Competition/Cooperation Models With Seasonal Succession And Dynamics For Singularly Perturbed Monotone Systems With Respect To 2-cones

In this thesis,we are concerned with the dynamics of competition and coopera-tion models with seasonal succession,as well as the generic dynamics of singularly perturbed monotone systems with respect to cones of rank 2.Firstly,we established an n-dimensional Lotka-Volterra competition model with seasonal succession.It is proved that there exists an(n-1)-dimensional carrying simplex which attracts every nontrivial orbit of the system.By the theory of the car-rying simplex,we reconsider the two-dimensional Lotka-Volterra competition model with seasonal succession.We also obtain the complete classification of global dynam-ics.Our approach avoids the complicated estimates for the Floquet multipliers of the positive periodic solutions.Meanwhile,a new approach is presented for investigating competition models with seasonal succession.Secondly,we investigate a three-dimensional Lotka-Volterra competition model with seasonal succession and obtain the local dynamics of all periodic solutions on the boundary of the carrying simplex.We further focus on the stability of heteroclinic cycles of the symmetric May-Leonard competition model with seasonal succession.Sufficient conditions for stability of heteroclinic cycles are obtained.Meanwhile,we present the explicit expression of the carrying simplex in a special case.By numerical simulation,we find that there are rich dynamics for the system as ? varies,which provides inter-esting features of such a system.Thirdly,we analyze a three-dimensional Lotka-Volterra cooperation model with seasonal succession.The Floquet multipliers of all periodic solutions of the system are thoroughly estimated.Besides,we rigorously prove that the system admits a unique positive periodic solution under certain conditions.Meanwhile,we also obtain a com-plete dynamics of global coexistence and extinction,which extends previous results with respect to three-dimensional cooperation models and enriches the researches of Lotka-Volterra cooperation models with seasonal succession.Finally,we further gain insights into the generic dynamics for singularly perturbed monotone systems with respect to cones of rank 2.We obtain the so called Generic Poincare-Bendixson theorem for such perturbed systems.That is,for a bounded posi-tively invariant set,there exists an open and dense subset P(?)such that for each z?P(?),the ?-limit set ?(z)that contains no equilibrium points is a closed orbit.This result extends the previous Generic Poincare-Bendixson theorem for monotone flow with re-spect to cones of rank 2 into singularly perturbed monotone systems.