An approximation for prey-predator models with time delay

A prototype prey-predator (P-D) model in which the effective size of the predator population interacting with its prey follows an instantaneous time-delay tau regarding its total size is considered here. A simplified model was derived after substituting the approximation D(t - tau) approximate to D(t) - tau(D) over dot(t) into the above time-delay model. In order to assess the reliability of the simplified model, we performed a comparative study of both models under a wide range of parameter values, focusing on the effect of tau on two issues: (i) the boundary (in parameter space) between the regions leading either to stable fixed points or to limit cycles, and (ii) periods and amplitudes. We have found that, for small enough values of tau compared with the period characteristic of the non-delay model, both the boundary and the periods and amplitudes obtained for the time-delay model can be fairly approximated by the corresponding results for the simplified model.