Dynamical Analysis For The Competitive Model Of Enterprises With Lower Threshold

With the rapid development of the global economy,commercial ecosystems such as the enterprise interactional model have been attracted extensive attention and research.Based on the research of related fields in both domestic and international,we focuse on the competitive model of enterprises with lower threshold.As regard to the phenomenon of enterprise information delays and economic cycle,the autonomous with two delays and nonautonomous competitive model of enterprises with lower threshold are established.Then,the dynamic properties of models,which can describe the basic characteristics of the system,are studied.By such research,people can understand the business rules better.The main works of this paper is summarized as follows:Firstly,by considering the influence of information delays on output,the time delays?₁^*and?₂^*are introduced into the competitive model of enterprises with lower threshold.And then,we establish a competitive model of enterprises with two delays and lower threshold,which can reflect the competitive relationship well.Secondly,the dynamic properties of the competitive model of enterprises with two delays and lower threshold are analyzed.To begin with,the distribution of the eigenvalues are discussed by eigenvalue analysis method with time delays?₁^*and?₂^*as the parameters.The sufficient conditions for the local stability of the equilibrium and Hopf bifurcation are obtained.Furthermore,the central manifold theorem and the Poincarécanonical form method are used to obtain the direction of the bifurcation and the stability of the periodic solutions.Some numerical simulations supporting our theoretical predictions are included.As a result,we find that small changes in the delays have an important impact on the stability of the output.Meanwhile,the trajectory of the output and phrase space are presented with different time delays.A more intuitive observation of the development trend of enterprises is given to reveal the change rules of the output.Thirdly,according to the phenomenon of economic cycle,it is assumed that all parameters of the equation are positive periodic functions.A nonautonomous competitive model of enterprises with lower threshold is established,and the existence of periodic behavior of the model is researched by continuous theorem of coincidence degree theory.Consequently,the sufficient conditions for the existence of at least one positive periodic solution are obtained.Finally,we summarize this article in brief,and give the directions which are worthy of further investigation.