Applications Of Impulsive Differential Equations In Microorganism Cultures And Population Controls

Mathematical models of differential equations play an important role in describing popula-tion dynamic behavior. Mathematically, these models explain all kinds of population dynamicbehavior, which allows people to understand population dynamics scientifically so that someinteractions of population can be intend to control. Especially, impulsive differential equationsdescribe population dynamic models, which is more reasonable and precise on reflecting all kindsof change orderliness, since many life phenomena and human exploitation are almost impulsivein the natural world. In this dissertation, population dynamic models are established to con-sider several problems in microorganism cultures and population controls by means of the theoryand method of impulsive differential equations. Numerical simulations are used to investigatedynamic behavior including the stability of equilibrium, the existence of periodic solution, thepermanence and extinction of system and the complexity of system. The main results of thisdissertation mav be summarized as follows:In Chapter 2, microorganism cultures are investigated. Asymptotic behavior in the ratio-dependent chemostat model with variable yield is studied in Section 2.1. In the model, weassume that the yield is a linear function of the nutrient concentration and the microbial growthrate is a ratio-dependent type function. Thus, we have developed the classical Monod model. Itis shown that system is permanent if and only if it has a positive equilibrium by the qualitativetheory of ordinary differential equation. The sufficient conditions of existence of limit cyclesand globally asymptotic stability of the positive equilibrium for the model are given. In Section2.2, dynamic behaviors of Monod type chemostat model with impulsive input on the nutrientconcentration. Using Floquet theory and small amplitude perturbation method, we prove thatthe microorganism-eradication periodic solution is asymptotically stable if the impulsive periodsatisfies some conditions. Moreover, the permanence of the system is discussed in analyticalmethod. Finally, we verify the main results by numerical simulation. Using the similar methodof Section 2.2, complex dynamics of a chemostat with variable yield and periodically impulsiveperturbation on the substrate is studied in Section 2.3.In Chapter 3, the existence of periodic solutions of nonautonomous periodic systems isconsidered. In Section 3.1, the existence of periodic solutions of a predator-prey system withthe Beddington-DeAngelis functional response and impulsive perturbations is studied. By thecontinuation theorem of the coincidence degree theory, the sufficient conditions of the existenceof positive periodic solution are obtained. Finally, an example is given to illustrate the influence of the impulse on population dynamics by numerical simulation. In Section 3.2, using the con-tinuation theory for k-set contractions, the sufficient conditions of existence of positive periodicsolution of the impulsive delay Logistic model are given. In Section 3.3, the problem of existenceof positive periodic solution of the predator-prey system with functional response is analyzed bythe continuation theorem of the coincidence degree theory, the sufficient conditions of existenceof positive periodic solution are obained and the main result is simulated by computer.In Chapter 4, the effect of impulsive perturbation on predator-prey models is investigated.On the theory of controlling pests using epidemic, the problem on optimal controlling pestsby infected pests is studied in Section 4.1. We assume that the releases of infected pests arecontinuous and impulsive. Thus, the corresponding models are the ordinary differential equationsand the impulsive differential equations, which are studied by the qualitative theory of ordinarydifferential equations and the theory of impulsive differential equations. Mathematically, thetheoretical evidence of the controlling pests using epidemic in the integrated pest managementis given. In Section 4.2, the dynamic behaviors of a Holling-type predator-prey system withimpulsive immigration on the predator is considered. It is shown that the system is permanent.By means of numerical simulation, we illustrate that with the increasing of the immigrationnumber, the system exhibits complex dynamics.