Mathematical and dynamic analysis of an ecological model with an impulsive control strategy and distributed time delay

Abstract

In this paper, on the basis of the theories and methods of ecology and ordinary differential equations, an ecological model with an impulsive control strategy and distributed time delay is established. By using impulsive equation theories, small amplitude perturbation skills and the comparison technique, we get the condition which guarantees the global asymptotical stability of the prey ( x ) and predator ( y ) eradication periodic solutions. Further, the influences of impulsive perturbations on the inherent oscillations are studied numerically; these show rich dynamics features, such as period-halving bifurcation, a chaotic band, a periodic window, chaotic crises, etc. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of the strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems.