ON THEORETICAL MODELS FOR COMPETITIVE AND PREDATORY BIOLOGICAL SYSTEMS

Differences of opinion are obviously possible on the degree to which admittedly oversimplified theoretical models can explain some of the complex observational phenomena to be found in nature. Criticisms from biologists of the mathematical work of Lotka (1925), Volterra (1926) and subsequent writers on the growth and interaction of biological populations have, however, sometimes been justified and sometimes unjustified, for in spite of inevitable limitations such work constitutes a permanent contribution to the understanding of how populations may behave. A significant constructive survey was made by Gause (1934) when he attempted to bridge the gap between theoretical models and natural biological phenomena by controlled laboratory experiments in animal ecology. While experiments also have limitations as representations of nature, the role of both theory and experiment in the physical sciences might be recalled by any biologists inclined to be sceptical of the value of either. The interrelation of these approaches in biology may be illustrated in the field of epidemiology. Here the vicissitudes of infected populations have been studied in the laboratory as well as in the field; but, as I have emphasized elsewhere (Bartlett, 1956, 1957), the properties of theoretical models indicate, among other things, the extent to which population size may sometimes be crucial in the probable sequence of events, and thus indicate to what extent laboratory observations will have any similarity to larger-scale field observations even if the same model applies to both. An essential point is that recent theoretical formulations explicitly recognize the discrete character of populations and the stochastic or random aspect of changes, as distinct from strictly deterministic formulations. The need for this in ecology, which was already perhaps envisaged by Gause (1934, p. 124), is quite apparent in the experiments by Park with the flour-beetle Tribolium, in which one of two species together in a container survived not every time, but with a definite probability (e.g. 30 % of times), that could be estimated by replication and changed by changing the environment (see, for example, Neyman, Park & Scott, 1956). There is now no mathematical difficulty in the formulation of stochastic models (see, for example, Bartlett, 1955 a, 1956), and such formulations have already been made for typical ecological models by Chin Long Chiang (see Kempthorne et al. 1954). The greater intractability of even the simplest of these is, however, a serious obstacle to progress, especially in animal ecology, where even in the deterministic formulations of Lotka and Volterra many simplifications, such as neglect of age structure or of other heterogeneity, were made. One aim of the ensuing discussion is to indicate the enhanced value of deterministic formulations of population dynamics when properly interpreted within more comprehensive stochastic