Existence And Attractivity Of Periodic Solutions For Three Models In Mathematical Biology

In this paper, by employing a fixed point theorem in regular cone and two con-tinuation theorems in coincidence theory, we study the existence of one or multipleperiodic solutions of three models in mathematical biology. Meanwhile, by meansof discussing the oscillation and non-oscillation of positive solutions in one system,we study global attractivity of unique positive periodic solution of this system.This paper consists of four chapters:In Chapter 1, we introduce the backgrounds and significance of our studies,main work of this paper, preparing knowledge and some notions.In Chapter 2, we study a discrete hematopoiesis model. First, we demon-strate that all positive solutions of this system are permanent. Then by applying afixed point theorem in regular cone, we obtain su?cient conditions which guaranteeunique positive periodic solution of this system. At last, by discussing the oscil-lation and non-oscillation of positive solutions in this model, we obtain su?cientconditions which guarantee global attractivity of unique positive periodic solutionof this system.In Chapter 3, we study a Lotka-Volterra cooperative model with harvesting.By applying a continuation theorem in coincidence theory, we obtain su?cientconditions which guarantee the existence of at least four positive periodic solutionsof this system.In Chapter 4, we study a discrete predator-prey model with nonmonotonicfunctional response and harvesting. By applying another continuation theorem incoincidence degree, we obtain su?cient conditions which guarantee the existenceof at least four positive periodic solutions of this system.For models with harvesting, from the works which have already done in theliterature, we can see that they often present complex dynamical behaviors. Inthis paper we study two nonautonomous systems with harvesting and obtain someconcise su?cient conditions which guarantee the existence of four positive peri-odic solutions, the results we obtained re?ect the complexity of these two systemsto certain extent. When we employ the continuation theorems in coincidence de-gree theory to study the existence of four positive periodic solutions in di?erentialsystems and delay di?erence systems, we need to construct four bounded open sets and compute the Brouwer degrees in these sets. In this paper we apply somenew inequalities and variable transformations to construct proper bounded opensets, and simplify the computation of degree by using the proposition of homotopicinvariance of Brouwer degree. Actually our results are concise and e?ective.