Prey Arrested In Fresh Reaction Diffusion Model Of Patterns And Traveling Wave Solutions

Recently, spatial dynamic behaviors are concerned vastly in predator-prey systems. Inthis paper, we mainly investigate the turing pattern structures and traveling wave solutionsof spatial extended predator-prey systems.In chapter 2, the pattern formation of the reaction diffusion system based on classicalBazykin model in spatial two dimensional domain is researched. we derive the conditionsfor turing instability in detail and obtain the turing space, in which the spatial system canemerge turing pattern. Furthermore, we simulate the pattern structure using the periodicalboundary condition. Our results show that the Bazykin system stabilizes to a striplikepattern structure when diffusion is present.In chapter 3, the pattern structures of the spatially extended Holling-Tanner model areinvestigated. By linear stability and bifurcation analysis, we present the dispersion relationdiagrams and the four different Turing spaces. Choosing appropriate parameter values inthese spaces, we obtain rich pattern structures in different Turing regions, which are spot,stripe and spot coexist pattern structures and so on. Furthermore, we study the Turing-Hopfbifurcation and also present labyrinthine patterns in the related region. Finally, we discussthe choice problem of the parameter a and find that these patterns can emerge only if thehalf saturation is much smaller than the carrying capacity.In chapter 4, the existence of traveling wave solutions for two reaction diffusion systems,which are based on the diffusive Holling-Tanner model for predator and prey interactions,is established. From dynamic, the traveling wave solution is equivalent to a heteroclinicorbit in 3-dimensional phase space. The proof of existence uses Wazewski's Theorem, theStable Manifold Theorem, and LaSalle's Invariance Principle. We also discuss some possiblebiological implications of the existence of these waves.