Dynamic Analysis Of Two Kinds Of Predator-prey Models

Predator-prey model is a very important mathematical model in population dynamics,which has been paid close attention by ecologists and mathematicians.By analyzing the important frontier problems such as various equilibrium points,periodic solutions,Hopf bifurcation,Turing instability and long-term dynamic behavior of the system,we can not only understand the natural law of population growth,but also help to maintain the balance of the ecosystem.This thesis mainly studies these two kinds of mathematical models by using nonlinear analysis and differential equation theory: The first is the predator-prey model with artificial linear harvest term and Holling type II functional response function;The second kind is the predator-prey model with modified Leslie Gower diffusion.For the first kind of model,the main contents of the research include the existence,stability and uniqueness of the positive solution of the model.The methods and theories used include comparison principle,implicit function theorem,fixed point index theory and bifurcation theory.We give the critical conditions of Hopf bifurcation at the system and the calculation expression of the first Lyapunov coefficient.Finally,we verify it by numerical simulation with MATLAB.For the second kind of model,under the condition of homogeneous Neumann boundary,firstly,the global attractor and persistence of the positive solution of the system are discussed by using the comparison principle and linearization method;Secondly,the local stability and global asymptotic stability of the constant equilibrium solution of the system are studied by eigenvalue analysis and Lyapunov function;Using the maximum principle and Harnack inequality,a priori estimate of the positive solution of the system equilibrium is given;Finally,using the theories of Young inequality and energy integration method,the conditions for the nonexistence of non-constant positive solutions of the system are obtained.