## Simple Models

The simplest model of a stochastic population growth process probabilistically leading to extinction is the simple birth-death process. The simple birth-death process supposes that for each individual in the population in a small interval of time At the chance that it reproduces one more individual is given by bAt and the chance that the individual dies is given by dAt, where b and d are parameters that are referred to as the birth rate and death rate, respectively. This model assumes that the birth rate and death rate are the same for all individuals at all times. These are unrealistic assumptions. For instance, each individual has the same chance of reproducing or dying, regardless of age, sex, or physiological status. Similarly, the assumption that the birth rate and death rate are constant implies that the population could grow indefinitely. Further, many species reproduce by multiple births and all species exhibit some period of embryonic development, so the assumptions of individual births and equal probabilities of reproduction in all consecutive intervals of time are violated. Nonetheless, this model represents a reasonable starting point for understanding the effects of chance demographic events (i.e., births and deaths) on population extinction. For some populations it could even be a reasonable approximation to a more realistic model, at least for forecasting growth of small populations over relatively short intervals during which the birth and death rates remain approximately constant.

Representing the size of the population by N and the probability that the population is size N at time t by pN(t), a differential equation for the change in pN(t) is dpN^ = b(N - 1)PN - l(t) - (b + d )NpN (t)

over N = 0,1,2,.. . and t> 0. There are several ways to approach deriving this equation. For an example, see the book by Eric Renshaw in the list of further reading. An exact expression for the solution of this equation can be obtained but is too complicated to be very useful. However, some summary results are more intuitive. For instance, the population size at time t averaged over all possible trajectories, M(Nt), is given by

where no is the size of the population at time o, while the variance in N(t) is

Renshaw observes that, unlike the average population size, the variance in population size depends on the magnitudes of the birth and death rates, not just the difference between them. This is reasonable because predictions about the future population size should be less precise when births and deaths occur in rapid succession than when the processes underlying growth and decline are relatively slow. Returning to the focus of PVA on the chance of extinction, the time-dependent probability that the population is at size 0 (i.e., that it is extinct) is obtained as

From this equation it is evident that (1) the chance of extinction declines exponentially with population size, and (2) the chance of extinction declines with increasing intrinsic rate of increase r = b — d. Indeed, even for populations exhibiting only modest population growth at very small population sizes, the chance of extinction due to demographic stochasticity alone is very small.

Already it is clear that even a very simple stochastic model where births and deaths are chance events results in considerable variation in the possible future trajectories of a population. As a step toward a more realistic model of population growth, we make two further generalizations. First, we allow that the environment may fluctuate randomly over time. In contrast to the birth-death process which assumes that the population reproduces continuously, most models of 'environmental stochasticity' start from the assumption that survival and reproduction can be represented in discrete time. This assumption is reasonable for many species that reproduce on an annual cycle. Then, it is assumed that annual birth and death rates vary a small amount from year to year. It is well known that populations under such circumstances grow approximately multiplicatively. However, if we consider instead the growth of the transformed variable x = loge(N) the population will grow by the addition of a random quantity at each time step. This transformed variable can then be approximated by a very well-understood stochastic process, Brownian motion, and is therefore sometimes referred to as a diffusion approximation. If the population starts at an initial population size x0, the chance of extinction within a time t is approximately po(t) = $

is the standard normal distribution function.

The parameter m refers to the 'infinitesimal mean', which governs the average growth ofthe stochastic process, while g2 refers to the 'infinitesimal variance', which describes the typical random fluctuations around the mean. It can be seen from the above equations that the chance of extinction decreases as the population increases in size, consistent with the birth-death process. However, in this case the chance of extinction also increases with the variance of fluctuations in the average growth rate, that is, the rate of extinction increases with environmental variability. In a landmark paper published in 1991, Brian Dennis and colleagues showed how the parameters m and G could be estimated from a time series of population abundances using simple regression models. Subsequently, other methods for model estimation have been proposed to deal with issues pertaining to data quality and to relax some of the more restrictive model assumptions. Recent extensions of these models have considered additional phenomena exhibited by real populations including population regulation, Allee effects, catastrophic population loss, and sexual reproduction. See the papers by Dennis et al., Holmes, and Lande and Orzack and the books by Caswell and Tuljapurkar for further details.

One extension of the diffusion approximation that has received considerable attention concerns populations with age or stage structure. As most endangered species for which PVA is performed exhibit clear population structure, this extension adds considerable realism to theoretical models. However, the primary advantage ofstructured models is that elasticity and sensitivity analysis on the stage-structured model can guide management by highlighting particular cohorts or developmental stages to target for intervention. Both elasticity and sensitivity analysis are concerned with the effect of changes in the vital rates of different age or stage classes on the average population growth rate. Sensitivity refers to the change in the population growth rate with respect to some vital rate, whereas elasticity refers to a proportional effect of a change in a vital rate. Effectively, whereas sensitivities describe absolute changes in growth rate with respect to changes in vital rates, elasticities are just normalized sensitivities that account for the fact that different vital rates (e.g., fecundity and survival) have different units of measurement and scale very differently (e.g., y e

fecundity can range from 0 to thousands or tens of thousands while individual survival rates range from 0 to 1). Generally, elasticity and sensitivity are calculated with respect to the dominant eigenvalue of the Leslie matrix or Lefkovitch matrix describing structured population growth, a quantity that is referred to as the asymptotic population growth rate and describes the eventual rate of the growth of the total population. Importantly, while the addition of structure does not change the general conclusions that extinction risk declines with population size and decreased environmental variation, the precision of forecasts can be improved by accounting for population inertia. For these reasons, most PVAs are of this structured variety. The books listed in the section titled 'Further reading' describe these methods in detail.

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