Dynamic Studies On Some Mathematical Models With Taxis

Chemotaxis is a chemosensitive movement of biological species which detects and responds to chemical substances in the environment.By judging whether biological species are close to the chemical signal or far away from it,they can be divided into attraction or repulsion.Biological aggregation is the significant characteristic of chemotaxis.In ad-dition to the random diffusion of predator and prey,the spatiotemporal variations of the predator velocity are affected by the prey gradient,that is prey-taxis.Therefore,study-ing the diffusion-reaction predator-prey model with prey-taxis and going on considering chemotaxis models is very necessary.In this thesis,the dynamical properties of several kinds of mathematical models with taxis have been studied.The global existence and boundedness of the classical solutions of these models are proved.For the predaor-prey systems,we have made further research on it,and obtained some properties of steady state of this model.Moreover,the results of the study are applied to several specific examples.The main contents can be summarized as follows:Firstly,this part proves the global existence and boundedness of solutions to a gen-eral reaction-diffusion predator-prey system with prey-taxis defined on a smooth bounded domain with no-flux boundary condition.The result holds for domains in arbitrary spatial dimension and small prey-taxis sensitivity coefficient.This part also proves the existence of a global attractor and the uniform persistence of the system under some additional con-ditions.Especially,these results have been applied to three special models:for the diffu-sive Rosenzweig-MacArthur predator-prey model with prey-taxis,we have obtained the global existence and boundedness of the classical solutions,a global attractor and uniform persistence of the model;for the diffusive predator-prey model with strong Allee effect in prey growth and prey-taxis,the global existence and boundedness of the classical solu-tions of the model were proved by using our methods,and since(0,0)is always locally asymptotically stable,for the system uniform persistence never holds;for the model with taxis,we have improved the results in the case of n = 1.Secondly,an attraction-repulsion chemotaxis model with nonlinear chemotactic sen-sitivity functions and growth source is considered.The global-in-time existence and bound-edness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function by energy estimates.Our results improve the earlier ones for the linear sensitivity functions.Finally,this part considers the attraction-repulsion chemotaxis model with homo-geneous Neumann boundary conditions in a smooth,bounded,convex domain.In this model,when the scaling constant is zero and the chemotactic sensitivity functions are nonlinear,we prove that this system possesses a unique global classical solution that is uniformly bounded under some assumptions by constructing auxiliary functions;when the scaling constant is one and one of the chemotactic sensitivity functions is nonlinear,we also obtain a unique bounded global classical solution under some assumptions by energy estimates.