Analysis of mathematics and dynamics in a food web system with impulsive perturbations and distributed time delay

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, a food web system with impulsive perturbations and distributed time delay is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the prey and intermediate predator eradication periodic solution. On this basis, we get that the food web system is permanent if some parameters are satisfied with certain conditions. In order to show that these conditions are effective, the influences of impulsive perturbations on the inherent oscillation and distributed time delay are studied numerically; these show rich dynamics, such as period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems.