The analysis of higher-order cascade systems with separation conditions pivoting on the slow components: Application to a model of migration for survival of the species

Cascade systems, characterized by highly diversified time responses, are considered in this paper. Singular perturbation principles, which have been used to analyze relaxation oscillations in second-order dynamical systems, will be extended here to accommodate nonlinear systems in which more state variables are involved in multiscale interactions. Separation conditions will be derived for the identification of limit cycle behavior in a higher-dimensional (n ≥ 4) cascade system. It is found that when appropriate regularity and boundedness requirements are met by the slow components of the dynamical system, pivoting on the slow components can lead to separation conditions which identify limit cycle behavior as well as other dynamic behavior permitted by the model. The principle is then applied to a model of two communities coupled by migration. Through such analysis, we can examine how the mechanisms of migration, variations in reproduction, recruitment, mortality, and feeding success, exploited by interacting species, may achieve survival and coexistence of the populations concerned.