The Study About Two Kinds Of Mathematical Models For The Dynamic Behavior Of The Biology

Mathematical biology is an important branch of biomathematics. It mainly studies on the dynamic behaviors of the biological dynamic system, which includes population dynamic models, neural network models, microbial continuous culture and chemostat models, environmental pollution models and so on. In recent years, more and more experts and scholars become greatly interested in the study of mathematical models of biology. And they have got a lot of research achievements, especially in the aspect of the neural network models. Furthermore, these research achievements have been widely applied to various aspects such as various complicated mechanics, physics, chemistry, electromagnetism and biology, image processing, automatic control field and so on. The thesis mainly uses the theory of coincidence degree, the methods of differential inequality, the methods of constructing Lyapunov-Kratowski functional, the linear matrix inequality techniques to study the problem about the existence of periodic solutions of neutral type cellular neural networks with S type distributed delays and the problem about the global exponential stability of high-order neural networks with s-type distributed delays and markov jump parameters. The criterion of the existence of periodic solutions and the criterion of the global exponential stability of the equilibrium point have been gotten. The accuracy of the proposed results has been demonstrated by a numerical example.The thesis is divided into five chapters, its main contents and arrangements are as follows:The research background and research situation at home and abroad is mainly introduced in the first chapter. Then the present research situation, purpose and meaning about the problem are given. Finally, the main contents and arrangements are summarized.The basic theories and knowledge which will be used in the process of the study in the following two chapters are mainly introduced in the second chapter. It mainly includes the definition of k-set compression mapping, the definition of zero index of the operator of Fredholm, the definition of the global exponential stability of equilibrium point in the mean of square sense, some important lemmas which are used in the study of the stability of the neural network.The problem about the existence of periodic solutions of neutral type cellular neural networks with S type distributed delays by using the theory of coincidence degree and the methods of differential inequality is mainly studied in the third chapter. The criterion of the existence of periodic solutions has been gotten and the accuracy of the proposed results by a numerical example has been demonstrated.The problem about the global exponential stability of high-order neural networks with s-type distributed delays and markov jump parameters by using the method of constructing Lyapunov-Kratowski functional and the linear matrix inequality technique is mainly studied in the fourth chapter. A simple criterion of the global exponential stability of the equilibrium point has been gotten.A conclusion to the main contents and some innovative results is summarized in the fifth chapter. The problems which remain to be researched have been also pointed out.