Oscillations and Single Species Models with Delays

Abstract

The first four chapters have dealt with the stability of equilibria of ecological models which incorporate time delays either in the growth rate response to interactions with other species in a community or in the self-inhibitory response of a species to resource limitations. In this chapter (and the next) we will consider a few topics dealing with oscillatory behavior of solutions. Oscillations in an ecological community could arise of course from a great number of conceivable causes, including such things as the form of the growth rate response itself as it depends on the density of the species; variations in the system’s parameters (e.g. birth and death rates, carrying capacities, etc.) due to periodic fluctuations in the environment caused by, for example, seasonal or daily changes in temperature, rain fall, etc.; seasonal or periodic harvesting, seeding, immigration or emigration; random fluctuations of the model parameters; and many other causes. Another source of oscillations can also be delays present in growth rate response to changes in species densities.