DYNAMIC COMPLEXITIES IN RATIO-DEPENDENT PREDATOR–PREY ECOSYSTEM MODELS WITH BIRTH PULSE AND PESTICIDE PULSE

In many models of pest control, increases in pest population due to birth are assumed to be continuous, but in fact, pest population reproduces only during a single period; at the same time, pesticides are often applied during the period. So in this paper we propose a ratio-dependent predator–prey model with birth pulse and pesticide pulse. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which have Ricker functions or Beverton–Holt functions, and obtain the threshold conditions for their stability. Above the threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the ratio-dependent predator–prey model with birth pulse and pesticide pulse are very complex, including small-amplitude oscillations, large-amplitude cycles and chaos. This suggests that birth pulse and pesticide pulse, in effect, provide a natural period or cyclicity that allows for period-doubling bifurcation and period-halving bifurcation route to chaos.