Global Behavior Of Solutions For A Predator-prey Chemotaxis Model

Chemotaxis model and we use it not only in the biological process at microscopic scale, but also in the population dynamics at macroscopic scale.Due to the existence and important effects of chemotaxis, more and more researchers have begun to study the effects of its on the global existence and long time behavior of solutions and pattern formation. But the nonlinearity of chemotaxis make the investigation very di?cult. In particular, researches on two-species or multi-species chemotaxis models have not been done too much so far. In 2012,Tania et al.(Proc. Natl. Acad. Sci. USA, 2012, 109(28): 11228-11233.) have been proposed and studied the two-predator and one-prey predator-prey chemotaxis system. Their work focus on the exist pattern formation chemotaxis models instead of discussing the other nonlinear dynamic behavior of solutions.This paper is mainly divided into four parts, which are devoted to studying the impact of chemotaxis on the global existence and long time behavior of solutions and pattern formation. There are three kinds of topics: Global existence and global boundedness of solutions; Long time behavior, including long time convergence and convergence rate, to the global solutions; and pattern formation.Firstly, by using the operator semigroups theory and Banach contraction mapping principle, we prove the existence and uniqueness of local solutions for the threespecies chemotaxis model for reasonably regular initial values. Afterwards, global bounded solutions under small inital data condition in a two dimension predatorprey chemotaxis system which has the defensive switching property of predationavoidance are shown.Secondly, based on the unequal diffusion coe?cients, nonlinear dynamics near an unstable constant equilibrium for the three-species predator-prey chemotaxis model in a three-dimensional is discussed by applying the embedding theorem, the energy estimates and the bootstrap arguments. Our results indeed provide a rigorous quantitative characterization for the nonlinear evolution of early spatiotemporal pattern formation on the unstable positive constant equilibrium.Thirdly, this paper considers global behavior of solutions in the higher-dimensional a three-species predator-prey chemotaxis model without reaction term for the predator. The result reveals that under small initial data condition, the solutions of the system is global in time and bounded and approaches the steady state solution exponentially as time tends to in?nity in the smooth bounded domain; Moreover, we shall that under small initial data condition the global existence of the solutions on the whole space.Finally, this paper considers global behavior of solutions for a three-species predator-prey chemotaxis model with reaction term. To begin with, we obtain the existence and uniform boundedness of the global solutions based on Maximal Sobolev Regularity. Next, if the ratio a1/χ1, a2/χ3is suitably large, then the unique positive spatially homogeneous equilibrium is globally asymptotically stable. The result implies that three species can coexist.