Qualitative Analysis Of Some Predator-prey Models

At the beginning of this thesis,three predator-prey systems are proposed.The first system is a harvested model with Holling typeâ…¡response.When the harvesting term is linear,it is proved that all the solutions of the system are upper bounded and the sufficient condition is obtained for the existence and uniqueness of stable limit cycle.Furthermore,it is shown that local asymptotic stability of the positive equilibrium implies its global asymptotic stability.As an application,we also examine some special case of the system to confirm our main results through numerical simulation. When it is the system with constant rate harvesting,it is shown that the positive equilibrium can exhibit the Hopf bifurcation or Heteroclinic bifurcation when parameters vary in a small neighborhood of the values of parameters as well as the uniqueness and the number of the stable or unstable limit cycle.The results illustrates that the use of harvesting efforts as a control to obtain strategies for the control of a prey-predator system with Hollingâ…¡functional response.The second system is a radio-dependent predator-prey model with disease in predator. It is obtained the sufficient condition for local stability of every equilibrium in system. Meanwhile,it is constructed Lyapunov function to prove the global stability of equilibria on the axis under some condition,this provides the parameter condition to avoid the extinction of predator.The third system is about a nonautonomous radio-dependent food-chain.It is proved that the permanence and global attractivity of the solution of the model.Furthermore, it is shown that the existence and uniqueness of periodic solution as well as its global asymptotic stable.In the second part of this thesis,the existence and uniqueness of limit cycle for a class of higher degree polynomial system is considered,as well as the discussion of bifurcation of system when m=0.The result is also applied in the research of number and distribution of limit cycle for more general quadratic system.The conclusions improve entirely the relevant research results given before.