Categories - Predator Predator-prey System With Impulsive Control Model Research

Biomathematics, which has been developing for near one hundred years, is a borderline science of biology and mathematics, and whose subfields include math-ematical ecology, mathematical medicine, biostatistics and quantitative genetics. Mathematical ecology, a fast developing subfield, mainly uses mathematical meth-ods to study the behavior of a population ecology system. Population ecology studies the interactions between population and environment, and the dynamical behavior of population systems, models various relations such as cooperative relation, com-petitive relation or predator and prey relation between populations. The researchers establish and study the models of population systems, and predict and adjust the development of population systems according to practical needs.In the early research of population dynamics, the population development of a system was modelled by an ordinary differential equation system generally. How-ever, in addition to the inherent law, many population systems subject to human interference or the effect of other external factors so that the variables or the in-creasing law can be changed instantly. If this is the case, it is very difficult using an ordinary differential equation to describe this phenomenon, but an impulsive differential equation works. Driven by some practical applications, impulsive differ-ential equation has been rapidly developing, and now has become a mathematical field which has integral theory and wide application prospects. It has been widely used to describe population dynamical systems, to optimize biological resources, to control pest ecosystems and so on.Using the theorem of impulsive differential equations and population dynam-ical system, several predator-prey models with impulsive control are studied. We focus on discussing the existence of the periodic solution, the globally asymptotically stability of pest-free periodic solution and the permanence of the system. Further-more, we investigate the complicated dynamics behaviors of the system by numerical simulation.The following problems are discussed and results are obtained in this thesis: (1) A pest control model with natural enemy releases is investigated, the mathe-matical problem involves a predator-prey system with periodic coefficients and pulse conditions. First, we show that the system has a unique pest-free periodic solution, which is globally asymptotically stable under certain condition by the comparison theorem of impulsive differential equations. Furthermore, we prove that the system is permanent if the pest-free periodic solution is unstable. The results generalize several previous theorems on persistence for the system with constant coefficients.(2) A stage-structured predator-prey model with time delay and pulse toxicant input in a polluted environment is discussed. The system is in a periodically chang-ing polluted environment with impulsive toxicant input. The sufficient condition for global attractivity of the prey-free periodic solution is obtained by the stroboscopic mapping of discrete dynamic system. Moreover, using the theory of delay differ-ential equation, we also show that the system is permanent under the appropriate condition.(3) The permanence of a large class of impulsive predator-prey systems in a periodic environment is studied. The mathematical model is described by a peri-odic impulsive differential equation system. By comparison theorem for impulsive differential equations and some analysis techniques, we prove a series of propositions and lemmas, and in final, we prove that the system is permanent under certain con-ditions. Since the system is of more general form, the obtained results can be used to provide decision-making for a large class of ecosystems for the ecological balance and the sustainable development of population system.