Effects Of Delays On Dynamical Systems In Biology

Mathematical models of delay differential equations play an important role in describing bio-dynamic behavior. Mathematically, these models explain many biodynamic behaviors of among populations and between population and environment, which helps people to understand biody-narnics scientifically so that some interactions of population and interactions of between population and environment can be intend to control. In this dissertation, by means of the theory and method of delay functional differential equations and impulsive differential equations, biodynamic models are established to consider several problems in the almost period of delay population dynamic systems and population controls on delay impulsive dynamic systems. The investigated dynamic behaviors include the existence and stability of almost periodic solution, the existence and attractivity of semi-trivial periodic solution, the permanence and extinction of systems, and their biological significance is discussed. The main results of this dissertation may be summarized as follows:In Chapter 2, nonautonomous delay almost periodic population dynamic systems are investigated. Section 2.1 studies a nonautonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays. By means of the theory and method of delay functional differential equations, the effect of time delays on permanence of systems is shown. By constructing suitable Lyapunov functional, it is shown that the system is globally asymptotically stable. Using almost periodic functional hull theory, the sufficient conditions for the existence and uniqueness of almost periodic solution are obtained. This solves the difficulty that multidimensional and multidelayed dispersal dynamic systems can not be investigated by the former methods. Moreover, some former results are improved. Dynamic behavior in a pure delay integrodifferential Logistic almost periodic system is studied in Section 2.2. By means of the qualitative theory and method of delay differential equations, the weak conditions for the bound-edness of the system are obtained. By use of differential mean value theorem and computational techniques on delay differential equations, we show that the system is globally asymptotically stable under condition for the boundedness. By lemma on almost periodic functional hull equation, we directly analyze the righthand functions of system to discuss the existence and uniqueness of strictly positive almost periodic solution, remove some restrained conditions of the former results, and answer an open question raised by G. Seifert.In Chapter 3, the delay impulsive population dynamic systems are discussed. In Section 3.1, we investigate a robust predator-prey system with a generic functional response function, periodic harvesting for the prey and stage structure for the predator with maturation delay. Using the discrete dynamical system determined by the stroboscopic map, we obtain 'predator-extinction' periodic solution. By use of comparison theorem and differential inequalities for delay impulsive differential equations, we show that the system is globally attractive and give the conditions with time delay for the permanence of system. We show that natural enemy becames extinct firstly when pest is harvested largely, which lead to pest increase rapidly and overran ultimately. The influence of time delay and impulse on population dynamics is simulated by computer. In Section 3.2, an S type Holling functional response predator-prey model is investigated, in which natural enemy is released impulsively and pest has stage structure and maturation delay. Using basic theorem of delay impulsive differential equations, we obtain the condition with time delay for the extinction of pest by impulsive releasing natural enemy and the minimal releasing natural enemy and maximum impulsive period when pest population is controled under the economic threshold level. We illustrate the academic results and the application on pest management by numerical simulation. In Section 3.3, we develop the classical Monod model, then consider a new Monod type Chemostat model with delayed growth response and pulsed input in a polluted environment. The effect of impulsive input of the nutrient concentration, time delay for growth response and impulsive input of the toxicant on dynamic behaviors of Chemostat model is analyzed. Whether the microorganism is extinct or not is determined completely by the input amount of the substrate and concentration of the toxicant at the fixed impulsive period nT. The results show that the environment with no pollution is in favor of living of the microorganism species and that the polluted environment can lead to the microorganism species be extinct. This shows that the input concentration of the toxicant greatly affects the dynamics behaviors of the model.In Chapter 4, the influence of pulse vaccination, time delays and vertical transmission on SEIR and SEIRS epidemic models is investigated. In Section 4.1 and Section 4.2, delay SEIR and SEIRS epidemic disease models with vertical transmission and pulse vaccination are discussed, respectively. We develop the classical SEIR and SEIRS epidemic models and win through the mixing impulsive conditions. In Section 4.1 and Section 4.2, using the discrete-dynamical system determined by the stroboscopic map, we obtain an 'infection-free' periodic solution, discuss global attractivity of the periodic solution and give the condition for the disease to be eradicated, respectively. When the disease generates endemic, we use qualitative analysis method to prove the permanence of systems and discuss pulse vaccination strategy. The numerical simulation shows the influence of pulse vaccination, two delays and vertical transmission on the dynamics of systems.