Applications Of Impulsive Control In Mathematical Biology

In resent years, the basic theory and method of impulsive differential equa-tions has been fully developed. Some natural phenomena cannot be considered as continuous or discrete since they jump at certain moments of time only; this is, for instance, the case for animals that reproduce seasonally or animals that move during a certain period of their life cycle. And the impulsive equation is suitable to describe these phenomena. The first impulsive approach was put forward by Beverton and Holt (1957) in their construction of a discrete-time model analogous to the continuous-time logistic model on the basis of a semi-discrete model. Since then, a large body of literature has proposed impulsive models in almost every field of the life sciences. In this paper, two kinds of applications of impulsive control modelling in life sciences have been briefly discussed. The first application is that by using the persistence and the branch of nontrivial periodic solution of the system, we propose a complete control strategy to control the population of the prey in a predator-prey mod-el; the second one is that we consider one continuous model and one impulsive model to show how the release of sterile insects may manage or eradiate the wild insects in a general predator-prey model.The paper have four chapters.In the first chapter, the basic theory of the impulsive differential equation has been introduced, including the defination, common classify, stability and so on. Specially, we introduced the comparison theorem and the floquet the-orem of the impulsive differential equation. After that, the different kinds of applications of impulsive modelling in life sciences have been briefly reviewed. So far, impulsive differential equations have been widely applied to various aspects of epidemiology, medicine and population models.In the second chapter, a Beddington-DeAngelis interference model with impulsive biological control is studied. In the model with Beddington-DeAngelis interference, interference penalizes predominantly the predation ability, and hence appears in the trophic response. S. Nundlolla, L. Mailleretb and F. Grognarda(2010) have established the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. And we investigate the permanence of the model and seek its nontrivial periodic solutions. The pest-free periodic solution is local asymp-totically stable if the impulsive control rate is larger than a critical value or the release period is smaller than another critical value. Conditions for per-manence of the model are established. The existence of nontrivial periodic solution is established when the pest-free periodic solution loses its stability. At last, we propose a complete control strategy to control the population of the prey by using the stability of the trivial solution and the nontrivial periodic solution.In the third chapter, we consider one model to show how the release of ster-ile insects may manage or eradiate the wild insects in a general predator-prey model. We study the feasibility of the SIT in a general continuous predator-prey model. This model is under the framework of Murray’s model. In his work, the population of sterile insects is kept as a constant. We extend Mur-ray’s model to a general predator-prey model and carry out a theoretical study, discuss the dynamical behaviour of the model, and compute the crit-ical conditions for eradication of wild insects. These features in the context of a SIT model with predation lead to rich, interesting, and complex dynam- ics that include but are not limit to multi-stability (hysteresis), saddle node bifurcation, Hopf bifurcation. It has two important threshold:the fixed SIT threshold and the predation threshold. Both of them decide the effort of the predation in SIT model.In the fourth chapter, we study the application of the impulsive control in the Sterile Insects release Technology. In order to account for the release of sterile insects, we introduce the impulsive model, a periodic or pulsed re-lease method, which is fairly well modelled by an impulsive system of ordinary differential equations. The impulsive model is widely used in the area of epi-demiology and population dynamics. Compared to the continuous system, the pulsed SIT is easy to operate. We calculate a critical condition for eradica-tion of wild insects and get the global stability of the trivial solution of the impulsive SIT model.