Periodic Time-Dependent Predator-Prey Systems

The general system of differential equations describing predator-prey dynamics is modified by the assumption that the coefficients are periodic functions of time. By use of standard techniques of bifurcation theory, as well as a recent global result of Rabinowitz, it is shown that this system has a periodic solution (in place of an equilibrium) provided the long term time average, of the predator’s net, unihibited death rate is in a suitable range. The bifurcation is from the periodic solution of the time-dependent logistic equation for the prey (which results in the absence of any predator). Numerical results which clearly show this bifurcation phenomenon are briefly discussed.