variance, at the maximum error variance of 0·4 there was a 5% chance that the observed rate would differ by 75% or more from the mean rate in any given observation.

Incorporating process error into the birth, death, and catastrophe model we used resulted in quantitative changes in our results, primarily reducing the persistence of small and intermediate sized populations. However, while there were quantitative effects, the qualitative results of our model did not change, and thus we chose to exclude environmental stochasticity from our analysis. It is important to note that the time required for calculating the expectation of the inverse matrices can be prohibitive if the number of realizations of the process error is very large.