Permanence On Delays Systems With Feedback Controls

In the early 20th century, with the development of biology and mathematics, there appeared a new interdisciplinary subject-biomathematics. Biomathematics is the subject to make use of some mathematics theories to solve some practical problems in biology. Among all these theories, the theories of differential equation were extensively used, and have solved many important biology issues. Mathusian equation, Logistic equation and Lotka-Volterra equation, for example, reflected the ecological relationship in different species. Thus, it is of great necessity and importance to make a further study on these equations and their characteristics of permanence, stability, periodicity, extinctive and global asymptotic. In recent years, with more and more scholars devoted in this aspect, many representative products have been achieved. This paper depends on the existent theories and expands them on some typical ecological models.This paper mainly studies eco-system's feature of permanence. It consists of four chap-ters.Chapter 1 consists of 2 parts:1. Literature reviews.2. Structure of the paper.Chapter 2 mainly studies the single-species system with delays and feedback controls: The author makes use of differential inequality, comparison principle and the nature of global attractivity, and gets the sufficient condition of permanence, and expands it into further aspect.Chapter 3 introduces the predator-prey systems of Lotka-Volterra: In this chapter, the writer makes use of the existent conclusion and comparison principle, combined with global attractivity, and concludes sufficient conditions of eco-system's feature of permanence and improves the predator-prey systems of Lotka-Volterra.Chapter 4 studies the permanence in non-autonomous cooperative systems with delays and feedback controls:This paper, based on the method of the referenced paper [l]and comparison principle, esti-mates the solution and concludes the sufficient condition of the system. In addition, with the consideration of continuous delays and feedback, this paper extends discrete delays Lotka-Volterra from two into three, and the result is satisfactory.