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

Existing results on the existence of the periodic system often fall into one ofthe following three categories: (1) the results of the applications of the contractionprinciple or ?uctuation principle,which establish both the existence and attractiv-ity of periodic solution in periodic equations with time delay; (2) the existencesimply follows the observation that periodic solution exists when there is no timedelay and this periodic solution remains so when time delay is a multiple of theperiodic equation; (3) the results of the application of the Horn asymptotic fixedpoint theorem.While these methods often allow the investigator to address the sta-bility issues of the periodic solutions,the conditions for the existence part are oftenunnecessarily numerous,tedious,stringent,and di?cult to satisfy.Specifically,all ofthe above methods are ill-suited to problems with state-dependent delay requireonly a set of natural and easily verifiable conditions.These conditions are read-ily satisfied in many realistic population models.Such an approach was adoptedin two dimension population models.Topological theory is a strongly tool of non-linearity operator qualitative theory,from this we can obtain many famous fixedpoint theorem.So we obtained the existence of the periodic solution.As is wellknow,periodic phenomena is widely distributed in nature.Then these phenomenaoften lead us to study the existence of periodic solution of functional di?erentialeqution.Specifically,in order to make the models more practical,we also considerthe existence of positive periodic solution.In this paper,we establish the existence of periodic solution and the attractiv-ity for some di?erential and di?erence equation by using continuation theorem ofdegree theory.Recently,there are many paper which obtained the existence of pe-riodic solution for the population system by using continuation theorem of degreetheory and good results was obtained.Firstly,we present the background and necessity for the study of periodic solu-tion of di?erential equations,and some lemmas and our main results in this paper.Secondly,a discrete two-species competive model with stage structure is con-sidered.By using coincidence degree theory,some su?cient condition are obtainedensuring the existence of positive periodic solution for the system.Thirdly,by employing the continuation theorem of coincidence degree the-ory,the existence of a positive periodic solution for a prey-predator model with sex-structure.Further,by using one of Theorem of Zhao and constructing a V func-tional and using the result of periodic solution,the attractivity of positive periodicsolution for above system is obtained.Finally,we study a Lotka-Volterra type and chain with structuring and timedelay,some su?cient condition are obtained ensuring the existence of multiple pos-itive periodic solution for the system.