Dynamic Properties For Several Kinds Of Reaction-difusion Predator-prey Systems With Delay Efect

In this thesis, we consider the dynamic properties for several kinds of reaction-difusion predator-prey systems with delay efect via the stability theory and qualitativetheory of diferential equations, obtain several new conclusions and extend the corre-sponding results in the existed papers.The thesis is divided into four sections according to the content.Chapter1briefly introduces the researched background of problems and the mainwork of the paper.Chapter2studies a reaction-difusion predator-prey system with Beddington-DeAngelis functional response and delay efect. By analyzing the linearized systemof the positive equilibrium point and the corresponding characteristic equation, westudy the asymptotic stability of the positive equilibrium point and the conditions forthe existence of Hopf bifurcation of the system. By discussing the influence on the Hopfbifurcation, we obtain large difusion has no efect on the Hopf bifurcation, while smalldifusion can lead to the fact that the system bifurcates a spatially inhomogeneous pe-riodic solutions at the positive equilibrium point. By applying the normal form theoryfor partial functional diferential equations, the formula determining the direction ofHopf bifurcations and stability of bifurcating periodic solutions are obtained.Chapter3studies a difusive predator-prey system with Holling II functional re-sponse and nonlocal delay. By analyzing the corresponding characteristic equations,the local stability of each of steady states and the conditions for the existence of Hopfbifurcation of positive steady state are established. By using the technique of upper-lower solutions and monotone iteration scheme, the sufcient conditions for the globalstability of the positive steady state and the semi-trivial steady state are derived.By using the geometric singular perturbation theory, the existence of travelling wavesolution of the system is established.Chapter4studies a reaction-difusion predator-prey system with stage structureand nonlocal delay. By using the cross iteration method and constructing a pair ofsuitable upper-lower solutions, we reduce the existence of travelling wave solutions, which connecting the trivial steady state to the positive steady state of the system. Byapplying energy function method, the sufcient conditions for the global attractivityof positive steady state are derived, which improves the corresponding results of Xuand Chaplain.