Predicting The Global Dynamics In Predator-prey Models

The predator-prey model as one of the typical ecosystems has played significant roles in the theoretical studies of biology and mathematics.The establishment and analysis of predator-prey models can help us sufficiently understand the characteristics and development trend of biological population,and then effectively implement necessary biological control.Due to the complexity of the interactions between organisms,mathematical models describing population dynamics often contain many meaningful parameters.If the qualitative or topological structure of model changes as parameters varied in the neighborhood of one or more parameter values,the model is said to have bifurcation phenomenon,and the corresponding parameter values are called bifurcation values.The bifurcation theory of dynamical systems has a wide and profound application background,which originates from the study of some instability phenomena in many disciplines such as fluid mechanics,elastic mechanics,celestial mechanics and nonlinear vibration.Since the 1970s,with the continuous research and exploration of dynamic systems,nonlinear analysis,numerical simulation and so on,the mathematical theories and methods of bifurcation gradually formed.They were widely used in many disciplines such as biology,ecology,control engineering technology,physics,economics and sociology.The bifurcations mainly contain equilibrium bifurcation(saddle-node bifurcation,Hopf bifurcation,cusp bifurcation,BogdanovTakens bifurcation,zero-Hopf bifurcation,double Hopf bifurcation,generalized Hopf bifurcation,etc.),limit cycle bifurcation(saddle-node bifurcation of limit cycles,period doubling bifurcation,cusp bifurcation of limit cycles,etc.)and homoclinic(heteroclinic)bifurcation.Local and global bifurcation analysis on predator-prey models will facilitate our effective understanding of the dynamic characteristic of ecosystems and contribute to the implementation of biological control.The functional response in the model usually appears in different forms due to the differences in predation effects.In addition,there are many factors influencing the dynamical behavior of the population,such as anti-predator behavior,Allee effect,harvesting or stocking,refuge,intraspecific(interspecific)competition,etc.In recent years,different types of predator-prey models have received extensive research and attention.However,the bifurcation research of models with anti-predator behavior,additive Allee effect,harvesting or stocking in both populations and the mite predator-prey model of Leslie type with Holling Ⅳ functional response,especially the high codimension bifurcation analysis,is not exhaustive.Aim at above issues,the following innovative works were presented in this doctoral thesis.Firstly,the high codimension bifurcations are investigated for a generalized Gause predatorprey model with anti-predator behavior.Through the seven-step transformation method,We raise the codimension of Bogdanov-Takens bifurcation in this model to codimension 3.In addition,the codimension of Hopf bifurcation is increased to codimension 2,which shows the co-existence of stable and unstable limit cycles in the system.In particular,the specific effects of anti-predator behavior on system dynamics are elucidated.By numerical bifurcation analysis,we reveal the interactions between different parameters in the model,and demonstrate the existence of bistability and tristability states.These results indicate that anti-predator behavior has a great influence on the dynamic behavior of predator-prey systems.Secondly,the bifurcation dynamics of a modified Rosenzweig-MacArthur model involving constant harvesting or stocking rate in both populations are investigated.We show the rich dynamical behaviors including saddle-node bifurcation,cusp of codimensions 2 and 3,Bogdanov-Takens bifurcation of codimensions 2 and 3,as well as degenerate Hopf bifurcation of codimension 2.In particular,a codimension-2 cusp of limit cycles is found,which indicates the co-existence of three limit cycles.Moreover,an interesting and novel scenario is first discovered:two Bogdanov-Takens bifurcation points are not always connected by a continuous homoclinic bifurcation curve.Further,the positive effects of harvesting or stocking are elaborated.By choosing different parameter values and initial states,we can stabilize the system to different coexistence states.This work contributes to a deeper understanding of the dynamics of ecosystems when harvesting or stocking occurs simultaneously,allowing for effective biological control.Thirdly,the bifurcation behavior of a Leslie type predator-prey model with additive Allee effect is investigated.In this part,the asymptotic dynamics near the origin are studied by blasting transformation.Using dynamical system methods,we analyze the existence and types of equilibria,and emphasizes the rich bifurcation phenomenons such as saddle-node bifurcation,BogdanovTakens bifurcation of codimensions 2 and 3,degenerate Hopf bifurcation of codimension 2,homoclinic bifurcation,saddle-node bifurcation of limit cycles,codimension-2 cusp of limit cycles and isola bifurcation of limit cycles.Through numerical simulations,we verify the theoretical results,reveal the co-existence of two and three limit cycles,and highlight the significance of additive Allee effect in causing more complex dynamics.Notably,we observe the isola bifurcation of limit cycles for the first time,suggesting a new mechanism for sustained oscillations.Finally,the bifurcation behavior is analyzed for a mite predator-prey model of Leslie type with Holling IV functional response.By using theories such as canonical forms of dynamical systems,this model is shown to exhibit rich bifurcation dynamics,including subcritical and supercritical Hopf bifurcations,degenerate Hopf bifurcation,cusp and focus types degenerate Bogdanov-Takens bifurcations of codimension 3,in which Bogdanov-Takens bifurcations act as the organizing center for the whole bifurcation set.Moreover,through numerical simulation,we clearly explain the existence region of two limit cycles,and elucidate the co-existence of multiple equilibria and limit cycles.Our work extends some crucial results in the literatures.