Analysis Of A Plant-herbivore Model With Time-delay And Toxin-determined Functional Response

In this paper, we study a plant-herbivore model with delay and toxin-determined functional response. We modify the traditional Holling type 2 response by taking into the analysis of the negative effect of toxin on herbivore growth and the delay about the conversion of consumed plant biomass into new herbivore biomass. The uniform permanence of the system is proved by using the dissipative theory. Then we discuss the stability of the equilibrium points. Finally, we study the system without consideration of delay and obtain the global attractivity of the positive equilibrium point.The content of this article is divided into six chapters:In chapter one, we introduce the historical background of this issue and the significance of research; then we review the present research in both domestic and international, and propose the research ideas and the structure of this article.After the introduction of the plant-herbivore system model studied by Chen and Liu, we point out its deficiencies (They make an assumption that predators are immediately getting energy after they have swallowed plants, it is clearly inconsistent with the reality) and improve it in the second chapter. In order to be more closer to reality, we take into the consideration of the delay about the conversion of consumed plant biomass into new herbivore biomass at the new improved model. Under the assumption that predators are not immediately getting energy after they have swallowed plants and they will get growth and reproduction after a period of time for digestion and absorption.In chapter three, we prove that the system is persistent by the dissipative theory, and we also obtain the sufficient condition of its persistence.In chapter four, by discussing the stability of equilibrium points we find that the system with delay exists some particular phenomenon which we called it stability switches, it shows that as delay changes, the stability of positive equilibrium point may also change:it changes from stable to unstable, and then stable thereafter; at last, it legs into unstable after limited cycles. In chapter five, we obtain the global attractivity of positive equilibrium point by constructing suitable Lyapunov function.Chapter six is the last chapter of this article; in this chapter, we summarize our research work, the problems and direction of further study are also pointed out in this dissertation.Ⅲ...