Applications Of Boundary Value Problem And Impulsive Effects To Dynamical Systems In Biology

Based on the qualitative results of continuous dynamical models in biology,the long-term and short-term managements of species are investigated by means of the theory of impulsive differential equations and boundary value problem.The short-term managements can be described by the systems with boundary value problem,and the long-term managements can be investigated by discussing the effects of the impulse at fixed moments and the state impulse on the dynamical systems in biology.The thesis has 6 chapters.Chapter 1 gives the biological backgrounds of the long-term and short-term managements of species,briefly states the present studies of impulsive differential equation and its applications to the dynamical systems in biology.Chapter 2 introduces the basic theories and preliminaries of impulsive differential equation and boundary value problem.In Chapter 3,the applications of boundary value problem to the system with Allee effect and impulsive effect are given.According to the initial density of a single-species with Allee effect,Chapter 3 presents two kinds of time-limited management models:the model with impulsive release and the model with impulsive harvest.By means of the comparison principle and the method of upper and lower solution,the corresponding sufficient conditions under which the models have a solution or no solution are obtained. If other parameters are given,the controllable parameters such as the population of release and the times of impulsive harvest are estimated respectively.In Chapter 4,the applications of boundary value problem to the system of two species with impulsive control in finite time are investigated.Section 4.1 presents a kind of timelimited pest control of a Lotka-Volterra predator-prey model with impulsive harvest.By the comparison principle,the conditions under which the model has a solution are found by a series of the upper solutions and the conditions under which the model has no solution are also given by a series of the lower solutions.Furthermore,the times of harvesting pest in the given time is estimated.Section 4.2 presents a kind of time-limited control model of a Lotka-Volterra competition system with impulsive harvest.The existence of solution of the model is discussed.Similarly,by the comparison principle,the conditions under which the model has a solution or no solution are found.Finally,the practical meanings of those conditions are explained.As an example of theoretical results,if other parameters are given,the times of impulsive control is estimated and the theoretical results are verified by numerical simulations.In Chapter 5,the effects of three kinds of impulsive control strategies on predatorprey system are investigated by the Floquet's theory and the comparison theorem of impulsive differential equation.By means of chain transform,Section 5.1 investigates the impulsive effects on a kind of predator-prey system with distributed time delay.The thresholds between permanence and extinction are obtained as functions of model parameters. It is proved that the impulsive period and the proportion of the impulsive harvest will ultimately affect the fate of each species.Section 5.2 discusses the effects of impulsive release on a kind of one-prey two-predator system with Ivlev's and Beddington-DeAngelis' functional response.It is proved that the prey-free periodic solution is locally asymptotically stable when the period of impulsive release is less than a critical value. Furthermore,the conditions for the permanence of the system are obtained.Numerical simulations show that the system has complex properties,including periodic solution, period-doubling bifurcation,chaos,etc.Finally,a brief discussions on the relationships between the continuous system and the impulsive system are given.Section 5.3 discusses a kind of impulsive control to an omnivorous species by means of the dynamics of infectious disease and obtains the sufficient conditions under which the susceptible predator will be extinct or which the species coexist.The numerical simulations verify the theoretical results and show the system has complex dynamical behaviors.In Chapter 6,the periodicity of continuous culture of microorganism in the turbidostat is discussed by investigating the existence of periodic solution of a kind of trubidostat model with impulsive state feedback control and Monod growth rate.Based on the qualitative analysis,the conditions for the existence of periodic solution of order one are obtained by the existence theorem of periodic solution of a general planar impulsive autonomous system.It is shown that the system either tends to a stable state or has a periodic solution,which depends on the feedback state,the control parameter of the dilution rate and the initial concentrations of microorganisms and substrate.The period and the initial point of the periodic solution are given.Finally,the theoretical results are verified by numerical simulations.