Stability and Multi-Species Interactions with Delays

Abstract

The purpose of this chapter is to explore briefly some models for species interactions when time delays are present in at least some of the growth rate responses to interactions with either members of other species or the same species. We will confine ourselves mostly to mathematical investigations of the stability of equilibria for models which involve only quadratic interaction terms (i.e. for models (1.1) with f_i linear in N_j). Many ecological models have been proposed and studied in which these response functions f_i are not linear, both for models with delay and (even more so) without. For example models with Michaelis-Menten type terms and with delays have been studied by Caperon (1969) and MacDonald (1976) and with In N_j. terms with delays by Gomatam and MacDonald (1976). Caswell (1972) numerically studied a delay model involving complicated rational expressions in N_j. In principle however the linearization techniques we use would of course apply to these and any other more general model as well and in fact the results would be identical (allowing for differences in parameter interpretations) for models with identical linearizations. We will also upon occasion consider models in which the response functions are not necessarily linear.