Stable And Permanent Properties Of Solutions Of Impulsive And Nonautonomous Mathematical Biological Models

Mathematical models of differential equations play an important role in describing population dynamics. Mathematically speaking, these models explain all kinds of population dynamic behavior, which make persons to understand population dynamics scientifically so that some interactions of all individuals of the population can be intend to control. In the natural world, since biological and environmental systems are naturally subject to fluctuation in time and the subrogation of the four seasons, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. People also find that continuous biological dynamical systems can not describe seasonal reproduction, human exploit behavior such as planting and harvesting etc. More realistic and interesting models should take into account both time-dependent differential equations and impulsive differential equations. Our three classes of models in this dissertation which have different application background respectively are of these two kinds of differential equations. Our results of this dissertation can be organized as follows.In Chapter 2, toxicant-population models are investigated. In Section 2.1, some sufficient conditions on uniform persistence of the model have been obtained. It is also shown that the number of prey population and predator population change periodically under appropriate assumption. Numerical simulation is then presented. In Section 2.2, a single species model in a polluted environment with pulse toxicant input is proposed and analyzed. It assume that the toxin in the organism is impulsively excreted by some factors. It also assume that the toxicant emission and toxin excretion are used in the simultaneous periodicity. By using the comparison results of impulsive differential equations and some known results of a single species Logistic model, it is observed that the population is extinct on condition that the impulsive period is less than some threshold value. When the impulsive period is larger than this critical value, the population is shown to be permanent. It is also proved that the permanent conditions assure that there exists a unique positive periodic solutions which is globally asymptotically stable. In Section 2.3, a single species model in a polluted water body environment in which all individuals of the species are synchronously impul- sively affected is considered. By employing strict k-set contractive theorem, sufficient conditions of the existence of a positive periodic solution to the species are given. In Chapter 3, some epidemic models for pest management strategies are considered. In Section 3.1, we propose two impulsive differential systems concerning biological and, respectively, integrated pest management strategies. In each case, by using the Floquet theory and small amplitude perturbation method, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution. The system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its stability. Further, the existence of a non-trivial periodic solution is also studied by means of numerical simulations. In Section 3.2, we propose two mathematical models concerning continuous and, respectively, impulsive pest control strategies. In the case in which a continuous control is used, it is shown that the model admits a globally asymptotically stable positive equilibrium under appropriate conditions which involve parameter estimations. As a result, the global asymptotic stability of the unique positive equilibrium is used to establish a procedure to maintain the pests at an acceptably low level in the long term. In the case in which an impulsive control is used, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution. The system is shown to be permanent. Further, the existence of a nontrivial periodic solution is also studied by means of numerical simulation. Finally, the efficiency of continuous and impulsive control policies is compared. Section 3.3 deals with an impulsive delay epidemic disease model with stage-structure and a general form of the incidence rate concerning pest control strategy, in which the pest population is subdivided into three subgroups: pest eggs, susceptible pests, infectious pests that do not attack crops. We obtain the exact periodic susceptible pest-eradication solution of the system and observe that the susceptible pest-eradication periodic solution is globally attractive. The system is shown to be permanent. Our results indicate that besides the release amount of infective pests, the incidence rate, time delay and impulsive period can have great effects on the dynamics of our system.In Chapter 4, microorganism cultures are investigated. In Section 4.1, a system of periodic coefficients functional differential equations is used to model the single microorganism in the chemostat environment with a periodic nutrient and antibiotic input. Furthermore, the total toxic action on the microorganism expressed by an integral term is considered in our system. Based on the technique of analysis, we obtain sufficient conditions which guarantee the permanence of the system and extinction of the microorganism. In Section 4.2, we consider a variable yield model of a single species growth in a well-stirred tank containing fresh medium, assuming the moments of time in which the nutrient refilling process and the removal of microorganisms by the uptake of lethal external antibiotic are initiated as triggering factors. It is also assumed that the periodic nutrient refilling and the periodic antibiotic injection occur with the same periodicity, but not simultaneously. The model is then formulated in terms of autonomous differential equations subject to impulsive perturbations. It is observed that either the population of microorganisms essentially washes out, or more favorably, the system is permanent. To describe this dichotomy, some integral conditions having biological significance are introduced. Further, in a certain critical situation, a nontrivial periodic solution emerges via a bifurcation phenomenon. Finally, the dynamics of the model is also illustrated by means of some numerical experiments and simulations.