Asymptotic Properties Of Discontinuous Biological Mathematical Models

From the perspective of natural law and human behavior,there are still a large number of discontinuous dynamical systems,but the properties of differential equation solutions in the classical sense are not applicable to such systems,so this paper will discuss some of these discontinuous biological mathematical models.Due to the importance of the study of discontinuous biometrics models,this paper refers to the definition of solutions of ordinary differential equations with discontinuous right-hand sides in the sense of Filippov,and discusses the existence,uniqueness,stability and convergence of solutions of differential equations with discontinuous right-hand sides for several classical derivative models.For a class of Nicholson's blowflies model with patch structure and discontinuous harvesting,the existence and uniqueness of the pseudo-almost periodic solutions are proved by fixed point theorem and exponential dichotomy theory,and then we construct a proper Lyapunov function to obtain the exponential convergence of the pseudo-almost periodic solution.Similarly,for a class of Lasota-Wazewska model with infinite delay and discontinuous harvesting,the expression of the pseudo-almost periodic solution is firstly given by exponential dichotomy,then the existence and uniqueness of the solution is proved by fixed point theorem,and the exponential convergence of pseudo almost periodic solution of the equation is studied by constructing appropriate Lyapunov function.Furthermore,for a class of SEIRS epidemic model with discontinuous treatment,first analyze the boundedness and uniqueness of the system solution,and determine the expression of the basic reproductive number R₀.Then the local stability of disease-free equilibrium and endemic equilibrium is discussed by Jacobi matrix and Hurwitz criterion,and the persistence and extinction are discussed by using Lyapunov function.Finally,using Matlab to test the theoretical results of each model.The innovation of this paper is to propose a more biological derivative model with discontinuous right-hand sides,and to extend the asymptotic properties of solutions in continuous cases to discontinuous models.Particularly,in the proof of the existence and uniqueness of Nicholson's blowflies model,the condition that the supremum norm of the corresponding matrix is less than 1 is optimized to the spectral radius is less than 1,which weakens the restriction of the system on the parameters.Moreover,the exponential convergence also can be obtained without other additional conditions,which extends the conclusions of the existing classical literature.