| Delayed differential equations are differential equations with time delays, which are mainly used to describe dynamical systems in which the future status depend upon the present and past ones. In recent years, there has been an extensive and deep understanding for ecosystems by constructing and investigating the corresponding mathematical models describing the ecosystems. In this paper, the mathematical model describing the dynamic development of the input and output of two enterprises is improved based on the viewpoint of ecology and the dynamical behaviors of the model are also investigated in view of the fundamental theory of retarded functional differential equations.The goal of this thesis is to study the effect of time delays on the dynamical behaviors of delayed differential equations by using the following model of interactions of two enterprises as a research object The stability of the unique positive equilibrium and the existence of Hopf bifurcation are analyzed by choosing the sum ? of two delays ?1 and ?2 as the bifurcation parameter and employing the Hopf bifurcation theory. It is found that when ? is less than a certain critical value, the positive equilibrium is locally asymptotically stable while it becomes unstable when ? is greater than the above critical value. In addition, the system can bifurcate a family of nontrivial periodic solutions from the positive equilibrium when ? crosses increasingly through a sequence of critical values containing the above critical value. In particular, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained according to the normal form theory and the center manifold theorem for delay differential equations. Finally, some numerical simulations supporting our theoretical predictions are included by applying the step method of differential difference equations and the soft packages MATLAB and XPPAUT. |
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