STABILITY ANALYSIS OF PREY-PREDATOR SYSTEM USING TAKAGI-SUGENO APPROACH

Abstract

Predator-Prey model is a relationship between two species living in the same space. It gives the e ect on population between the two species. When two species are living in same habitat they share some resources such as food resource and eco logical niche. A predator-prey interaction has been described rstly by two pioneers Lotka and Volterra in two independent works. After them, more realistic prey predator models were introduced by Holling suggesting three kinds of functional responses for di erent species to model the phenomena of predation. Almost all of the physical dynamical systems in real life cannot be represented by linear di erential equations. The non-linear model is analyzed with the help of Takagi-Sugeno Fuzzy model. The fuzzy model proposed by Takagi and Sugeno is described by fuzzy IF-THEN rules which represents local input-output relations of a nonlinear system. This has motivated the work in this thesis, where an attempt has been made to study the stability of Lotka-Volterra predator-prey system with fuzzy impulsive control. The thesis has four parts, which consists of ten chapters. Part-I consists of a single chapter (chapter 1) which gives an introduction to the problems discussed in this thesis and it provides motivation to the study carried out. A survey of pertinent literature is presented to show the signi cance of the problems considered. Part-II contains three chapters, 2, 3 and 4, which deals with the stability of interaction dy namics of two and three species prey- predator system without infection. Part-III deals with the the stability of prey and predator system with infection. It consists of ve chapters, namely 5, 6, 7, 8, and 9. In all the above chapters, mathematical models are considered to study the re lationship among preys and predators. We have two, three species Lotka-Volterra predator-prey models with imprecise biological parameters. To improve the model's reality we analyze the global and asymptotic stability of this model with the help of the Takagi-Sugeno (T-S) model. The T-S impulsive control model and the fuzzy impulsive control models were used to explore the stability of the Lotka-Volterra predator-prey system. The impulsive control technique, which is analyzed in the framework of the fuzzy systems based on T-S model, is found appropriate for very complex and non-linear system with impulsive e ects. vii Part-IV consists of a single chapter 10, which presents the summary of the thesis with main conclusions and point out various problems which are yet to be solved in this area of research