The Applications Of The Delay Differential Equations In The Ecology

As a branch of the mathematics, the theory of delay differential equations was applied in many fields, such as auto-control, physics, biology, medicine, economics, chemistry and so on. The last years, many new developments had occurred with the cross-subject appearance. In this paper, the application of the delay differential equations in ecology was studied, to research two practical problems: â‘  The evolution laws of the population; â‘¡ The dynamical characteristics of the forest system and how could people take some controls to manage and utilize the forest system when they got profits from the forest.In fact, when the stability of the delay differential equations (DDE) was discussed, the characteristic equation of the linear section of the DDE had to be considered. But the characteristic equation usually was a transcendental equation, so it was important to study that. In the third chapter, a simple method to judge the stability of the roots of a class transcendental equation was given out. According the Rouche theorem and the characteristics of the solutions of the transcendental equation in the complex plane, the existence of the stable solutions and the Hopf bifurcation for the different forms of the class equation was analyzed. Based on those conclusions, this paper expounded the existence of the pure imaginary roots, the transversal condition, and the existence of Hopf bifurcation for this class equation.In the fourth and the fifth chapters, two concrete Bio-mathematics models were researched by the analysis method of the third chapter. In the fourth chapter, a three-dimensional eco-epidemic model with delay was studied, where those three variables represented population densities at time tof prey, susceptible predator and infected predator. By studying the linear stability of the model, it was found that there existed Hopf bifurcations when the delay passed a sequence of critical values. Then the explicit algorithm was derived for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form method and center manifold theorem. By those researches, the development laws of the similar ecology system were found.In the fifth chapter, the dynamic behavior of a kind of red pine woods ecological system with delay was discussed by using the systematic analysis method of the ecology. The main factors that affected the evolution of red pine woods ecological system were red pine nut, rats and seedlings. This section mainly studied the stable conditions on the delay system, the change of the stability about the equilibrium and the periodic solutions and appearance of the Hopf bifurcation. The conclusions proved that it was important to make use of the forest system when people got profits from the forest.