Dynamical Behavior Of Some Biological Systemswith Time Delays And Impulsive Effects

In the natural world, many change rates of system state variables are not only related to the current system variables state, but also depend on the past certain point or a period of previous time. Thus, instead of ordinary differential equations, it is much reasonable to model this kind of systems by introducing delay differential equations or functional differential equations. On the other hand, there are many natural or man-made factors in the real world to rapidly affect the inherent laws of the system. Since these effects are usually appears very shortly, then the system modeling is considered in a fixed time. As a result, the system is not continuous any more. It is therefore much reasonable that the semi-continuous impulsive differential equations are applied to replace the continuous dynamical system for modeling. In addition, change rates of system state variables in the real world are also affected by the environmental noises, and then the determined differential equations will lose the efficiency to describe the details of the corresponding system. Instead, it is more reasonable to use the stochastic differential equations to deal with these problems.Therefore, in terms of the background mentioned above, four kinds of hybrid bio-logical systems with time delay, impulsive effects and stochastic perturbations are studied and discussed in this thesis, respectively. And main contents are listed as follows.1. A class of chemostat model with periodic impulsive input and distribute time de-lay is introduced and studied in chapter 2, in which nutrient recycling is also considered. Using comparison theorem, Floquet theory and small amplitude skills in the impulsive differential equation, sufficient conditions are derived to guarantee the global asymptotic stability of the microorganism-eradication periodic solution. On the other hand, by con-structing suitable Liapunov functions, the boundary of the system is also obtained. Fur-thermore, by making full use of the skills of mathematical analysis, sufficient conditions for the continuous cultivation are also established. Finally, some numerical examples are demonstrated to verify our theoretical results and further discuss the influence of some parameters variations on the extinction or the permanence of the microorganism.2. A food-chain system with digest delay and periodic harvesting for the prey is studied in chapter 3. By using comparison theorem, skills of small amplitude and differ- ential inequalities, sufficient conditions for the existence and the global attractivity for the predators-eradication periodic solution, sufficient conditions for the permanence of the system are established. In addition, numerical examples and simulations are presented to support the correctness and feasibility of the theoretical results. Moreover, we suggest some control strategies for decreasing the pest population to a lower economic threshold level.3. A nonautonomous almost periodic prey-predator system with impulsive effects and multiple delays is considered. Applying the mean-value theorem of multiple vari-ables, integral inequalities, differential inequalities and some other mathematical analysis skills, sufficient conditions which guarantee the permanence of the system are developed. Meanwhile, by constructing a series of Liapunov functionals, it proved that there exists a unique uniformly asymptotically stable almost periodic solution of the system. Numeri-cal examples and simulations are also presented to prove the correctness and effectiveness of the achieved theoretical results. Also the dynamical behavior with different impulsive perturbations or different delays are investigated finally.4. A nonautonomous stochastic food-chain system with functional response and impulsive perturbations is studied. By using Ito’s integral formula, exponential martin-gale inequality, differential inequality and the related mathematical skills, some sufficient conditions for the extinction, nonpersistence in the mean, persistence in the mean, and stochastic permanence of the system are established. Furthermore, some asymptotic prop-erties of the solutions are also investigated. Finally, a series of numerical examples are presented to support the theoretical results, and the effects of different intensities of white noises perturbations and impulsive effects are analyzed and discussed in the simulation evaluations.