Existence And Global Attractivity Of Positive Periodic Solutions For Ecological Mathematics Models With Impulses

This Ph.D.thesis is divided into five chapters and main contents are as follows:In Chapter 1, we give a survey to the developments of population ecology, delay differential equations and impulsive differential equations. Then we intro-duce the background of problems, the main results of this dissertation and some preliminaries are also summarized.In Chapter 2, we discuss the predator-prey model with diffusion and im-pulse. In the section 2.1, we obtain some sufficient conditions which guarantee the existence of at least one positive periodic solution for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays by using Mawhin's continuation theorem of coincidence degree theory, the result not only improves but also generalizes the known one. In the section 2.2, by applying Mawhin's continuation theorem of coincidence degree theory and a suitable Lyapunov func-tional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness and global stability of positive periodic solution for a delayed predator-prey model with dispersion and impulses. Some known results are im-proved and generalized. As an application, we also give some examples to illustrate the feasibility of our main results.In Chapter 3, we investigate a delayed predator-prey model with monotonic or non-monotonic functional response and impulse. By using Mawhin's continuation theorem of coincidence degree theory, we obtain some sufficient conditions which guarantee the existence of multiple positive periodic solutions for the model. In particular, the presented criteria improve and extend some previous results.In Chapter 4, we consider two classes of non-autonomous delay n-species com-petitive models with impulses. In the section 4.1, we obtain some sufficient and realistic conditions for the existence of positive periodic solutions of a general neutral delay n-species competitive model with impulses by using some analysis techniques and a new existence theorem, which is different from Mawhin's contin-uation theorem of coincidence degree theory and abstract continuation theory for k-set contraction. As an application, we also examine some special cases which have been studied extensively in the literature, some known results are improved and generalized. In the section 4.2, we apply the fixed point theorem in a cone of Banach space to obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solution of generalized n-species Gilpin-Ayala type competition system with multiple delays and impulses, which improves and generalizes the known ones.In Chapter 5, we study some classes of Logarithmic population models with multi delays and impulses. In the section 5.1, we use the theory of abstract contin-uous theorem of k-set contractive operator and some inequality techniques to get sufficient and realistic conditions which are established for the existence, global attractivity of positive periodic solution to a neutral single-species Logarithmic population model with multi delays and impulse. The results improve and gen-eralize the known ones. As an application, we give an example to illustrate the feasibility of our main results. In the section 5.2, by using the contraction map-ping principle and some analysis techniques, we establish a set of easily applicable criteria for the existence, uniqueness and global attractivity of positive periodic solution for a multi-species Logarithmic population system with impulses. The conditions we obtained are weaker than the previously known ones, which can be easily reduced to several special cases. At last, we also give an example to illus-trate the feasibility of our main results. In the section 5.3, we investigate a neutral multi-species Logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence,uniqueness and global attractivity of positive periodic solution is established. The results improve and generalize the known ones. At last, we also give an example to illustrate the applicability of our results.