Study On Predator-prey Dynamic Models With Herd Behavior

Population dynamics is one of the most grown-up and widespread usedsubdiscipline in biomathematics. Population dynamics mainly study the interactiverelationships between the species and environment and interspecies in ecology. All thetime, the relationship between predator and prey has attracted attention in interactiverelationships among species. Therefore, the predator-prey model has become animportant problem attractting many scholars to research. In this paper, we investigatethe dynamical behaviors of three predator-prey models with herd behavior and discussthe effects of time delay and diffusion to dynamical behaviors. The outline of this paperis as follows:In the first chapter, we give a summary of relevant background and researchfulsignificance of predator-prey model and introduce some newly developments in thisfiled. At the end, we briefly state the main results obtained in this paper.In the second chapter, a delayed predator-prey model with herd behavior isinvestigated, where we use time delay to model the lag caused by nutrient conversion.Firstly, choosing the time delay as a control parameter, we obtain the conditions of thelocal asymptotic stability of the positive equilibrium and the existence of Hopfbifurcation by the qualitative analysis of characteristic equation. Making use of theNormal form theory and central manifold theorem, we obtain the formula to determinethe direction, stability and period of bifurcating periodic solution. Finally, numericalsimulations are carried out to illustrate our results.In the third chapter, a model of reaction-diffusion predator-prey model with herdbehavior is discussed, where we assume that both predator and prey population canrandom walk in two-dimensional bounded domains. The parameter space in whichTuring pattern possibly takes place is obtained by analyzing linear stability. By themultiple-scale analysis we deduce the amplitude equation of wave vector nearby theTuring bifurcate point, and obtain the expression which is used to determine thestability of the pattern. Furthermore, by numerical simulations, we find that the systemexists three kinds of pattern structure: H hexagons, stripes and hexagons-stripemixtures.In the fourth chapter, a delayed model of reaction-diffusion predator-prey modelwith herd behavior is investigated, where both time delay and diffusion factors are takeninto account. Firstly, we obtain the conditions of local asymptotic stability of positive equilibrium and the existence conditions of Hopf bifurcation, including the conditionsof spatial homogeneous and nonhomogeneous periodic solution respectively nearby thepositive equilibrium. Secondly, we obtain the formula which determines the directionand properties of bifurcating periodic solution by Normal theory and center manifoldtheorem of partial functional differential equations.In the fifth chapter, we give the conclusion of our present work and the outlook forfurther investigation.