Global Analysis Of A Predator-prey System With Harvesting

Human harvesting is an important means to regulate the balance of the ecosystem.However,the phenomenon of species extinction resulted from overharvesting has been common,which destroys the stability of the ecosystem.Therefore,it is the key factor of keeping the stability of the ecosystem to find a suitable harvesting rate.The effect of harvesting on ecosystem can be investigated by establishing the differential equations model and analyzing its dynamical behavior.In this paper we study the globally dynamical behavior of a Lesile-Gower predator-prey system with prey harvesting.In chapter 2,we analyze the local dynamics of the extinction equilibrium(origin)in the system.Furthermore,by constructing the normal sector,generalized normal sector or open sector,the information of orbits near origin is discussed completely.It is shown that the topological structures in a neighborhood of the origin consist of the parabolic sector,the hyperbolic sector,the elliptic sector and any combination of them in the different parameters.In chapter 3,using the geometric singular perturbation theory,we investigate the existence of limit cycles and the cyclicity on all types of limit periodic sets,such as canard slow-fast cycle,contact point and singular slow-fast cycle.Firstly,the cyclicity of contact point is proved by the slow-fast normal form theory.Then,using the slow divergence integral theory,blow up technique and entry-exit function,we discuss the cyclicity of the non-degenerate canard slow-fast cycle,the canard slow-fast cycle with a hyperbolic saddle on the slow manifold and singular slow-fast cycle,respectively.However,there is no general theory to treat the slow-fast cycle with two canard mechanisms,transitory canard slow-fast cycle and slow-fast cycle with a degenerate singularity on the slow manifold.For the transitory canards and slow-fast cycle with two canard mechanisms,the slow-fast cycle,which can bifurcate limit cycles,is detected by the cylindrical blow-up.Then the relation between full divergence integral and transition time is studied along the closed orbit of the system and it is found that the cyclicity is at most one.For the slow-fast cycle with a degenerate singularity on the slow manifold,it is proved that system can’t generate limit cycles in most conditions of parameters by constructing the normal sector and generalized normal sector.In chapter 4,we study the global dynamics of system and achieve some interesting dynamical behaviors,such as relaxation oscillation and canard explosion.Finally,we give numerical simulations and biological interpretation for the theoretical results.In the last chapter,we statement some problems encountered when we analyze dynamics of the system,and some topics of further study.