Dynamics Of Biological Systems Under Different Discontinuous Control Strategies

In recent years,discontinuous systems have been widely used in the context of engineering,biology,and physics.Aiming at the problems the study of theo-ries for discontinuous differential system and its application in biological systems.This thesis studies discontinuous biological systems in three respects.First,by employing set-valued mapping theory,differential inclusion,non-smooth analysis and inequality techniques,especially discontinuous functional differential equation theory,several biological systems are studied under several types of discontinuous harvesting management policies.The development and improvement of related discontinuous functional differential equation theory mainly include the existence of periodic solutions,different kinds of stability and convergence behaviors for solutions(e.g.,asymptotic,exponential,finite time).Secondly,using non-smooth analysis,Filippov convex method,Bendixson-Dulac’s method,inequality tech-niques,LaSalle’s invariance principle,discontinuous loop domain theorem,the Filippov biological system under economic threshold is studied.The dynamics include the sliding mode domain of the system,the sliding mode dynamic equa-tion of system,the regularity of the solution,boundedness of the solution,and various equilibria of system(e.g.,real equilibrium,virtual equilibrium,pseud-equilibrium,tangent point,boundary point)and different kinds of sliding bifur-cations(e,g.,boundary-node bifurcation,boundary-focus bifurcation,real-virtual equilibrium bifurcation).Finally,by using the Brouwer’s fixed point principle,inequality techniques,Poincare’s mapping,vector field analysis,bifurcation the-ory and piecewise integration techniques,several biological systems with impulse control strategies are studied.The research issues mainly include the positivi-ty of the solution,the boundedness of the solution,the properties of the order-1 and order-2 periodic solutions(existence,uniqueness,orbits asymptotic stability,finite time convergence)and global stability dynamics of system.These results not only contribute to the further development of mathematics,but also provide a reliable theoretical basis and effective key technologies and methods for scientific and engineering applications.This thesis is consists of five chapters.In the first chapter,we briefly introduce the historical background and devel-opment of the theory of biological systems under discontinuous control strategies.Meanwhile,the history and development of the discontinuous biological dynamical systems under three type control strategy also are introduced.Finally,the main content and structure arrangements of this thesis are summarized.In the second chapter,we give some basic theoretical knowledge.The rest of this thesis presents our main research results and is organized as follows.In the third chapter,we investigate the dynamical behaviors of the biological systems under discontinuous harvesting management policies.Firstly,we give the definition of solutions in sense of Filippov for biological systems with discontinu-ous harvesting management policies via appropriate Filippov differential inclusions.Based on the fixed point theory in set-value analysis,topological degree theory in set-value analysis,non-smooth analysis theory with generalized Lyappunov ap-proach,we mainly establish a series of useful criteria on existence,uniqueness and global asymptotic stability of the positive periodic solution for the delayed biolog-ical systems with discontinuous right-hand sides.Moreover,we also discuss the global convergence in measure of harvesting solution.Finally,numerical simula-tions are used to illustrate the theoretical results.In the fourth chapter,a biological system under the discontinuous thresh-old control strategy is considered.By using non-smooth analysis,Filippov con-vex method,Bendixson-Dulac’s method,discontinuous loop domain theorem,we mainly study some dynamic behaviors include the sliding mode domain of the discontinuous dynamic system,the sliding mode dynamic equation of system,the regularity of the solution,the boundedness of the solution,different kinds of e-quilibria(e.g.,real equilibrium,virtual equilibrium,pseudo equilibrium,tangent point,boundary point).By using numerical methods,we investigate local sliding bifurcation of discontinuous system(e.g.,boundary node bifurcation,boundary fo-cus bifurcation,boundary saddle bifurcation,real-virtual equilibrium bifurcation).Finally,numerical simulations are used to illustrate the theoretical results.In the fifth chapter,we consider the dynamical behaviors of the predator-prey system under state impulsive control.Our methods to be used involve the Brouwer’s fixed point principle,Poincare’s mapping,bifurcation theory.Firstly,the positivity and boundedness of solutions with impulsive equations are discussed.Secondly,when only a boundary equilibrium exists in the system,it is shown that the boundary equilibrium is globally asymptotically stable.Thirdly,when a local equilibrium point is invisible,it is found that there exists an order-k periodic solution by using Brouwer’s fixed-point theorem.The solutions of the system will converge to the boundary equilibrium when the control threshold is below the critical value,and there exist stable order-1 and order-2 periodic solutions when the control threshold is beyond the critical value.Furthermore,when the local equilibrium is visible,the solutions of the system converges to this local point.The sufficient conditions of existence and the stability of the order-1 or order-2 periodic solutions are obtained by using the comparison method and the fixed-point principle.Finally,numerical simulations are used to illustrate the theoretical results.