| This paper consists of three parts:Firstly,a nonlinear discrete competitive system of two species with the effect of toxic substances is proposed.After that,by constructing a suitable Lyapunov-type extinction function,we obtain the sufficient conditions which guarantee that one of the components is driven to extinction while the other one is globally attractive with any positive solution of a discrete equation.Two examples together with their numerical simulations illustrate the feasibility of our main results.The results not only improve but also complement some known results.Secondly,we propose a nonautonomous competitive system with infinite delays and feedback control.By giving the subtle analysis of the right-hand side of the system,a set of sufficient conditions which guarantee the permanence of the system is obtained.By constructing a suitable Lyapunov type extinction function,we obtain the sufficient conditions which guarantee that one of the components is driven to extinction.Examples together with their numerical simulations illustrate the feasibility of our main results.Thirdly,an autonomous stage-structured competitive system with the effect of toxic substances is investigated.Sufficient conditions which guarantee the global attractivity of the system and the extinction of the partial species are obtained,respectively.Examples together with their numerical simulations illustrate the feasibility of our main results.Our results supplement and compliment one of the main results of Liu and Li[Global stability analysis of a nonautonomous stage-structured competitive system with toxic effect and double maturation delays,and Applied Analysis,Volume 2014,Article ID 689573,15 pages]. |
