Singular perturbation analysis of a model for the effect of toxicants in single-species systems

We consider a mathematical model for the effect of toxicant levels on a single-species ecosystem in the case where there is an initial instantaneous introduction of a toxicant into the environment. The population birthrate as well as the carrying capacity are assumed to be directly affected by the level of toxicant in the environment as it is absorbed by the population. The toxicant level in the population can be depleted at a constant specific rate, a part of which may return to the environment. Through a singular perturbation analysis, we are able to identify different dynamical behavior which may be possible to the system, including the existence of sustained oscillation in the levels of toxicant in the population and the environment.