On The Stability And Hopf Bifurcation For Predator-prey Models With Time Delays

There are many factors that afect the development of species in the nature, such astime-delay, stage-structure, interaction between species, patches, refuge, disease and soon. These factors are important to the species. In this paper, we establish exact predator-prey models based on the above factors and study the stability and Hopf bifurcation ofthese models.This paper is divided into four chapters.Chapter1briefly introduces the researched background of problems and the mainwork of this paper.Chapter2studies a stage-structured predator-prey model with time delays and preda-tor consuming immature prey only. By analyzing the stage-structure predator-prey modeland using the standard comparison theorem, sufcient conditions for permanence of thesystem are derived. The existence and the locally asymptotic stability of nonnegativeequilibria are considered which made us obtain the important result of this paper thatthe variation of predator stage structure afect the stability of the boundary equilib-rium. By means of an iteration technique, sufcient conditions are obtained for the globalasymptotic stability of the positive equilibrium.Chapter3studies a delayed predator-prey model with prey refuge and difusion. Byanalyzing the corresponding characteristic equations, the local stability of the equilibria isinvestigated and Hopf bifurcations occurring at the positive equilibrium under sufcientconditions are demonstrated. According the result of that the positive equilibrium islocally asymptotically stable under some conditions, we know that the prey refuge anddifusion afect the stability of the predator-prey system. The stability and the direction ofthe Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are determinedby using the normal form theory and center manifold theory.Chapter4studies a delayed predator-prey model with disease in the prey only. Byanalyzing the corresponding characteristic of the positive equilibrium, we know that thepredator can afect the diseased prey. The stability and the direction of the Hopf bifur-cation periodic solutions bifurcating from Hopf bifurcations are determined by using thenormal form theory and center manifold theory.