| In today,s world, our environment is more serious day and day. So research on changes of population,s survival condition in a polluted environment becomes a hot topic to many scholars. In addition to pollution, capture is also a important factor that affect the survival of population. Especially, the impact of constant capture is more difficult to control. In this paper, Smith model and Gallopin mod-el which live in polluted environment are studied by methods of integral in mean, comparison theorem and so on. So we can predict the development of population as well as the impact of man-made capture on population survival. Main results are as follows:Firstly, under the effect of the variance of the dead population on the internal and environment toxicant density, a kind of single species Smith model is established. This paper gives sufficient conditions of the existence and stability of equilibrium under the constant external toxin by comparison theorem, Ruth-Hurwitz criterion and complex matrix. Under changed external toxin it gives some sufficient conditions forβ- persistence andβ- extinction of the Smith model with no capture by tow methodds. One is looking for the right boundary of internal and environment toxicant density by comparison theorem. The other is introudcting control function and discussing its extreme range by integral in mean. For the model with constant capture, it gives the sufficient conditions forβ-persistence andβ-extinction and obtains the maximal harvesting rate when the population is persistence.Secondly, considering the influence of death on consumer population and the toxin density of environment, the original Gallopin model is improved. For the model with no capture, this paper gives the boundary of consumer population and resoures population. And the sufficient conditions forβ- persistence andβ-extinction are gives by differential inequality and comparison theorem. For the model with constant capture, the sufficient conditions forβ- persistence,β- extinction and zero-extinction are given by the same methods. And the maximal harvesting rate is obtained when the consumer population is persistence.The research about stability and persistence of the population can be used to guide people to protect the environment and control pollution. Those will have very extensive theoretical and practical significance to maintain the diversity of ecosystems and make full use of the nature. |
