Kinetic Analysis Of Two Types Of Population Models With Different Movement Strategies

The reaction-diffusion model is an important model to describe the expansion and evolution of biological population in ecosystem.However,this expansion behavior is sometimes affected by the environment,resulting in certain directionality,such as the horizontal or vertical flow of the river,the upward or downward trend of the water column under pressure,etc.,which can be described by the advection term.Therefore,the mechanism of the evolution and expansion of the aquatic population with mobility can be studied by the reaction-diffusion-advection model.Based on this,the movement strategies of species may be affected by the environment,resulting in diffusion movement and diffusion-advection movement.In this paper,by means of maximum principle,upper and lower solution method,fixed point index theory,degree theory,Crandall-Rabinowitz bifurcation theorem,space decomposition technique,implicit function theorem,the theory of monotone dynamical system and spectrum theory,we discuss the dynamical behaviors of oneprey and two-cooperative-predators model with diffusion mechanism and LotkaVolterra competition-diffusion-advection model under different environmental conditions.The main works are summarized as follows:The first chapter introduces the biological background and research progress of the two models,and expounds the motivation,significance,ideas and main contents of research in this paper.The second chapter is devoted to studying one-prey and two-cooperative-predators model with diffusion and C-M functional response under Dirichlet boundary condition.Firstly,we discuss the existence of positive steady states by the fixed point index theory and the degree theory.In the meantime,we analyze the uniqueness and stability of coexistence states under conditions that one predator's consumer rate is small and the effect of interference intensity of another predator is large.Then,steady-state bifurcations from two strong semi-trivial steady states(provided that they uniquely exist)and from one weak semi-trivial steady state are investigated by the Crandall-Rabinowitz bifurcation theorem,the technique of space decomposition and the implicit function theorem.In addition,we study the asymptotic behaviors of system including the extinction and permanence by the comparison principle,upper-lower solution method and monotone iteration scheme.Finally,some numerical simulations validate the previous theoretical analysis,and further clarify the impacts of some parameters on the three species.The third and fourth chapter investigate the dynamical behaviors of LotkaVolterra competition-diffusion-advection model under general boundary condition by taking the growth rates and advection rates of two species as the variable parameters,respectively.Firstly,by analyzing the stability of the semi-trivial steady states,we give a complete classification on the local dynamics of the system.It turns out that there always exist two critical curves in the plane of growth rates or advection rates of two species,which may separate competition outcomes into competitive exclusion,bistability and coexistence.As a further development,under the assumption that the ratio of advection and diffusion rates of one species is not less than that of the other species,the global dynamics is analyzed in detail,which shows that bistability does not happen,but coexistence and competitive exclusion may occur.These interesting results reveal that the growth ability and advective movements play an important role in the dynamical behaviors.In fifth chapter,supposing that the species cannot pass through the upstream boundary and do not return into the habitat after leaving the downstream end,we study the dynamical behaviors of Lotka-Volterra competition-diffusion system in open advective environment by continuing to regard advection rates of two species as variable parameters.By analyzing the properties of the principal eigenvalues of the linearization problem corresponding to the semi-trivial steady states,we find that there exist critical curves which separate the stable region of the semi-trivial steady states from the unstable one.Then,a complete classification on the local dynamics the system is presented by analyzing the properties of the critical curves.Moreover,we further analyze the global dynamics of the system,which suggest that the species with relatively strong advection will be completely replaced by the weak one.In particular,under the assumption that the advection and diffusion rates of two species are proportional,it turns out that competition outcomes of two species may shift between competitive exclusion and coexistence.If the above assumption is not true,the global results for some special cases are established by using the perturbation theory and the properties of principal eigenvalues.These results indicate that advection movements are important factors affecting population dynamics.