Predator-Prey Theory and Variability

Field ecologists are often impressed by the variability of ecological processes (26). In spite of this, mathematical models for predator-prey interactions have been mostly deterministic. Stochastic components are included simply by modifying a deterministic model-often with considerable arbitrariness. There has been a ten­ dency to regard stochasticity either as noise obscuring a deterministic signal or simply as a destabilizing influence. However, variability (stochasticity) plays a fun­ damental role in predator-prey population processes; it helps to explain observable population phenomena. In this review we explore ways in which deterministic models are inadequate and suggest some methods of modelling variability. Variability can be classified according to its origin into five categories. These different kinds of variability affect population dynamics in different ways. Especially interesting is the effect of variability in the presence of population subdivision-i.e. when the populations are not homogeneously interacting because movements are not unlimited. In this situation it can be shown that within-individual variation, or demographic stochasticity as it is usually called, does not have a vanishingly small effect as the population gets large. This is contrary to current opinion (41,57) based on models of populations that interact homogeneously. The idea of a structural model is introduced below. Such models make qualitative assumptions and are designed to allow different models to be compared within a single organizing framework. Structural models also yield qualitative predictions depending only on their qualitative assumptions and are therefore especially useful for testing ideas in ecology. Current modelling procedures place too great an empha­ sis on specific models. More general approaches are needed, which depend less on particular models. Stochastic systems require stochastic stability concepts. The idea of stochastic boundedness is an outgrowth of the concept of tightness of a class of probability measures. As a stability concept, stochastic boundedness seems more applicable to