The Space-time Dynamic Analysis In Predator-prey System With Holling Type Ⅲ Functional Response

Mathematical biology is a subject that discusses the interaction of species and surrounding environment, and the dynamical behavior of predator-prey system has been one of the important topics in ecology and mathematical biology. Along with the development of modern mathematics, we analyze the predator-prey system by using qualitative analysis and stability theory. Research on the qualitative theory of dynamical systems has become important in recent years, and the existence and stability of equilibrium and periodic solution are among important branches of the research of dynamical systems.In the real world predator-prey model, not only we should consider the time evolution of the population, but we also need to consider the spatial diffusive effect. In the field of nonlinear partial differential equations, there exists an important nonlinear phenomenon, which is bifurcation. Bifurcation phenomenon means that, when the parameters cross through certain critical values, there exist changes of some structural properties in the system. Bifurcation could be local bifurcation, semi-local bifurcation and global bifurcation. The study of bifurcation of partial dif-ferential equations is not only related to the theories of classical dynamical systems, but also related to the other knowledge such as topology, algebra and functional analysis. The study is of great theoretical significance and practical background.This paper mainly performs ODE equational system and reaction-diffusion equations of the predator-prey system with Holling typeⅢfunctional response by using local linear analysis, Poincare-Bendixson theorem, constructing the Lya-punov function, global steady state bifurcation theorems and a few other mathemat-ical theory and method. Obtained results including the local stability analysis of equilibrium solutions, analysis of Hopf bifurcations, global steady state bifurcations and so on. It also provides some theoretical basis for future research.The paper considers a general equation in the following form: whereΩis a bounded domain, A, B, C, D, H are positive constants, p≥2. The main content of the paper isas follows:First, we give the dynamical analysis to the according ODE system by using local linear analysis and constructing the Lyapunov function;Second, using local Hopf bifurcation theorem of the semilinear partial differential equations, this paper mainly performs local Hopf bifurcation analysis to this reaction diffusion system; Lastly, a priori estimates of solutions to the corresponding steady state system are obtained by using the theory of elliptic equations regularity and analyzing the global steady state bifurcation.