Mathematical models of predator/prey/plant interactions in a patch environment

Several tritrophic systems are characterized by local over-exploitation of the food source. Interactions between predatory mites, spider mites and their host plants are an example of such systems: either the spider mites over-exploit local patches of host plants or the spider mites are exterminated by predatory mites. It is often stated that modelling the overall population dynamics of such systems in a realistic way would soon lead to an unmanageable edifice. We advocate, however, the use of physiologically structured population models as a both general and formal mathematical framework. The advantage is that analytically tractable models may be obtained from the complex ‘master’ model by time-scale arguments or special choices of model ingredients. In this way a network of models can be derived, each concentrating on a particular aspect, all inadequate to cover the entire spectrum, but together (we hope) providing a coherent set of insights the relative importance of which can be assessed by computer experiments on the ‘master’ model. In this paper a rather realistic model of predator/prey interactions in an ensemble of host-plant patches is presented and, as an example of our approach, some special cases are derived from that model. Their analysis provided some first, useful insights. It is shown that prolonged duration of the prey-dispersal phase and prey dispersal from predator (-invaded prey) patches may result in a stable steady state, whereas a humped plant-production function may — under certain conditions — result in two stable steady states.