Positive Periodic Solution For N-species Competition System With Time Delay And Feedback Control

In 1925, the ecologist D'Ancona proposed a problem regarding the shark of the Mediterranean sea when he studied the variation rule of the fim, and then, V. Volterra published a paper explaining how the fim of the harbor Finme to change. By using dynamical theories, he solved this issue successfully. Since then the theory of dynamical systems had been extensively applied to mathematical biological modelling. At the same time, the Lotka-Volterra models, such as predator-prey type, competitive type and reciprocal type to which were paid high attention by ecologists.A typical character of the population dynamics theory is the fluctuation phenomenon on the density of ecological species. The periodical fluctuation of the environments is one of the main reasons which would lead to the variation of the species density. Hence, the influence of varying environments on the growth of species have been the focus of the population dynamics research. However, if we consider the effect of the environments, it must be assumed that the parameters of the model are periodic or almost periodic. Mathematically, it may inevitably result into the nonautonomous Lotka-Volterra systems, in which require introducing time-dependent parameters. Only by this way, can one research the dynamical characters of the ecological species, but this will increase the difficulty of the model analysis. Therefore, it is a meaningful but challenging work to investigate the periodical Lotka-Volterra systems. Obviously, the positive periodic solution of the nonautonomous models are equivalent to the equilibrium solution of the corresponding autonomous ones. Thus when we study the periodical multi-species models, it is natural to ask that under what conditions the systems could be permanent, and under what conditions the periodical models exist positive periodic solution and furthermore this periodic solution is global asymptotic stability. It is of great help for us to deal with man, biological species and natural environments when studying such problems. Nevertheless, these theory results may provide beneficial guidance for human's practical activities. However, with the increasing application of the nonlinear functional analysis theories, for instance, Mawhin coincidence theory, k-set contraction operator theory, semigroup operator theory, Lyapunov functional and fixed point methods, to population dynamical models, the research on periodical population dynamical systems have taken great progress in recent years, and a great amount of excellent keeping worlds ahead results have been obtained.In this paper, we mainly discuss the periodical n-species Lotka-Volterra competition systems. On the bases of the classical Lotka-Volterra competitive model, we take into account the factors such as nonlinear intra-specific interference and interspecific interfer- ence, physiological stage structure, multi-kind time delays and various forms of feedback control, and then present three types of generalized models. By using comparison arguments, coincidence degree theory and Lyapunov functional methods, we provide a series of new sufficient conditions to guarantee the permanence, existence, uniqueness and global asymptotic stability of positive periodic solutions for these systems, respectively. Some existing results are improved and generalized.