Research On Bifurcation Dynamics Of Predator-Prey System With Delay And Prey Harvesting

The predator-prey models are arguably the building blocks of the mathematical andbiological modeling. Based on the biological background, the predator-prey models assistus in understanding and predicting the dynamical behaviors of real bio-systems by thetheoretical and numerical analysis. Hence, the study is of great practical significance.This thesis aims at the dynamical analysis on the predator-prey models with timedelay and Michaelis-Menten prey harvesting. From the point of view of bifurcations, weinvestigate the dynamics of predator-prey models as varying of the time delay and theharvesting rate, such as existence and stability of equilibrium points, periodic orbits andquasi-periodic orbits, and the attraction of quasi-periodic, homoclinic, heteroclinic orbits.In addition, we also study the long-term effect of the above bifurcations on the model.The key theoretical foundations of this thesis are the stability theory, center manifold the-orem, normal form theory, Hopf bifurcation and high co-dimensional bifurcation theoryof functional differential equations and partial differential equations. The details are asfollows:1. We consider a modified Leslie-Gower predator-prey model with time delay andMichaelis-Menten type harvesting in prey. Taking the time delay and harvesting rateas bifurcation parameters, we derive that the system undergoes saddle-node bifurcation,Hopf-bifurcation, and saddle-node-Hopf bifurcation at the coexistence equilibrium point.Furthermore, we calculate the normal forms up to the third order based on the originalparameters of the model, analyze and obtain the entire bifurcation set and phase portraitsof the saddle-node-Hopf bifurcation, and derive that there exist attractive quasi-periodicsolutions and attractive heteroclinic orbits near the coexistence equilibrium point.2. We study the effect of the maturation time delay and the Michaelis-Menten typeprey harvesting on the dynamical behavior of the modified Leslie-Gower predator-preymodel. By the stability analysis of the equilibrium points, we obtain the critical conditionsfor the fold bifurcation and Bogdanov-Takens bifurcation. Furthermore, from the point ofview of Bogdanov-Takens bifurcation, we investigate the stability of the periodic solutionsand attractiveness of the homoclinic orbits bifurcated at the coexistence equilibrium pointby using the second order unfolding equation.3. We investigate a phytoplankton-zooplankton model in the presence of Holling IIfunctional response. Considering the time delay due to toxin-producing and the Michaelis-Menten harvesting in phytoplankton, we derive that at the coexistence equilibrium point,the system undergoes stability switch, Hopf bifurcation, fold bifurcation, fold-Hopf b-ifurcation and double Hopf bifurcation. Through the theoretical analysis of the Hopfbifurcation, we find that there exist stable periodic solutions on the unstable regions ofthe stability switch. By the analysis of the double Hopf bifurcation, we develop that thephytoplankton-zooplankton model has stable periodic solutions and attractive homoclinicorbits. In particular, we find a quasi-periodic solution in numerical simulation.4. We consider a predator-prey model with the time delay, Michaelis-Menten typeprey harvesting and Holling II functional response. We study the effect of the diffusion onthe stability of the periodic solutions with homogeneous Neumann boundary conditions.As varying of the time delay, the system undergoes a Hopf bifurcation at the constantequilibrium point. In particular, the critical bifurcation values are ordered, the spatiallyinhomogeneous periodic solutions are unstable, and stable periodic solutions must bespatially homogeneous periodic solutions. Due to this specific characterization of themodel, we prove the global existence of the spatially periodic solutions of the predator-prey model using the global Hopf bifurcation theory of functional differential equation.