DISCRETE DYNAMICAL SYSTEM AND THEIR APPLICATION

Abstract

This study of the dynamics of some ecosystem models as newlineapplication to the discrete dynamical systems consists of two parts, newlinethe first part focuses on bifurcation analysis and stability of the newlineecosystem models with two dimensional systems. This part comes in newlinethree chapters, namely chapter two, chapter three and chapter four. newlineIn chapter two, the Lotka-Voltera discrete model is investigated, the newlineexistence and stability of all possible fixed points analyze and the newlinenumerical simulation is given. In chapter three, the discrete two newlinespecies predator-prey system with Holling type II with the death rate newlineis obtained, the possible equilibrium points are observed, their local newlinestability analyzed and the bifurcation diagrams and phase portraits newlinehave been obtained for selected rang of different parameters. As newlinesome parameters varied, this model exhibit chaos as long as time newlinebehavior. Also, the fractal dimensions are presented. In chapter four, newlinethe dynamics of discrete-time prey-predator model are investigated, newlinethe result indicates that the model undergo a flip bifurcation which newlinefound by using center manifold theorem and bifurcation theory. newlineNumerical simulation not only illustrate our results, but also exhibit newlinethe complex dynamic behavior, such as the periodic doubling in newlineperiod-2, -4 -8, quasi- periodic orbits and chaotic set. newlineThe second part comprises two chapters that focus on stability newlineconditions of the ecosystem models with three dimensions. Again, newlinethis part comes in two chapters: chapter five and chapter six. In newlinechapter five, the discrete- time food chain interaction model is newlineproposed; steady state implies all possible equilibrium points, newlinestability conditions of arising equilibrium points are analyzed with newlinenumerical examples. Further, the dynamical behavior of the newlinecoexistence equilibrium point is obtained. In chapter six, the newlinediscrete- time prey-predator species with scavenger model is newlineproposed, all possible equilibrium points are found, stability newlineconditions of arising equilibrium points are analyzed with numerical newlineexamples.