On the behavior of solutions of predator-prey equations with hereditary terms

Some global properties of solutions of the classical integrodifferential systems, introduced by Volterra in his study of two species predator-prey populations, are studied. It is shown for large logistic loads that the predator goes to extinction and the prey tends to its carrying capacity. By use of a nonlinear approximation it is shown that for smaller logistic loads a "critical point" is asymptotically stable, while for sufficiently small logistic loads this point is unstable. These cases are demonstrated numerically for the original integrodifferential system using parameters which were computed on the basis of experimental data of S. Utida for bean-weevil vs. braconid-wasp interactions. Moreover, numerical solutions suggest further varied behavior of solutions of this system.