Dynamical Analysis Of Some Planar Filippov Biosystems

In recent years,Filippov systems have been widely used in the mathematical modeling of practical problems such as prevention and treatment of infectious diseases,integrated management of plant diseases,integrated management of agricultural pests and the coexistence of predators and preys in nature.This paper focuses on the dynamical analysis of some planar Filippov biosystems,including the effect of integrated disease management on the dynamical properties of plant disease models,the effect of the refuge on the dynamical properties of prey-predatory models and the effect of on the dynamical properties of integrated pest management on pest-natural enemy models.Firstly,we establish two kinds of plant disease models with threshold control strategies and we analyze the global dynamics of the two kinds of plant disease models respectively by employing Poincare mapping and the qualitative theory of Filippov systems.Secondly,we propose a general discontinuous predator-prey model incorporating a threshold policy by extending a general continuous predator-prey model and further study the model.It is worth to mention that we can obtain a globally asymptotically pseudo-equilibrium and a globally finite-time stable sliding cycle under certain conditions,which can not be achieved in a continuous predator-prey model.Finally,based on the integrated pest management,we study the pest-natural enemy model and by using the qualitative analysis theory of Filippov system and we analyze the global dynamics of the model.This paper is mainly divided into five chapters.In the first chapter,we briefly introduce the research background and the development of several planar Filippov biological systems.In the second chapter,we give some basic knowledge needed in this paper.In the third chapter,we first consider a Filippov-type plant disease model with the switching line to be a oblique line.When the interaction ratio of the numbers of infected plants and susceptible plants is above the critical threshold,the control strategies including replanting or roguing are carried out,otherwise the control strategies are not necessary.By employing Poincare maps,our analysis reveals rich dynamics including a global attractor bounded by a touching closed orbit which is convergent in finite time from its outside,a global attractor bounded by two touching closed orbits and a pseudo-saddle,and a globally asymptotically stable pseudo-node.Then based on the plant disease model with the switching line to be a oblique line,a Filippov-type plant disease model with two thresholds are established.The threshold policy control causes the switching line to be a broken line,whose structure leads to giving birth to two pieces of sliding mode regions and two pseudo-equilibria.Meanwhile,one more sliding segment or pseudo-equilibrium brings more complex and rich global dynamics.By employing Filippov qualitative theory,the global dynamics of the model are analyzed,including the existence of one piece or two pieces of sliding mode regions on the switching line and their sliding mode dynamics.The complex topological structure near the break point of the switching line is analyzed,and the special function is constructed to exclude the existence of sliding limit cycle and crossing limit cycle.In the fourth chapter,we analyze global dynamics of a general predator-prey model with a refuge.Due to the hiding behavior of the prey,the model has piecewise smooth response function.This work reveals the rich global dynamics of the general non-smooth model,including sliding mode region,sliding mode dynamics,a globally asymptotically stable pseudo-equilibrium and a globally asymptotically stable sliding cycle.By employing the qualitative analysis theory related to Filippov systems,the necessary and sufficient conditions for the global asymptotical stability of a standard cycle,a touching cycle and a sliding cycle are obtained respectively.Furthermore,the sliding cycle is globally finite-time stable.Especially,several kinds of sliding bifurcations including boundary node bifurcation,boundary focus bifurcation and grazing bifurcation are studied.Moreover,two specific models are provided to verify the main results obtained from the general model.In the fifth chapter,we propose and analyze a Filippov pest-natural enemy model for integrated pest management.The feature of the model is that two threshold values are involved,one(xT)for pest and the other(yT)for natural enemy.By applying Filippov theory,we distinguish some cases to investigate the sliding mode dynamics and global dynamics according to the different values of xT and yT.It is shown that,for different parameter values,the proposed system can admit coexistence of multi-attractors including the real equilibrium,the pseudoequilibrium,and the point(xT,yT)These theoretical results are also illustrated with numerical simulations.Our findings indicate that proper control strategies can prevent the outbreak of pest disaster in ecological agriculture.