Population Growth in Random Environments: Some Issues Concerning Stochastic Differential Equation Models
by
Carlos A. Braumann
CIMA-UE (Centro de Investigação em Matemática e Aplicações da Universidade de Évora)

Stochastic differential equation models for population growth in a random environment have been around since the pioneer work of Levins (1969, Proc. Natl. Acad. Sci. USA 62: 1061-1065). However, although little is known about the growth rate functional form, models in the literature assume specific functional forms. Therefore, conclusions may be a property of the model rather than of populations. We have been studying the properties of general models, where the growth rate, instead of assuming a specific functional form, is a general function satisfying reasonable assumptions dictated by biological considerations, so that we can reach conclusions that reflect properties of populations.

Let N=N(t) be the population size at time t. We model the per capita growth rate (1/N)dN/dt as an “average” per capita growth rate g(N) plus fluctuations induced by environmental randomness, which we assume can be approximated by a white noise. For N > 0, the growth rate g(N) is assumed to be a C¹ decreasing function of population size (reflecting intraspecific competition) that is negative for sufficiently high population sizes. We also assume that the limit G(0+) exists and is equal to zero, where G(N)=Ng(N). These are assumptions that are biologically reasonable for every population (without Allee effects). The model takes the form

(1/N) dN/dt=g(N)+se(t),

where s > 0 is an intensity parameter and e(t) is a standard white noise.

For these models we determine conditions for extinction and non-extinction, as well as conditions for existence of a stationary density (a kind of stochastic equilibrium with ergodicity).

The conditions depend on the stochastic calculus (Ito or Stratonovich) used. In particular, the qualitative results concerning extinction differ according to the calculus used. That led to a considerable controversy in the literature. We resolve the controversy by showing that the “average” growth rate g(N) indeed means a different type of average according to the calculus used (arithmetic for Ito, geometric for Stratonovich), and that, taking into account the difference between the two averages, the two calculi yield exactly the same result.

Date received: July 27, 2006