Periodic Solutions And Almost Periodic Solutions For Functional Differential Systems In Biology

This Ph.D.thesis is divided into six chapters and main contents are as follows:In Chapter 1, we give a survey to the developments of periodic solut-ions,almost periodic solutions of functional differential equations and impulsive differential equations. Then we introduce the background of problems, the main results of this dissertation and some preliminaries are also summarized.In Chapter 2, we study respectively the existence of positive periodic solutions for a delay neutral population model with impulses and a delay neutral population model with feedback control, by means of an abstract continuous theorem of k-set contractive operator and some analysis techniques. A set of easily applicable criteria are obtained for the existence of positive periodic solutions of the two models, which is another answer to the open problem9.2in [118]. The two models containing several kinds of special forms, thus our conclusions extend and improve some existing ones.In Chapter 3, a neutral single-species Logarithmic population with multiple delays model is considered. By using of an abstract continuous theorem of k-set contractive operator, sufficient conditions are established for the existence and the global asymptotical stability of positive periodic solution under variable delay for the model. Also, our results improve the known ones.In Chapter 4, we considered a periodic predator--prey system with impulse. Sufficient conditions for the existence of at least two positive periodic solutions are established by applying the theory of coincidence degree. To our knowledge, this is the first work for the existence and multiplicity of positive periodic solutions for above impulsive system. Even under the non-pulse circumstances, this conclusions have greatly improved the known results in [54] by the introduction of an important lemma [Lemma 1.5.4].In Chapter 5, we consider a class of nonlinear first order neutral functional differential equations. Sufficient conditions are obtained on the existence of at least three positive periodic solutions, using of Leggett--Williams fixed point theorem. Finally, we apply the main results to several special types of functional differential equations and obtain some useful corollaries. Our conclusions extend some known results in [216].In Chapter 6, a delayed n-species Lotka-Volterra competitive system with feedback control is investigated. By using of comparison theorem and an suitable Lyapunov functional, sufficient conditions are establish-ed for the existence and uniqueness of positive almost periodic solution of the system.