Establishment And Research Of Spatio-temporal Population Model With Nonlocal Term

Classical models of population dynamics usually consider population individuals competing only with individuals in the same position,i.e.local competition.But in reality,individuals often compete for resources not only in their territory but also in a wider area.Therefore,this paper introduces the nonlocal competition term to study the influence of nonlocal competition on the Allee effect and the fear effect of the delay model.Firstly,a diffused predation model with nonlocal prey competition and Allee effect is established.The conditions for the coexistence equilibrium point to remain stable and to undergo spatially inhomogeneous Hopf bifurcation and Turing bifurcation have been studied.In the absence of Allee effect,we find that the coexistence equilibrium point of the system is locally asymptotically stable independent of the nonlocal competition.In the presence of Allee effect,nonlocal prey competition can destabilize the coexistence equilibrium point.Numerical simulations are carried out to illustrate the theoretical results.The amplitude of oscillation solution for nonlocal prey competition system is larger than local prey competition system until oscil lation solution evolves to periodic solution.Also,nonlocal prey competition term can drive a spatially inhomogeneous Hopf bifurcation,and the spatially inhomogeneous periodic solution emerges.Moreover,it is showed that when the habitat domain is larger,comparing with local prey competition system,the prey diffusion coefficient of system with nonlocal prey competition needs to be larger for two species coexistence in the spatially homogeneous form.Secondly,a time delay diffusion model of nonlocal prey competition and fear effect is established.The stability of coexistence equilibrium point and the existence of space inhomogeneous periodic solutions are studied.In the nonlocal prey competition model,the critical delay threshold increases with the increasing of the fear level or the intra-prey competition coefficient.In addition,in the case that the intra-prey competition coefficient is less than the threshold,local and nonlocal prey competition models admit the same critical delay threshold.However,the critical delay threshold for nonlocal prey competition is less than local prey competition in the case that the intra-prey competition coefficient is beyond the threshold.Nonlocal prey competition term can drive Hopf bifurcation for spatially inhomogeneous form,and the spatially inhomogeneous periodic solution emerges.It is worth noting that in the absence of delay,nonlocal prey competition model can undergo spatially inhomogeneous Hopf bifurcation and Turing instability by diffusion,but local prey competition can not occur.Numerical simulations are the validation of the theoretical analysis.Under the influence of nonlocal effect,the amplitude of the spatially homogeneous periodic solution becomes larger.Meanwhile,nonlocal effect increases the risk of extinction for two species to a certain extent.Finally,the paper summarizes the research results,innovation points and shortcomings,and puts forward the prospect of future work.