Stability and Instability in Predator-prey Models with Growth Rate Response Delays

We consider a general model for predator-prey interactions in which the instantaneous per unit growth rate j i — ft(Nv N2)(t) of each species at any time t is a functional of species densities N^s) at previous times s ^ t. We assume that the equation for prey density Nx obtained from the model in the absence of predators (N2 — 0) possesses at least one positive equilibrium c > 0 (which may or may not be stable). Our goal is to study the stability properties of positive equilibria Ni — ei > 0 of the predator-prey system as they are functions of c. Suppose the prey's growth rate response functional has the property that there exists at least one^osi t ive prey equilibrium c when N2 — 0 (i.e., f1 = 0 for JVj = c, N2 = 0) and that this parameter c is made explicit in the functionals fv Thus, we consider j i to be a function of c as well as of N{(s)9 s = t. Specifically we assume