Study Of The Microorganism Fermentation And Stage-structured Population Models

With the development of science and technology, biomathematics has been used in many domains such as biological technology, medicine dynamics, physics, economy, popu-lation dynamics and epidemiology. Mathematical models of differential equations play an important role in describing biological dynamics. Mathematically, these models explain all kinds of biological behaviors, which allows people to understand biological complexity scientifically so that some interactions of population can be intend to control. Impulsive differential equations are suitable for the mathematical simulation of the evolutionary process in which the parameters undergo relatively long period of smooth variation fol-lowed by a short-term rapid change in their values. The solutions of the impulsive systems are continuous between two impulses and discontinuous at the impulse, which makes the theory of the impulsive systems more complicated than that of the corresponding contin-uous systems. Based on the basic theory of the biochemical reaction kinetics,population dynamics,impulsive differential equations and delay differential equations, microorgan-ism fermentation and stage-structured population models have been established to study the effects of impulses on the these models including the existence and stability of equi-libria, the existence of periodic solution and its global attractivity, the permanence and extinction of system.The dissertation has five chapters and the main results of this dissertation may be summarized as follows:Chapter 1 and Chapter 2 give the biological backgrounds of the microorganism fer-mentation and pest control, basic theories and preliminaries of impulsive differential equa-tions, including the existence of the solution, the continuity of the solution and Floquet's theory and so on.Lactic acid is one of the organic acids, which has many applications in various types of industry and agriculture. Chapter 3 studies the dynamical behaviors of the lactic acid fermentation. Firstly, we present the lactic acid fermentation model with the continuous input. The globally asymptotical stability of the positive equilibrium has been obtained by using the qualitative analysis method. Secondly, we give the lactic acid fermentation model with impulsive input. The sufficient condition for the existence and stability of the biomass-free periodic solution is gotten by using the Floquet's theory. There exists a unique positive periodic solution via bifurcation theory, which implies the substrate, biomass and lactic acid oscillate with a positive amplitude. The numerical simulations verify the theoretical results and give some biological explanation.The chemostat can be used for representing all kinds of microorganism systems, mostly because these are the simplest and most easily constructed types of continuous cultures. In Chapter 4, we investigate the dynamics of the chemostat model during the microorganism culture, which can give a theoretical guidance. Firstly, we present dynamics of two-nutrient and one-microorganism chemostat model with time delay and pulsed input. By means of the principle of the impulsive differential equations and delay differential equations, the sufficient condition for the existence and attractivity of the microorganism-free periodic solution is obtained. In addition, the sufficient condition for the permanence of the system is also obtained. Next, we discuss the extinction and permanence of chemostat model with pulsed input in a polluted environment. Finally, The dynamical behavior of the annular chemostat model including competition relation and predator-prey relation is investigated. By using the theories about the impulsive differential equations and difference equations, the sufficient condition for the existence and stability of the boundary periodic solution is obtained. When the boundary periodic solution losses its stability, numerical simulation shows there is a characteristic sequence of bifurcation, leading to a chaotic dynamics, which implies that this system has more complex dynamics including period-doubling bifurcation, chaos and strange attractors.In Chapter 5, It is assumed that the predator population has two stages including immaturity and maturity and prey has a generic growth rare. Using the discrete dynam-ical system determined by the stroboscopic map, we obtain predator-extinction periodic solution. By use of the basic theorem related to impulsive differential equations and delay differential equations, sufficient condition for the global attractivity of predator-extinction periodic solution and permanence of the system is obtained. Furthermore, the results are confirmed by numeric simulations and complex dynamics is also investigated in view of the some parameter variations.