Stability Analysis Of Ratio-dependent Predator-prey Systems With Diffusion And Time Delay

In the natural environment of the real predator, based on ratio-dependent functional response function can be more accurate to characterize the changes in predator ratio. Delay and the phenomenon of diffusion often occurs in the ecosystem. Through differential equation theory and construction of Liapunov function method, based on ratio-dependent predator-prey systems with diffusion and time delay are studied in this paper. The principal tasks are as follows:First of all, taking the impacts caused by delay factors into species, predator-prey systems with diffusion, discrete and distributed time delay is considered, and improved only with a discrete time delay of the predator-prey systems. The criterio for boundedness and uniformly persistent of system is given through using of differential equations the comparison theorem; by doing Poincare mapping transform the existence of positive periodic solutions into argument for existence of fixed-point, and discussed the existence of periodic solution of problems through arguments for continuity of mapping and bounded closed convex set; items time delay were offset by constructing Liapunov function with constant coefficients, Makes the Liapunov function upper right derivative is negative, and use Lemma Barbalart to find sufficient condition for the global stability of systems positive solution.Secondly, taking into account a more realistic predator environment, based on the ratio of predator functional response function, set up a predator and two prey of three species based on ratio-dependent predator-prey systems with diffusion and time delay. Solution boundedness and uniformly persistent criterion of system are obtained by differential inequalities, Increase of time delay has increased the difficulty of inequality scaling, chosen the appropriate coefficients of Liapunov functionals through asymptotic methods, A sufficient condition for global stability of systems solution is obtained through the right upper derivativewhich is zoomed appropriatly and integraled on both sides, sufficient condition for global stability of positive solutions is obtained. Finally, taking into account the complexity and links of predator environment, extended conclusions from a predator and two prey of three species to more general the food chain systems. Discuss based on ratio-dependent food chain system with diffusion and time delay. The condition of solutions boundedness and uniformly persistent is given through differential inequality; through the asymptotic method to construct Liapunov function, in the constructor on a variety of attempts to construct containing two constant coefficients of the Liapunov functional, appropriate scaling the upper-right derivative through using conclusions of boundedness and uniform persistence, a sufficient condition for global stability of systems positive solutions is given through using Lemma Barbalart.Through research of persistence and stability of predator-prey systems with diffusion and time delay, allowing people to better protection of nature biological species, right to create a green environment and the rational development and utilization of natural resources play an important role.