Qualitative Analysis For Some Chemotaxis Systems And Nonlocal Diffusion Models

Except random diffusion,the species including cell tends to move to some special places.The most famous oriented movement is that the species move towards the area with higher density of signals such as food or chemoattractant,which is called taxis.Chemo-taxis models(prey-taxis models)involve both the random diffusion and biased movement.Compared with simple structure of the random diffusion system,the taxis term that de-scribes the biased movement couples the higher gradient term of food or prey,and hence make the analysis works more difficult.Investigating the qualitative property of various chemotaxis system(prey taxis system)is very important for understanding the roles of the taxis effect play in ecology system.The usual random diffusion models the local diffusion in the nearby area,however,it is noticed that there exist nonlocal diffusions in nature,and the knowledge of nonlocal diffusion is rare when compared with local diffusion.The function used to represent non-local diffusion is an integral that consist of some kernel function and the density function of the population.And this function is lack of the important regularity properties which are known to exist for the local diffusion operator ?,hence it would be very challenging to study nonlocal diffusion model mathematically.This thesis concerns with the qualitative properties of some chemotaxis models and nonlocal diffusion systems with free boundary.The main contents are as follows:Firstly,we introduce the background,significance,investigative situation and main contents of this thesis.Chapter 2 studies the global solvability of a higher-dimensional Keller-Segel mod-el with signal-dependent motility and logistic growth.By introducing a step function and analysing a special quantity,the difficulty that the diffusion may be degenerate is overcome,it is shown that the strong logistic damping ensures the global existence and boundedness of the classical solution in the higher dimensions.Chapter 3 concerns with the global existence and boundedness of classical solution of the higher-dimensional forager-exploiter model.For cases where there are no forager and exploiter growth sources,it will be shown that if either the initial data and the pro-duction rate of nutrient are small,or taxis effects are small,then the classical solution exists globally and is bounded.For the case that only the forager has growth source,in two space dimension,it will be shown that if the taxis effect of exploiter is small then the classical solution exists globally and is bounded.For the case that both the forager and exploiter have growth restrictions,in two space dimension,we find a condition for the lo-gistic degradation rates that ensures the global existence and boundedness of the classical solution.In Chapter 4,we investigate two kind of prey-taxis models.For the prey-taxis system with unbounded growth property for the prey,we prove the global existence of the solution in 2,3 dimensions.For the prey-taxis system with two predator species,we establish the global asymptotic stabilities of nonnegative constant steady states.The fifth chapter is devoted to studying a predator-prey model with taxis mechanisms and stage structure for the predator.We first show the global existence and boundedness of the solution in one space dimension.We then investigate the linearized stabilities of the nonnegative constant steady states,and the emergence of the stationary patterns and time-periodic patterns.In Chapter 6,we investigate the dynamics of a predator-prey model with diffusion and indirect prey-taxis,in which the predator moves toward the gradient of concentration of some chemical released by prey instead of moving directly toward the higher densi-ty of prey.The first objective is to investigate the global existence and boundedness of the unique classical solution.It is shown that,compared to the prey-taxis,the indirec-t prey-taxis will prevent the growth of the predator to ensure the global existence and boundedness of the solution.Then,with the help of some C2+?,1+?/2 a-priori estimates,by constructing suitable Lyapunov functions and delicate discussions,we study the global asymptotic stabilities of nonnegative constant steady states and the convergence rates.Chapter 7 studies a diffusive Beddington-DeAngelis predator-prey model with non-linear prey-taxis and free boundary.Through a sequence of regularity estimates,we prove the global existence and boundedness of the classical solution by estimating the higher gradient term of prey taxis effect.And,we obtain the long-time behavior of the global solution.It is shown that,if the species cannot spread into the half space,then both of them shall die out.For the case that they spread successfully,we give some estimates for the large time behavior of the prey and the spreading speed.Moveover,some sufficient conditions for both spreading and vanishing are also established.Finally,we investigate a class of free boundary problems of ecological models with nonlocal and local diffusions.We prove that such kind of nonlocal and local diffusion problems has a unique global solution,and then show that a spreading-vanishing dichoto-my holds for the classical Lotka-Volterra competition and prey-predator models.More-over,criteria of spreading and vanishing are established.Our conclusions show that,the emergence of nonlocal diffusion operator does not influence the global solvability and the spreading-vanishing dichotomy,but it will change the criteria of spreading and vanishing.