Some Applications Of Impulsive Differential Equations In Biological Control Problems

In the area of biological control, there are a lot of systems which vary continuously and gradually during certain time intervals. However, system states also receive certain relatively short perturbations due to people’s some interference behaviors. Such systems cannot be well described by continuous or discrete dynamical systems alone due to these sudden changes, but we can do it by means of impulsive dynamical systems. Based on the theories and methods of impulsive differential equations, this thesis will focus on three biological control problems: the releases of sterile mosquitoes for the control of mosquito-borne infectious disease, the input of resource in a mixed system of predator-prey system and resources consumption system and the injection of insulin in the insulin pump for the therapy of diabetes mellitus. We will reveal the impact of relevant critical parameters on control effect with a view to providing mathematical theoretical foundation for the control strategies of corresponding practical problems.In the second chapter, To study the impact of releasing sterile mosquitoes into an environment on mosquito-borne disease transmissions, we propose two new strategies of releases and formulate mathematical models with impulsive releases of sterile mosquitoes to characterize the strategies. We consider periodic impulsive releases in the first model and obtain the existence, uniqueness, and globally stability of a wild-mosquito-eradication periodic solution. We establish thresholds for the control of the wild mosquito population by selecting the release rate and the release period based on the thresholds. In the second model, the impulsive sterile mosquito releases are determined by the closely monitored wild mosquito density. When the wild mosquito destiny reaches a predifined threshold value, a special number of sterile mosquitoed will be released into the environment. For this model, we prove the existence of an order one periodic solution and find a relatively small attraction region of the system, which ensures that the impulsive releases in such an automated way can make the wild mosquito population under control. We provide numerical analysis which shows that asmaller release rate and more frequent releases are more efficient in controlling the wild mosquito population for the periodic releases, but an early release of sterile mosquitoes is more effective for the state feedback impulsive releases.In chapter 3, For the interaction of predator and prey which is widely found in labratory,we also consider the resource consumption relation between the prey population and the resource, a novel mathematical model of stage-structure with negative binomial predation, two maturation delays and impulsive resource input is proposed in this paper. According to the degree of the density dependence of the prey population, we discussed the permanence of the system in two cases and gave the sufficient conditions that can guarantee all the species to coexist. Then by using two different fixed point theories, the existence of predator-extinct periodic solution and positive periodic solution is also obtained. Some simulation results are presented to illustrate the results. The simulation results show that, the influence between the resource consumption system and the predator-prey mostly reflects on the change of the density of the prey, and the increase of the number of the prey can promote the development of the predator and at the same time inhibit the resource. The relationship between the predator and the resource is so week that the change of the input method of the resource has very little impact on the predator.In chapter 4, in order to study the injection strategy of insulin in the open-loop control for glucose concentration, an impulsive differential equation model with delay is proposed.Several time delays exist in the system, these time delays are:(1) insulin transport time delay:the delays include the time needed for insulin from injection depot to transport to the interstitial compartment;(2) insulin secretion time delay: the time for the slow inhibition of the hepatic glucose production(HGP);(3) time delay of HGP: the time for insulin secretion caused by the elevated glucose concentration level from remaining functional pancreatic β-cells. None of them are negligible in the study of the therapy of diabetes. The model proposed in this paper incorporates these time delays. Our analytical studies show that all solutions are permanent, a periodic solution exists under certain conditions, and for the case of type 1 diabetes mellitus the periodic solution is unique and globally asymptotically stable. Numerically it has beenshown that moderate insulin transport delay in the system is beneficial in lowering blood sugar level rather than harmful but short HGP delay is helpful for the glucose control. Our studies also elucidate the noticeable inhibitory effect on HGP by the remaining functional β-cells.Similarly to our previous work, we demonstrated that a smaller dose with higher delivery frequency has a better effect on continuous subcutaneous insulin injection administration.