Analysis Of The Dynamical Properties Of Several Types Of Reaction-diffusion Predator-prey Systems

Population dynamics is the branch of biomathematics that uses dynamical systems to study population density and population structure.The study of population dynamics behavior is of great importance for the conservation,management and development of ecological environment.Predator-prey relationship is one of the very classical interactions in population dynamics.In biology,mathematics and ecology,it is of great importance to study the dynamic behavior between predator and prey.In this paper,the dynamical properties of three types of reaction-diffusion predator-prey systems are studied separately,and the main work is organized as follows:(1)In Section 2,we extend a Leslie-Gower-type predator-prey system with ratiodependent Holling Ⅲ functional response considering the cost of antipredator defence due to fear.We study the impact of the fear effect on the model,and we find that many interesting dynamical properties of the model can occur when the fear effect is present.The relationship between the fear coefficient and the positive equilibrium point,and the effect of some key bifurcation parameters on the existence of the Turing instability,the Hopf bifurcation,and the Turing-Hopf bifurcation are obtained.(2)The hydra effect,which is a paradoxical result in both theoretical and applied ecology,refers to the phenomenon in which an increase in population mortality enhances its own population size.In this paper,through bifurcation analysis and numerical simulations,we consider a reaction-diffusion predator-prey model with Holling Ⅱ functional response to analyze the existence of hydra effect and the relationship between mortality independent of predator density and different steady-state solutions of the system.We investigate the existence of the hydra effect when the positive equilibrium point is locally asymptotically stable and Turing unstable.Meanwhile,numerical simulations verify the existence of the hydra effect when the one-dimensional reaction-diffusion system has a spatially inhomogeneous steadystate solution.In addition,we introduce the existence of the Turing bifurcation,the Hopf bifurcation,and the Turing-Hopf bifurcation,respectively.And the complex spatio-temporal dynamics near the Turing-Hopf bifurcation point is shown by the normal form for the TuringHopf bifurcation.(3)In Section 4,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We study the effect of three different types of diffusion on the stability of this system.The results are as follows: in the absence of prey diffusion,diffusion-driven instability can occur;in the absence of predator diffusion,diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable;in the presence of both prey diffusion and predator diffusion,the system can occur diffusion-driven instability and Turing patterns.At the same time,we also get the existence conditions of the Hopf bifurcation and the Turing-Hopf bifurcation,and the normal form for the Turing-Hopf bifurcation.