The Dissipative Structure's Interpertation Of The Prey-Predator Model Based On Trophodynamics

In the world of today, our ability of changing the nature is greatly heightened. Because of over-exploitation, some species are close to extinction or already extinct. We destroy the ecological balance seriously. Recognize the reality, in this paper; we investigate a mathematical model of a predator-prey system in which both predator and prey have their respective carry capacities. Through the application of the dissipative structure theory, we have proven that the model possibly has the dissipative structure's characteristic under the different conditions. Consider the actual meaning of the system; we are mainly interested in the dynamical behavior of the system in the phase plane. In the phase plane, with the parameter changing, the system can achieve the different position of equilibrium. Discussing the system's limit-cycle and diffusion structure and combining with the phase portrait, we know the model's ecological significance.On the basis of the second chapter, in the third chapter, we draw up the limit-cycle in the phase plane with the MATLAB. When k, u₁, u₂, u'₂, u₃, x'_m and x'_m, respectively achieve some conditions, the system will vibrate. Between the prey and the predator it can have the limit-cycle, and at the time, this system will present the dissipative structure's character. When these parameters surpass a certain value, the system cannot appear the limitcycle. Through the various parameters' changing, it demonstrates the forming process of the dissipative structure. Thus it show the dissipative structure's characteristic of the trophodynamics predation model and point out the ecological significance of the various parameters in the model.On the fourth chapter, the spruce aphides and its predator model is a predator model. According to the change of the enemy's quantity and the qualitative theory of the differential equation, we research the spruce aphides and its predator system. At the same time, we discussion the spruce aphid multiplication model when the natural enemy quantity is a certain value and the periodic oscillating model whose the quantity of the natural enemy changes along with the time. Obtains the stability condition of the balance point and the limit-cycle existing condition, the system presents the dissipative structure's character. Finally it explains that this theoretical analysis has certain practical significance regarding the preventing and controlling harmful insect.