Global Attractivity And Multiplity Of Periodic Solution For Some Biological Mathematics Models

Existing results on the existence of the periodic system often fall into oneof the following three categories: (1) the results of the applications of the con-traction principle or ?uctuation principle, which establish both the existence andattractivity of periodic solutions in periodic equations with time delay; (2) theexistence simply follows the observation that periodic solution exists when thereis no time delay and this periodic solution remains so when time delay is a mul-tiple of the periodic equation; (3) the results of the application of the Horn'asymptotic fixed point theorem. While these methods often allow the investiga-tor to address the stability issues of the periodic solutions, the conditions for theexistence part are often unnecessarily numerous, tedious, stringent, and di?cultto satisfy. Specifically, all of the above methods are ill-suited to problems withstate-dependent delay require only a set of natural and easily verifiable condi-tions. These conditions are readily satisfied in many realistic population models.Such an approach was adopted in two dimension population models.Topologicaltheory is a strongly tool of nonlinearity operator qualitative theory, form this wecan obtain many famous fixed point theorem. So we obtained the existence ofthe periodic solution. As is well know, periodic phenomena is widely distributedin nature. Then these phenomena often lead us to study the existence of peri-odic solution of functional di?erential equation. Specifically, in order to makethe models more practical, we also consider the existence of positive periodicsolution.In this paper, we establish the existence of periodic solution and the attrac-tivity for some di?erential and di?erence equation by using continuation theoremof degree theory. Recently, there are many paper which obtained the existenceof periodic solution for the population system by using continuation theorem ofdegree theory and many good results was obtained.Firstly,we present the background and necessity for the study of periodicsolutions of di?erential equations.Then,some basic definitions are give.Secondly ,a nonautonomous stage-structured population dynamics systemwith delay and di?uion is considered.By using coincidence degree theory,somesu?cient condition are obtained ensuring the existence of positive periodic solu-tion for the system. Further,by constructing a Lyapunon functional and using...