On Relationships Among Various Types of Population Models

Mathematical models for communities of interacting species usually seek to relate the population growth rates to the various inter- and intraspecific interactions. If birth is a continuous process, so that the populations grow in a continuous manner, one ends up with a system of differential equations; conversely, if generations are discrete, so that population growth is a discrete process, the result is a system of difference equations. Corresponding to any particular differential equation system is an analogous difference equation system, which embodies identical biological assumptions except that time is a discrete rather than a continuous variable. I make explicit the relation between the stability properties of any such pair of models, showing in precisely what sense the populations in the difference equation system tend to be less stable than those in the differential equation model. Although this point is basically a commonplace one, some of its implications do not seem to be widely appreciated. By way of illustration, detailed comments are made about aspects of some particular models (e.g., Lotka-Volterra; Nicholson-Bailey).