| In recent decades,population dynamics relationship of predator-prey has caused a lot of scholars' research boom.Natural phenomena in ecology have different forms,so diverse models are used to study the interaction among the populations and the interaction between population and environment by scholars.Predator-prey model is a vital part of it.Research on predator-prey model can help people better under-stand natural issues in ecology and thus help to maintain the ecological balance.In this paper,we mainly discuss the properties of solutions for two kinds of predator-prey model under the first boundary condition.One is a modified Leslie-Gower predator-prey model with cross-diffusion(?)and the other is a predator-prey model with additive Allee effect(?)In chapter 1,we introduce the research background of predator-prey model,lead into the predator-prey models to be studied in this paper,and simply analyze their significance of research.In chapter 2,we discuss a predator-prey model with cross-diffusion.Firstly,we derive the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system.Secondly,applying maximum principle,a priori estimate of positive solutions is given.Finally,we gain a sufficient condition for the existence of positive solutions by means of degree theory.In chapter 3,a predator-prey model with additive Allee effect is investigated.In the first part,we give a priori estimate of positive solutions and analyze the non-existence of positive solutions.In the second part,applying local bifurcation theory,the existence of local bifurcation solutions is proved,the local bifurcation can be extended,and its trend is obtained.In the third part,using stability theory and spectral analysis method,we discuss the stability of local bifurcation solutions and semi-trivial solutions,respectively.In the fourth part,some theoretical results are verified by the numerical simulation technique. |
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