Periodic Solutions For Several Nonlinear Biological Models

We note that any biological or environmental parameters are naturally subject to fluctuation in time. Therefore, it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.).The purpose of this paper is to investigate persistence and existence of periodic solutions, and the global stability of periodic solutions for three kinds of nonlinear models with periodic coefficients.First, by employing a comparison theorem the easily verified sufficient condition of permanence is obtained for a periodic predator-prey model with modified Leslie-Gower and Holling-Type-II schemes. Furthermore, by employing the Brouwer fixed point theorem the existence of positive periodic solution is also obtained for the model.Second, we concern the existence of periodic for a neutral delay impulsive system. By using a abstract continuation theorem in degree theory, a set of sufficient conditions are obtained for the model. Some known results are generalized.Finally, sufficient conditions which ensure the existence of periodic solutions for a n-dimension nonautonomous discrete-time neural network with time-varying delays are obtained by employing Mawhin's continuation theorem. Furthermore, sufficient conditions for global attractivity of periodic solutions are also obtained. Some known results are extended and generalized.