Hopf Bifurcation For A Logistic Population Model In The Heterogeneous Environment

In the nature world,intraspecific competition is a common interspecific relationship,and the individuals of a species may compete for resources,such as nutrients,light sources,water sources,and so on.This kind of competition not only exists in the spatially homogeneous environment,but also exists in the spatially heterogeneous environment.The research on the nonlocal competition in the advective heterogeneous environment could help us analyze the population dynamics and related ecological process.In this paper,we consider a delayed reaction-diffusive-advection equation,which models the population dynamics in the advective heterogeneous environment.The main contents are as follows:Firstly,we consider the existence of the nonconstant positive steady states and the associated Hopf bifurcation.We can prove the existence of nonconstant positive steady-state solution by using the Grandall-Rabinowitz branch theorem.By linearizing the equation at the steady state,we obtained the characteristic equation.Then we obtained the distribution of the eigenvalues through the implicit function theory,and obtain the sufficient conditions for the stability of the steady states and the existence of Hopf bifurcation.Then,we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.A weighted inner product associated with the advection rate is introduced to compute the normal forms.Based on the center manifold theory and the normal form theory,we obtain that the direction of the Hopf bifurcation is forward,and the bifurcating periodic solutions from the first Hopf bifurcation value is asymptotically stable.Finally,we consider the effects of the spatial scale and advection on the Hopf bifurcation in the heterogeneous environment.And the population dynamics when the intrinsic growth rate is spatial homogeneity and spatial inhomogeneity are simulated by numerical simulation to support our obtained theoretical results.Hence,the folllowing conclusions can be drawn: Hopf bifurcation can be more likely to occur when the advection rate and spatial scale increase or decrease for different types of intrinsic growth rate.