ORDER, CHAOS, CONTROL AND SYNCHRONIZATION IN NONLINEAR ECOLOGICAL SYSTEMS

Abstract

Nonlinearity is all pervading in the ecological systems. In the recent times increasing attention has been focused on exploring real technological applications of nonlinear dynamics in ecology: emergence of chaos, controlling of chaos, synchronization of chaos to name but a few. Numerous mathematical ideas and techniques have been used to study nonlinear ecological systems and these , in turn have enriched the field of mathematics itself. In this thesis, attempts have been made to investigate the ecological system with respect to several factors that may be responsible for emergence of complex behaviors. Some mathematical models incorporate time delay(s) such as delay due to matura-tion, gestation and other kinds of negative feedback. The underlying system of delay differential equations are much more challenging to analyze compared to correspond-ing system of ODE's. These nonlinear delay models have lead to complexities in the system. The emphasis is to explore the complex dynamical behaviors including chaos in ecological models with respect to different control parameters. An effort has also been made to synchronize and suppress the complexity/chaos occurred in the systems. Chapter 1 gives a brief overview of the basic concepts related to the dynamics of ecological systems. Brief discussions on relevant tools/techniques are included. It also gives the review of literature related to this work. In chapter 2, the effects of an additional predator to top prey on Hastings Powell food chain dynamics has been studied. The proposed four species food web model is investigated to obtain different conditions for which system exhibits stability around various biologically feasible equilibrium points. The permanence of system is dis-cussed and global stability of an equilibrium point is established using geometrical approach of Li and Muldowney. Varieties of dynamical behaviors in the food web are possible depending upon the sharing of food between the two predators of the top prey. It is observed through numerical simulations that the addition of suitable predator may control the otherwise chaotic three species food chain, A delayed virally infected, toxin producing phytoplankton (TPP) and zooplankton system is investigated in chapter 3. The Hopf bifurcation analysis suggests that the otherwise stable system can be destabilized by the introduction of delay due to gestation. The infection in phytoplankton may control the chaos in the system when zooplankton predates only infected TPP. The numerical studies have revealed that delay can give rise to complex dynamical behaviors. In chapter 4, another dynamical model is proposed to study the delay effect on toxin producing phytoplankton-zooplankton system with Honing type-II functional response. The introduction of delay due to gestation in the model may lead to insta-bility in the system. A Hopf bifurcation analysis has been performed with respect to discrete delay which demonstrates the periodic solutions in the system. Global exis-tence of the periodic orbits is established. Properties of periodic solutions have also been determined using normal form theory and center manifold arguments. Numeri-cal simulations suggest that the system has rich dynamical behavior including chaos and limit cycles, which indicates the occurrence of algal blooms in TPP-Zooplankton system. In chapter 5, a modified Leslie-Gower type prey-predator model with Holling-type II functional response is investigated with two delays. The occurrence of Hopf bifurcation is shown for possible combinations of both delays. The combined effect of two delays has been observed and a two-parametric bifurcation diagram with respect to discrete delays is drawn. Accordingly, a domain of stability is obtained. Numerical ii iv