| In this paper,by the results of persistence or Mawhin's Continuation Theorem of the coincidence degree theory,we obtain the existence results of positive periodic solutions for several population models in Ecology .In addition, by constructing a Lyapunov functional (or a Razumikhin function), we study the global attractivity (or assymptotical stability) of a positive periodic solution for some models.The paper consists of five chapters.In chapter 1,we introduce the significance,background and the main work of our study as well as some lemmas and our main results in this paper.We consider a competitive and prey-predator model with stage-structure in chapter 2. By Independent-subsystem method, the results of existence and global attractivity of a positive periodic solution are obtained.In order to make the prior estimate of solution and calculate the upper right derivative of V(t) along the solution, we employ some transformation techniques on fraction.In chapter 3,by means of Comparison Theorem,we investigate the persistence of a prey-predator system with diffusion and delays .Further, we prove that the system has at least one positive periodic solution under the conditions that ensure it is persistent. Finally,by constructing a Razumikhin function , we get the sufficient conditions that ensure the positive periodic solution is globally asymptotically stable.In chapter 4,we study the existence of a variabal-delayed single-species model with stocking or harvesting. By employing Mawhin's Continuation Theorem ,the existence results of positive periodic solutions are obained.Furthermore,by means of constructing a Lyapunov functional,we attain to the sufficient conditions for the global attractivity of a positive periodic solution of this model.In the last chapter,by means of coincidence degree theory,we consider the existence of positive periodic solution for a generalized discrete prey-predator system with delay and functional responses. The means of analysis is employed during the prior estimation for the solutions,thus the verification of the conditions is more practical. Moreover,we consider the techniques in calculating the map's degree respectively in two cases that the functional responses function is monotone and differentiable or is nonmonotonic and undifferentiable.In this paper,we go deep into the computation of the map's degree especially... |
