INVESTIGATION OF REGULAR and CHAOTIC BEHAVIOUR IN SOME BIOLOGICALLY INTERACTIVE MODELS

Abstract

This thesis provides an examination of chaotic and regular patterns in some newlinebiologically interactive models through the conceptual lens of chaos theory. Chaos newlinetheory is chosen as an analytical tool because it allows us to reveal the patterns and newlineprocesses of complex systems as they move between order and disorder. newlineThe central question is that of how complexity, which is based in chaos theory, can newlinehighlight the ways to address macro and micro level problems in complex newlinebiological systems. newlineFour biological systems are discussed in terms of complexity. The first system newlineunder study was a three-tropic food chain model that has been investigated by newlinediscretizing the classical mathematical model for three-trophic food chains newlinemodified and used by Bo Deng (2006). Famous Euler s method has been employed newlineto discretize the differential equations used in the work of Bo Deng. Various newlinemeasurable quantities for emergence of chaos, like Lyapunov exponents, newlinetopological entropies, correlation dimensions, have been numerically calculated newlineand represented through plots. Finally, the chaos indicator, named Dynamic newlineLyapunov Indicator (DLI), has been used to identify clearly chaotic and regular newlinemotion. Bifurcation diagrams and various plots for LCEs, topological entropies, newlinecorrelation dimension etc. are interesting and provide help to properly analyze the newlineevolutionary behavior. newlineAnother problem on dynamics of two-gene Andrecut-Kauffman system has been newlinestudied for chaos and complexity. The model consists of nonlinear equations, in newlinecontext with biochemical phenomena obtained from chemical reactions appearing newlinein a two-gene model (Andrecut and Kaufmann, 2007). The chemical reactions are newlineassumed to correspond to gene expression and regulation. For this problem, studies newlinehave been performed carefully to understand chaotic phenomena during its newlineevolution together with complexities present in the system. newlinexv newlineThe third problem is based upon complexities in a Plant-Herbivore system. The newlinestudied work is based on the non-dimensional mathematical model proposed in a newlinerecent article (Abbott and Dwyer, 2007). Mathematical analysis and simulations of newlinethis model provide us with biological insights that may be used to devise control newlinestrategies to regulate the population of the herbivore. Since the herbivore newlinemovements are random, it is more appropriate to study a stochastic model instead newlineof deterministic one. Such realistic plant-herbivore system, would be our future aim newlineof investigation. newlineLastly, we have worked with a single-species model with stage structure for the newlinedynamics in a wild animal population for which births occur in a single pulse once newlineper time period. The measures, like Lyapunov exponents, topological entropies and newlinecorrelation dimensions, are obtained for this problem for discussion of evolutionary newlinephenomena. In the processes of study, we have discussed the stability criteria of the newlinesteady state solution. newlineIt is concluded that complexity, based on chaos theory, is a powerful framework for newlineunderstanding the interactions in biological systems. Chaos theory provides us with newlinemany tools which can indicate regular and turbulent patterns which could be of newlinegreat use if we wish to make some changes in the existing systems. This can answer newlinevarious questions regarding existence and extinction of various species or the newlineimpact of their social interactions on the environment. newline newline