ORDER AND CHAOS IN MULTISPECIES ECOLOGICAL SYSTEMS

Abstract

Most biological models of interacting species are nonlinear. Such nonlinear system can have several kinds of behavior not possible in linear system, including cycles, quasi-periodic, chaos. The idea that nonlinear dynamics and especially deterministic chaos play an important role in biology has become more influential in recent year. The terms quasi-periodic, chaos and strange attractor are becoming familiar to ecologists. The objective of this thesis is to obtain a better understanding of the order and chaos that occurs in models of multispecies ecological systems. Such a study is important since the natural ecological systems have all the necessary ingredients (nonlinearity, high dimensionality etc.) to be able to support chaos. A three species food chain model with ratio-dependent functional response and several food web models are investigated. Seasonality effect on the prey-predator models with ratio-dependent and predator-dependent functional response has also been studied. All these models are described by a system of ordinary differential equations. Analytical methods are used to study the stability of the models. An object oriented interactive program is developed to simulate the global dynamical behavior of the systems. Different tools for detecting chaotic dynamics e.g. bifurcation diagrams, Lyapunov exponent, Lyapunov dimension, poincare section, strange attractors are also included in the program. These tools are used extensively throughout the thesis. A brief introduction to the dynamics of ecological systems and the tools used for detecting chaos is given in chapter one. In the next two chapters a three species food chain of Holling-Tanner type ratio-dependent is investigated. The local stability analysis for each Kolmogorov subsystem and for the complete system is established. The irregular dynamical behavior is investigated through the bifurcation diagrams, Lyapunov exponents, Lyapunov dimension, and strange attractors. The model shows the complex dynamics in positive octant including chaos. The next four chapters are devoted to food webs. A food web consisting of two independent preys and a predator is investigated in chapter four. This food web is modified in chapter 5 to include the interspeciefic competition between the two preys species. Two predators competing over a single prey species is the model of chapter 6. However, chapter 7 deals with a food web consisting of a prey at the lower level, a specialist predator at the middle level, and a generalist predator at the highest level. The local stability analysis is established for all these food webs. Persistence of the system is discussed in most of these food webs. The possibility of existence of the complex dynamics, in these food webs, is detected using different tools. The final two chapters deal with the effects of periodicity on the prey-predator model. Chapter eight incorporates the effects of periodic forcing, in the intrinsic growth rate of the prey, on the Holling-Tanner ratio-dependent prey-predator system. While, in chapter nine, the effects of periodic variations on two parameters of prey-predator model with predator functional response are considered with and without phase difference. It is observed that, in both chapters, seasonality can very easily give rise to complex population dynamics.