Periodic Solutions And Stability Of Nonlinear Biological Dynamical System With Impulses

Many evolution processes in the real world, particularly some biological systems such as biological neural networks and bursting rhythm models in pathology, as well as optimal control models in economics, frequency-modulated signal processing systems, and flying object motions, are continuously distributed by unpredictable shorterm forces which can result in some abrupt changes of state. This is the familiar impulsive phenomena. Of practical interest in evolution processes is the question of whether or not an evolution process can withstand those unpredictable shorterm perturbations. To describe mathematically an evolution of this process, it is sometimes convenient to neglect the duration of the perturbation and to consider these perturbation to be "instantaneous". For such an idealization, it becomes necessary to study ecological systems with discontinuous trajectories. In the theory of impulsive differential equations, we call the discontinuous trajectories as impulses. In this dissertation, we mainly deal with the periodic solutions and stability of several kinds of nonlinear biology dynamical system with impulses. This dissertation is divided into three main chapters.In Chapter 2, we focus on establishing sufficient conditions for the existence and global attractivity of positive periodic solutions of two kinds of the delay differential system with impulses, The method involves the application of the Gaines and Mawhin's coincidence degree theory, a fixed point theorem in cone, the constructing suitable Lya-punov Functional and estimations of uniform upper bounds on solutions. When these results are applied to some special delay single population models, some new results are obtained, and some known results are generalized. In particular, our results indicate that under the appropriate linear periodic impulsive perturbations, the above impulsive delay differential equation preserves the original periodicity and global attractivity of the nonimpulsive delay differential equation.In Chapter 3, firstly, the autonomous Leslie-Gower predator-prey system with periodic constant impulsive perturbations is investigated. We show that there exists a stable predator-eradication periodic solution when the impulsive is less than some critical values. By using bifurcation theory, we show the existence a positive periodic solution. These results are quite different from those of the corresponding system without impulses. The results show that the system we consider has more complex dynamical behaviors. Secondly, sufficient conditions are obtained for the existence of periodic positive solutionsof a class of general neutral impulsive delay Lotka-Volterra systems. Our results generalize some known results. Our results of second part show that under the appropriate linear periodic impulsive perturbations, the impulsive multi-species system preserves the original periodicity of nonimpulsive multi-species system.In Chapter 4, we concern with the global stability characteristics of a system of equations modelling the dynamics of bidirectional associative memory neural networks with impulses. Sufficient conditions which guarantee the existence of a unique equilibrium and its exponential stability of the networks are obtained. Consequently, we study the existence and global exponential stability of periodic solution for high-order bidirectional associative memory (BAM) neural networks with and without impulses. Easily verifiable sufficient conditions are established. The method is based on coincidence degree theory as well as a priori estimates and Lyapunov functional. It is also shown that the above systems with impulses preserve the dynamics of the networks without impulses when we make some modifications and impose some additional conditions on the systems. Numerical simulation results are given to support the theoretical predictions of the existence and global exponential stability of periodic solution for high-order bidirectional associative memory (BAM) neural networks with and without impulses.