The Epidemic Models And Microbial Culture Models In Pest Management

The need for describing more actual natural system impels the evolution of mathe-matical biological models. In recent years, the researchs in mathematical biology which models by normal differential equations are mainly concerntrated on two branches:con-tinuous dynamical systems and impulsive semi-dynamical systems. Especially, the im-pulsive dynamical systems are suitable for the mathematical modelling of evolutionary processes which experience a change of state abruptly owing to instantaneous perturba-tions. The presence of impulses gives the system a mixed nature, both continuous and discrete. Therefore the theory of impulsive dynamical systems is much richer than the corresponding theory of dynamical systems without impulsive effects. In recent years, im-pulsive dynamical systems, have been widely used in biotechnology, physics, economics, population dynamics, epidemiology and so on. In this thesis, the infectious diseases mod-els and microbial culture models in pest management are established to consider several problems by means of the theory and method of impulsive differential equations. Dy-namic behaviors, including the existence and stability of equilibriums, the existence of periodic solution and its global attractivity, the permanence and extinction of system, are investigated. The thesis has five chapters:In Chapter 1,the backgrounds of the system investigated in the thesis are given. The relative researches are stated briefly.In Chapter 2, some preliminaries and the relative results of impulsive differential equations are introduced.In Chapter 3, viral infection dynamical models in pest management are formulated and investigated. In section 3.1, two models of continuous and impulsive release viruses are studied. In the case in which a continuous control is used, it is shown that the system admits a globally asymptotically stable positive equilibrium by means of analytical methods of ordinary differential equations. In the case in which an impulsive control is used, it is observed that the pest-eradication periodic solution is globally asymptotically stable by using Floquet theorem and comparison results of impulsive differential equations. Finally, the efficiency of continuous and impulsive control policies is compared by means of numerical simulation. Section 3.2 discusses the effects of impulsive state feedback control on the viral infection SV model. Based on the qualitative analysis, the conditions for the existence of periodic solution of order one are obtained by the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the viruses and the initial concentrations of viruses and pests. Finally, the theoretical results are verified by numerical simulations. In section 3.3, the SIV epidemic model is constructed and studied. It is assumed that the pests has two classes, susceptible pests and infected pests. In order to control the number of pest, virus particles are impulsively released at fixed time. So as to the susceptible pests can be infected, the number of the pests can be controlled effectively. The sufficient conditions of the susceptible pest-eradication periodic solution and the permanence of the system are obtained. Our results indicate that impulsive period and the release amount of the virus particles have great effects on the dynamics of our system.In Chapter 4, two stage-structured predator-prey models with infectious disease in the prey are studied. In section 4.1, a stage-structure delay prey-predator model with infectious disease in the prey is investigated. Using the discrete dynamical system de-termined by the stroboscopic map, we obtain the susceptible pest-eradication periodic solution. By use of comparison theorem and differential inequalities for delay impul-sive differential equations, we show that the period solution is globally attractive. By impulsive releasing infected pests and natural enemies, we obtain the minimal releasing infected pests and natural enemies and maximum impulsive period, the susceptible pests are controlled under the economic threshold level, that is the harm of the susceptible pests are no more than the super compensation point of the crops. The results pro-vide a reliable theoretical tactics for pest management and also indicate the important influence of time delay on population dynamics. In section 4.2, a prey-dependent con-sumption predator-prey(natural enemy-pest) model with age structure for the predators and infectious disease in the prey is studied. By using Floquet theorem, small-amplitude perturbation skills and comparison theorem, we obtain both the sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. Our results indicate that impulsive period and the release amount of infected pests and natural enemies have great effects on the dynamics of our system.In Chapter 5, microbial culture models are investigated. Section 5.1 consider a Monod-Haldane competitive chemostat model with delayed growth response and impul-sive input nutrient. The effect of impulsive input of the nutrient, time delay for growth response on dynamic behaviors of chemostat model is analyzed. Whether the microor-ganisms is extinct or not is determined completely by the input amount of the substrate, the length of impulsive period and the time delay of microbial growth and reproduction at fixed impulsive period nT. Section 5.2 consider a Monod competitive chemostat model with delayed growth response and impulsive input nutrient in a polluted environment. We also analyze the effect of impulsive input of the nutrient, time delay for growth response and impulsive input of the toxicant on dynamic behaviors of chemostat model. Whether the microorganism is extinct or not is determined completely by the input amount of the substrate and concentration of the toxicant at fixed impulsive period nT. The results show that the environment without pollution conducive to microbial culture and polluted environment may lead to the extinction of microorganism. This shows that the input con-centration of the toxicant greatly affects the dynamics behaviors of the model. Section 5.3 consider an application of microorganism cultured-the study of mathmatical model of ethanol fermentation with gas stripping. We consider continuous input and pulse input of nutrition respectively. For the case of continuous input substrate, we study the existence and local stability of two equilibria. According to Poincare-Bendixson Theorem, the suf-ficient condition of the globally asymptotical stability of positive equilibria is obtained. Which implies we can get stable ethanol product. For the case of impulsive input sub-strate, we obtain the sufficient condition of the local stability of cell-free periodic solution by using the Floquet's theory of impulsive equation and small amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges. Further, our main results are verified by means of numerical simulation and obtain the the impulsive input is more effective than continuous input.