Dynamics Of Several Models With Nonlinear Chemotaxis Effect

In the spatial activities of species,in addition to the random diffusion of species themselves,species also make movements due to the influence of the external environment,which is called chemotaxis.Chemotaxis can be divided into attraction and repulsion according to the direction of movement of species.In addition,the rate of species movement brought by chemotaxis is not constant in most cases,and the rate of chemotaxis may be related to population density,time and even space.Based on the above considerations,in order to more completely study the directional motion problems caused by natural phenomena in space,several kinds of nonlinear chemotactic reaction-diffusion equation models are studied in this paper.Our main contributions are as follows:Firstly,we investigate a predator-prey system with nonlinear prey-taxis under Neumann boundary condition.For a class of chemotactic sensitive functions,we obtain the existence and boundedness of global classical solutions for initial boundary value problem of the system.It should be emphasized that we do not need to make truncation assumption for chemotaxis sensitive function,nor do we need to limit the spatial dimension.In addition,we also study the local stability of the constant steady state solution,and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions.Furthermore,the steady state bifurcation,Hopf bifurcation and fold-Hopf Singularity are analyzed in detail by using LyapunovSchmidt reduction method.Secondly,in view of the research in the previous chapter,we investigate a predator-prey system with stage structure for the predator under Neumann boundary condition,which has not only the taxis mechanism caused by the interaction between mature predator and prey,but also contains the taxis mechanism generated by the interaction between mature predator and immature predator.Regardless of the strength of the chemotactic coefficient,the existence and boundedness of global classical solutions are investigated for initial boundary value problems in two-dimensional space.In addition,appropriate Lyapunov functions are constructed to obtain the large-time behavior of the steady state solution under different predation intensity.In particular,it is interesting to observe that intra-specific competition keeps the species alive rather than dying out.Finally,we investigate a Lotka-Volterra model with nonlinear cross diffusion.Under appropriate assumptions,the global existence of the generalized solution is established,which excludes the possibility of blow-up the system solution.It should be emphasized that our results apply to any dimensional space.Moreover,In addition,we study the nonexistence of the nonconstant steady states solutions for sufficiently large diffusion rates.At the same time,sufficient conditions ensuring the existence of non-constant steady states solutions are obtained by using LeraySchauder degree theory.Furthermore,steady-state bifurcation analysis is carried out in details by using Lyapunov-Schmidt reduction.In addition,the steadystate bifurcation analysis is carried out by using the Lyapunov-Schmidt reduction method,and the existence,multiplicity and stability of the nonconstant steady states solutions are obtained.