SLOW FAST LIMIT-CYCLES IN PREDATOR PREY MODELS

This paper is devoted to the analysis of the cyclic behavior of predator-prey systems under the assumption that the time responses of prey and predator are quite diversified. In such a case, a geometric approach (separation principle) based on the singular perturbation method can be applied to detect slow-fast limit cycles. The main characteristic of these cycles is that the fast component of the ecosystem is present at significantly high densities only during a fraction of the cycle. At the start and at the end points of this period the slow component of the ecosystem reaches its minimum and maximum densities, which are related to each other through an integral equation. This equation is specialized for the classical predator-prey model and is later used to study the cyclic behavior of more complex predator-prey systems, some of which were not fully understood up to now. The conclusion of the analysis is that the existence of slow-fast limit cycles can be ascertained by means of the separation principle, while the geometry of the cycle can be fully specified only by using the integral equation discussed in this paper.