Nonlinear Predator System Time And Space Complexity

In this paper, we investigate the emergence of predator-prey system with reaction-diffusion, especially three types of predator-prey system with reaction-diffusion—Ivlev-type, Michaelis–Menten-type with constant harvest rate and Holling-type IV. For the three predator-prey model, respectively, We study how diffusion affects the stability of predator-prey coexistence equilibrium and derive the conditions for Hopf and Turing bifurcation in the spatial domain with symbolic calculation. Based on the bifurcation analysis, we give the spatial pattern formation, the evolution process of the system near the coexistence equilibrium point, via numerical simulation. For Ivlev-type,we find that pure Hopf instability leads to the formation of spiral patterns and pure Turing instability destroys the spiral pattern and leads to the formation of chaotic spatial pattern.Furthermore, we perform three categories of initial perturbations which predators are introduced in a small domain to the prey-only state to illustrate the emergence of spatiotemporal patterns, we also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations. This indicates that for prey-dependent type predator-prey model,pattern formations do depend on the initial conditions, while for predator-dependent type they do not.For Michaelis–Menten- type with constant harvest,the results of spatial pattern analysis, via numerical simulations, typical spatial pattern formation is isolated groups, i.e.,stripe-like,patch-like and so on.We also obtain spatial pattern of the prey and the predator with the same coefficients,the results validates that the types of the prey pattern and the predator pattern are the same.For Holling- type IV,additional, we study the model with a color noise and external periodic forcing. From the numerical results, we know that noise or external periodic forcing can induce instability and enhance the oscillation of the species density, and the cooperation between noise and external periodic forces inherent to the deterministic dynamics of periodically driven models gives rise to the appearance of a rich transport phenomenology. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics. Our results show that modeling by reaction–diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics. Our results show that modeling by reactiondiffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.