The Studies On Biochemistry Systems And Crop Protection With Impulsive Effects

Impulsive differential equations are suitable for the mathematical simulation of evolutionary process in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change in their values. Continuous dynamical system, discrete dynamical system and impulsive dynamical system are three main kinds of dynamical systems. The solutions of impulsive systems are continuous between two impulses and discontinuous at the impulse, which makes the theory of impulsive systems richer and more complicated than that of the corresponding continuous system. Processes of this type are often investigated in various fields of science and technology. As an example of this we shall point out the investigations of biological technology, medicine dynamics, physics, economy, population dynamics and epidemiology. Consequently, studying population dynamical systems involving impulsive effect is of great practical importance. Recently, energy pulses in floodplain theory and nutrient pulses in tropical forests have already made many authors pay attention to significance of pulse nutrient release. Their results make it possible to investigate impulsive differential equations in the biochemistry systems. In the dissertation, we mainly research chemical and microbial kinetics systems and combine epidemic models and population dynamical systems.In chapter 2, the well-known classical system of Brusselator is remodeled owing to nutrient pulse and we prove this system is dissipative in the plane. When the quantity of nutrient pulse varies in time we picture the bifurcation, limit cycles, as well as periodic solutions and attractors. Based on the enzyme reaction in chemical kinetics, we research effects of constant input on the quasi-steady state assumptions for enzyme-catalyzed reactions and get the result that is the existence of periodic solutions and there is no limit cycle. Finally, we suppose the relation between microorganisms and nutrient is the mode of mass-action and establish the model which contains two competitive microorganisms and one of them is able to release toxins which can inhibit the growth of the other microorganism. Difference differential equations is used to solve the existence and stability of positive periodic solution and simulate the outcome of the competitive microorganisms in the model.In chapter 3, according to the basic epidemic model of SI, we extend epidemic models to plant pathology and propose to pull up the diseased plants in the pulse because plant is not the same as animal to be immune. Two impulsive models are used to picture differential process of unpin the diseased plants: one is impulsive differential equations at the fixed moments of an impulse effect in which we prove the local stability of periodic solution and get the relation between the max period and pull-up rate; the other is impulsive differential equations at the unfixed moments, in which we use the Brouwer theorem to prove the existence of periodic solutions.In chapter 4, we combine population dynamics and epidemic models to advance the model in which natural enemies together with diseased pests control the growth of pest in agriculture. Constant input and pulse release of diseased pests are considered and we get the global stability that is useful for biological control.