A Class Planktom The Phase G Of The Steady-state Solutions And Periodic Solutions

In the field of Mathematical Ecology, it is important to use Mathematics to study population dynamics. In 1897 it was Tomas Malthus who firstly construct differential equation to research population ecology. In 1926 Volterra proposed ordinary equation from predator-prey model, which is the classic Lotka-Volterra model. From then on plenty of work had been done to research the periodic solution and stability of ordinary differential equation. Those work explained some phenomenon of population, such as coexistence, existence and persistence.Time is only considered as the effect on population density in ordinary model. In fact, to gain abundant food in some spatial domain population move from high density region to low density region. In order to defend the disease and rivals population have the crossing diffusion. Lin Zhigui considered the effect of diffusion and constructed partial differential equation model. He used the efficient tool of partial differential equation to study the asymptotical behavior of population and obtained some good results, which could not obtained by ordinary equation. In predating or competition model the baby predator has a weak predating ability and the baby prey is more vulnerable than the mature prey. Some researchers introduce time delay to the model. Time delay problem is becoming a hot problem in mathematical ecology.An interesting observation is that the increased population of one species might affect the growth of another species or of several other species by the production of allelopathic toxins. A Volterra competing model that arises in plankton allelopathy is considered in this paper. A delay weakly coupled reaction diffusion system is proposed. The existence and uniqueness of global solution are given using upper and lower solutions. By a method of monotone iteration the asympotical behavior of stable solution is discussed. Some sufficient conditions for the stability of positive and semi-positive stable solution are obtained. Furthermore, if the coefficients of the system are periodic functions, the existence of periodic solution can be studied using upper and lower solutions. Some sufficient conditions for the existence of positive periodic solution are also given.