The Dynamics Of Several Biodynamic Systems With Time Delay

Bionic ammunition have been proposed in recent years, which can simulate the behavior of biological dynamics of a new concept of ammunition. Compared with conventional ammunition, munitions bionic dynamic behavior more dependent on the understanding of the biological system itself dynamical behav-ior. Biodynamic system in the study of biological dynamical system with time-delay characteristics of the biological world closer to reality, more truly reflect the kinetic properties of biological itself. Therefore, the dynamic behavior of biological power systems with time delay depth study for future research Bionic ammunition technology to lay a good foundation.This article discusses the dynamical behavior of biological systems with time-delay, including the spatial patterns. The detailed contents are as follows:In chapter 2, a predator-prey model with time delay and diffusion was considered. For the temporal model, we showed that there exists a threshold of time delay in predator-prey interactions; when time delay is below the threshold value, the positive equilibrium E* is stable. However, when time delay is above the threshold value, the positive equilibrium E* is unstable and period solution will emerge. For the spatiotem-poral model, By mathematical analysis, two different types of instability are found and the conditions for emerging Turing instability are given in detail. Furthermore, we illustrate the spatial patterns via numeri-cal simulations, which shows that the model dynamics exhibits a delay and diffusion controlled formation growth not only of stripe-like and spots patterns, but also of short stripe-like and spotted patterns emerge coexist.In chapter 3, a predator-prey model with time delay is considered. We give the conditions for emerging Turing instability in detail. In reaction-diffusion caused by unstable conditions, the spatial patterns via numerical simulations are illustrated, which show that the model dynamics exhibits rich parameter space Turing structures. In time-delay caused by unstable conditions, through a numerical simulation, black-eye patterns are got. The obtain results show that this system has rich dynamics, these patterns shows that it is useful for the diffusive predation model with a delay effect to reveal the spatial dynamics in the real model.In chapter 4, we study the dynamics of a delayed predator-prey model with double Allee effect. For the temporal model, we showed that there exists a threshold of time delay in predator-prey interactions; when time delay is below the threshold value, the positive equilibrium E* is stable. However, when time delay is above the threshold value, the positive equilibrium E* is unstable and period solution will emerge. For the spatiotemporal model, through numerical simulations, which show that the model dynamics exhibits rich parameter space Turing structures.In chapter 5, a predator-prey model with predator cannibalism and time delay is considered. By math-ematical analysis, different types of instability is found and the conditions for emerging Turing instability are given in detail. The spatial patterns via numerical simulations are illustrated, which show that the model dynamics exhibits rich parameter space Turing structures, respectively, spots, stripe-like patterns, and coex-istence of both.In chapter 6, a vegetation model with both cross diffusion and time delay is considered. For the temporal model, we showed that there exists a threshold of time delay in predator-prey interactions; when time delay is below the threshold value, the positive equilibrium E* is stable. However, when time delay is above the threshold value, the positive equilibrium E* is unstable and period solution will emerge. For the spatiotemporal model, Based upon a stability analysis, we demonstrate that the delay affects the stability and spatial patterns under some conditions. In addition, by numerical simulations, we obtain different spatial patterns.