Study On Long - Term Dynamic Behavior Of Competitive System

This paper focuses on investigating the long-term behavior of the discrete-time dynamical systems involving in competitive mappings, time recurrent nonautonomous systems and autonomous competitive systems, which includes three respects:A. Establishment of the carrying-simplex theory and the index formula for the general competitive mappings, defining an equivalence relation between two maps by the local dynamics of the boundary fixed points, and classifying the 3-dimensional Leslie/Gower map and Atkinson/Allen map.A criterion which can be easily checked is provided to guarantee the existence of the carrying simplex for competitive mappings. By this, we show that a family of competitive mappings possesses carrying simplex. In particular, the Leslie/Gower map and Atkinson/Allen map with arbitrary dimension are first to shown to possess carrying simplices unconditionally.Based on the carrying simplex theory, we present a formula on the sum of the indices of all fixed points on the carrying simplex for 3-dimensional competitive maps and analyze the stability of the boundary of the carrying simplex. Specifically, we provide the criteria on the stability of the heteroclinic cycle. Next, we define an equivalence relation between two maps by the local dynamics of the boundary fixed points, and then classify the 3-dimensional Leslie/Gower map and Atkinson/Allen map by the index formula. They both have 33 stable equivalence classes, respectively, and in classes 1- 18, every orbit tends to a fixed point. The other 15 classes may have relatively complicated dynamics. In particular, we mainly investigate the Neimark-Sacker bifurcations, cyclical fluctuation phenomenons, and the existence and stability of heteroclinic cycles in each of these classes.B. Establishment of the Decomposition Formula for nonautonomous LV systems with identical intrinsic growth rate perturbed by minimal functions, to describe the long-term dynamical behavior.We establish the Decomposition Formula: the solution of the Lotka-Volterra(LV) system with identical time-dependent intrinsic growth rate can be expressed in terms of the solution of the unperturbed autonomous LV system multiplied by an appropriate solution of the nonautonomous scalar-logistic-equation subject to the same perturbation. By virtue of such formula, one discovers that the dynamics of autonomous LV system can be inherited by the perturbed nonautonomous LV system. By the Decomposition Formula, we first provide the existence of quasi-periodic/almost periodic solutions and chaotic motions in nonautonomous competitive LV systems perturbed by periodic/almost periodic functions. Based on Zeeman’s nullcline equivalent classification for 3-dimensional competitive LV systems, we obtain a total of 37 topological classes in 3-dimensional competitive LV systems with identical intrinsic growth rate. Further, by the Decomposition Formula we provide the complete dynamical classification for the perturbed nonautonomous competitive LV systems via pull-back way in skew-product flow.C. To show that Zeeman’s nullcline equivalent classification for 3-dimensional competitive LV systems can be adapted to the 3-dimensional competitive systems admitting linearlydetermined nullclines.We develop Zeeman’s nullcline equivalent classification for 3-dimensional competitive LV systems for the applicability to 3-dimensional Kolmogorov competitive systems. We show that the total of stable nullcline classes by nullcline equivalence is always 33 for any of the 3-dimensional Kolmogorov competitive systems admitting linearly-determined nullclines. Classes 1- 25 have trivial dynamics, the class 27 has a heteroclinic cycle and the class 32 does not admit Hopf bifurcations. By analyzing Hopf bifurcations we find that for 3-dimensional continuous-time competitive Ricker model and Leslie/Gower model, Hopf bifurcations can occur in each of classes 26- 31,while for 3-dimensional continuous-time competitive Atkinson/Allen model and Gompertz model,Hopf bifurcations can only occur in classes 26- 27. We also compare the difference among these systems. We also provide examples to show that the Ricker model, Leslie/Gower model, and Gompertz model can admit two limit cycles. Such a discovery greatly extends Zeeman’s methods to the applications.