Dynamical Analysis Of Two Types Of Discrete And Continuous Mixed Ecosystems

The integrated management of agricultural and forestry pests has been widely used in agriculture and forestry related departments.The management methods are mainly chemical control and biological control,such as spraying chemical insecticides and releasing natural enemies.To evaluate the effectiveness of the integrated control strategy and design the optimal control scheme,it is necessary to build the corresponding mathematical model of the dynamic growth of the pest and natural enemy,and integrate the human intervention strategy including chemical and biological control measures.In recent years,researchers have established a series of continuous or discrete pest-natural enemy ecosystems to describe the interaction between populations,and analyzed the effects of intervention measures on system dynamics behavior and pest control by integrating integrated pest management strategies.The impulsive or Filippov non-smooth ecosystems(called as hybrid systems)based on impulse control and threshold control policy have been rapidly developed and widely applied due to their ability to accurately describe transient or intermittent interventions.In this paper,considering that the feedback control threshold depends not only on the population density of the current generation,but also on the population density of the next generation,and the influence of seasonal factors such as the periodic change of the external environment on the pest population,the discrete host-parasitoid system with switching strategy and the pestnatural enemy ecosystem with periodic perturbation and state dependent feedback control are developed respectively.In this paper,by developing the corresponding non-smooth theory and numerical analysis techniques,the dynamic properties of the two hybrid systems and the related biological conclusions are studied in depth.In the second chapter,based on the discrete host-parasitoid model,we consider that when the weighted density of the host population of two adjacent generations increases and exceeds the economic threshold,the IPM strategy is implemented,and when it is lower than the economic threshold,the corresponding control strategy is suspended.We establish a discrete switched host-parasitoid system with nonlinear threshold control strategy.Firstly,the dynamic behavior of the two subsystems is studied theoretically,and the existence and local stability of the equilibrium of the two subsystems are analyzed.By using the central manifold theorem and local bifurcation theory of the one-parameter family of discrete map,the one-parameter bifurcation of the equilibrium of two subsystems is investigated,including Flip bifurcation,Neimark-sacker bifurcation and the critical conditions for the transcritical bifurcation of the boundary equilibrium of the controlling subsystem.Then the types and existence of all possible equilibria of the whole switching system are discussed,and the existence and coexistence regions of different types of equilibria including regular and virtual equilibria are revealed numerically.In addition,by means of numerical analysis,the complex dynamic properties of the nonlinear switched system are discussed through one-parameter and two-parameter bifurcation diagrams of key parameters,and the phenomena of periodic solution,quasi-periodic solution,chaos,period-doubling bifurcation,period-halving bifurcation,period-adding bifurcation and coexistence of multiple attractors are obtained.Finally,the effects of the initial density of host and parasitic populations on the outbreak of host,the frequency of implementation of control strategies and the final stable state of the two populations were analyzed,which provided a basis for the design of reasonable pest management programs.In the third chapter,a non-smooth pest-natural enemy system with periodic forcing and nonlinear impulse feedback control is established to further study how limited resources affect pest control under seasonal fluctuation of environment.Firstly,the existence and stability of boundary periodic solutions are discussed by using the theory of impulse differential equations.Then the complex bifurcation patterns of the system are revealed numerically by using one or two dimensional bifurcation diagrams of key parameters.The numerical bifurcation results of onedimensional parameters show that the system has the phenomena of period-adding and period-decrease bifurcation without chaos,period-doubling,period-halving,the coexistence of order-k periodic solution and order-(k+1)periodic solution,and the coexistence of chaotic attractor and periodic attractor.In addition,some special fractal structures are found,that is,the order-k periodic solutions embedded in the chaotic window show the regular pattern of alternating change,and are arranged according to the Farey tree sequence.It is observed that the distribution region of periodic solutions of the system shows a "V" shape similar to Arnold's tongue and has certain self-similarity through the two-dimensional parameter bifurcation plane.The above phenomena not only indicate that periodic disturbance has a very important effect on the system dynamics,but also indicate that the design of optimal control scheme for integrated pest management is challenging.