Analysis Of Spatiotemporal Pattern Dynamics For Nonlocal Predator Prey Systems

The spatial nonlocal effect is a mechanism that reflects the intraspecific competition of species for limited resources.Studying the dynamics of a predator prey system with nonlocal competition is helpful to understand and predict the evolution of predators and prey species,which is of great significance to effectively prevent and control harmful species,protect endangered species and maintain the balance of the ecosystem.In this thesis,we study the spatiotemporal dynamics of predator-prey systems with nonlocal intraspecific competition of prey by applying linear stability analysis and bifurcation theory,including the stability of the positive constant steady state and the existence and stability of the coexistence state revealed by the double-Hopf bifurcation and Turing-Hopf bifurcation.The main contents are as follows:Firstly,to investigate the spatiotemporal dynamics near the double-Hopf singularity,we derive explicit formulas of the normal form up to the third-order term of double-Hopf bifurcation for general partial functional differential equations(PFDEs)with the nonlocal effect.It is formed by original parameters of the system and can be applied for both functional differential equations(FDEs)and partial differential equations(PDEs)with or without nonlocal effects.In addition,we also modify the formula for calculating the coefficients of the normal form of Turing-Hopf bifurcations from [1] for general PFDEs,including the resonance cases.The application of the formulas in this case is carried out for the first time.Secondly,considering a spatial average kernel function,we investigate the effect of nonlocal intraspecific prey competition on the spatiotemporal dynamics of HollingTanner predator-prey model with diffusion.We establish the criteria for Hopf bifurcation,Turing bifurcation,double-Hopf bifurcation,and Turing-Hopf bifurcation,and determine the stable and unstable conditions of the positive equilibrium.It turns out that in addition to Hopf bifurcation and Turing bifurcation,the positive equilibrium can destabilize through double-Hopf bifurcation,and Turing-Hopf bifurcation as well,and double-Hopf bifurcation is induced by the nonlocal competition.The normal form up to the third order is derived by our formulas,many spatio-temporal patterns are found by analyzing the corresponding normal form,including stable spatially homogeneous or nonhomogeneous periodic solutions,and stable spatially nonhomogeneous quasi-periodic solutions.Moreover,by calculating and analyzing the normal form truncated to the third-order of Turing-Hopf bifurcation,we derive that,with strong nonlocal interaction,the system exhibits the tristable phenomena,that is,the coexistence of a stable spatially nonhomogeneous periodic orbit and two nonconstant stable steady states,as well as the existence of periodic orbits with two spatial wave frequencies induced by the nonlocal interaction.Biologically,the emerging spatiotemporal patterns suggest that the global intraspecific competition can promote the coexistence of the prey and predator by allowing the prey maintain a critical total population size,which may provide an alternative approach in explaining the group formation of some prey species under the risk of predation.To compare the advantage of nonlocal and local competition,we introduce another prey species with local intraspecific competition.Numerical investigations show some coexistence patterns are destabilized leading to the coexistence of two preys and one predator,namely,the prey with nonlocal interaction is concentrated at a single spatial location,and the other prey is distributed uniformly in the rest of the habitat.Accordingly,the predator is forced to change its behavior as well.Finally,considering a spatial dependently kernel in the nonlocal effect,we investigate the periodic pattern formations with spatial multi-peaks through double-Hopf bifurcation.We first proof the existences of Hopf bifurcation,Turing bifurcation,Turing-Hopf bifurcation and double-Hopf bifurcation,and determine the stability of the positive equilibrium.It turns out that the stable parameter region for the positive equilibrium decreases with αincreasing,which implies that the parameter region of pattern formation for such kernel is smaller than the spatial average case.We calculate the normal form of double-Hopf bifurcation,which is expressed by the original parameters of the system.Via analyzing the normal form,some complex spatiotemporal patterns are found,including stable spatially nonhomogeneous periodic patterns,the bistability of such periodic solutions,as well as unstable spatially nonhomogeneous quasi-periodic solutions,all of them possess multiple spatial peaks.Interestingly,some possible strange attractors are found numerically near the double-Hopf singularity.Biologically,the emerging spatio-temporal patterns imply that such nonlocal intraspecific competition can promote the coexistence of the prey and predator species in the form of more complex periodic states.