| In this thesis, we mainly study the stability and bifurcations in two categories of predator-prey systems with Michaelis-Menten type harvesting. The main trunk of this thesis is completed in chapter 2 and chapter 3. The details are as summarized follows:We research the dynamical behavior for a Leslie-Gower type predator-prey system with Michaelis-Menten prey harvesting in chapter 2, including the stability of the system and the existence of some bifurcations in this system. The major theoretical bases of this part are qualitative theory and the stability theory of differential equation, bifurca-tion theory, normal form theory, and so on. First of all, the existence of all the available equilibria in the system is analyzed. By choosing the appropriate parameters, the ex-istence conditions of these equilibria are derived. Secondly, we study the stability of these equilibria under their specific existence conditions. According to different cases,we analyse the system by some specific method such as eigenvalue analysis method, lin-ear method, and so on. We show that the boundary equilibria are always unstable under their existence conditions but the positive equilibria may exist as sink, source, center,saddle - node, cusp under different concrete parameter conditions. Thirdly, some bifur-cations may occur in this system are studied, including saddle-node bifurcation, Hopf b-ifurcation and Bogdanov-Takens bifurcation. By choosing some appropriate parameters as the bifurcation parameters, we derive the concrete bifurcation point. The existence of saddle-node bifurcation and Hopf bifurcation are also strictly proved by verifying the transversality conditions. In addition, we determine the direction of Hopf bifurcation by using the first Lyapunov method. Furthermore, the occurence of Bogdanov - Takens bifurcation of codimension 2 in the system is proved by applying normal form theory.We derive the normal forms up to the second order and the expression of bifurcation curve.Besides, we study the stability and Hopf bifurcation for another predator-prey sys-tem with maturation delay on prey population and Michaelis-Menten type harvesting on predator population in chapter 3. The key theoretical bases of this part are the stability theory of differential equation, Hopf bifurcation theory and delay functional differential equations theory. We derive the parameter conditions when there is a positive equilibri-um exists alone in the system. The necessary and sufficient condition to guarantee local asymptotic stability for the unique positive equilibrium in the system is exploited by analyzing the characteristic equation and by using the Hurwitz criterion. Furthermore,we choose time delay as analysis parameter. The certain range to make sure the system local asymptotic stability and the concrete critical delay value are derived by analyzing the real part of the eigenvalue. At last, we show that Hopf bifurcation occurs as the delay pass through the critical value.Finally, some numerical simulations to support them are also presented successively after getting the main theoretical results in this thesis. |
