DYNAMICAL BEHAVIOR OF NONLINEAR MODELS IN ECOLOGICAL SYSTEMS

Abstract

The dynamics of nonlinear systems that occur in ecological systems has attracted the attention of mathematicians. The mathematical answers to the ecological issues of survival, existence, extinction and persistence lie in the stability of equilibrium points, limit cycles, and complex behavior of solutions of underlying nonlinear dynamical models consisting of system of coupled differential / difference equations. In this thesis, the complex dynamical behavior of some multi-species ecological systems has been investigated. Due to multi level interactions in food webs underlying nonlinear equations have complex dynamical behavior: Limit cycle, quasi-periodic behavior and chaos. The first chapter gives a brief introduction to the dynamics of ecological systems. A Brief discussion on tools / techniques used in the study is included. It also gives the review of work done in this area. The second and third chapters investigate the dynamical behavior of a food web with two prey communities and one predator species. In the model, the logistically growing preys with modified Holling type functional response are considered. The predator equation is written according to the Leslie-Gower scheme in which the conventional carrying capacity term is being replaced by the renewable resources for the predator. The model is analyzed mathematically. Global behavior is simulated numerically for biologically feasible choice of parameters. The biological parameters are varied to investigate the existence of non-chaotic and quasi-periodic behavior. Typical phase diagrams are obtained for suitable choice of parametric values. The long time persistence of the species is investigated under the biological feasible range of parameters. The existence of Hopf bifurcation has been established. Numerical simulations are carried out to show the loss of stability in a given range of the key parameter. The persistence in the form of local and global stability is investigated. The persistence in the form of periodic solution is observed. The fourth chapter deals with the effects of seasonal variation on prey predator interactions. In the first model the predator is having Leslie- Gower type growth. Asynchronous periodic variations on growth parameters are considered. Supercritical Hopf s bifurcation has been investigated with respect to a key parameter. The bifurcation diagrams show the passage to chaos through a sequence of period doublings. In the second model, the toxin producing phytoplankton-zooplankton interactions are considered. The co-existence of phytoplankton and zooplankton species in the form of stable non trivial equilibrium point has been obtained. Chaos and quasi periodic behavior is observed in the system which may be responsible for bloom. The fifth chapter is devoted to a food web consisting of two competing predators and a prey. The analysis of mathematical model shows that the persistence is not possible when any one of the boundary prey-predator planes has a stable equilibrium point. However, numerical simulations exhibit persistence in the presence of periodic solutions in the boundary planes. The system exhibits quasi-periodic behavior in the positive octant. ii The model in the sixth chapter considers the mutation between the two predator species. The analysis of the model shows that the strong persistence is possible for two mutating predators sharing a single prey species. The existence of limit cycle with the help of Hopf bifurcation theorem is established. The system exhibits quasi-periodic behavior in the positive octant. The last chapter considers harvesting in two types of models. The first model considers harvesting of predator feeding on two independent prey populations. Numerical simulation experiment depicts that the harvesting can control the chaos in the food web. The optimum and economic equilibrium of this food web have been obtained The Pontryagin's maximum principle has been used to optimize the net revenue. Numerical Computations are carried out to obtain the economic equilibrium for given set of parameters. In another model, two harvesting species are considered. The dynamics of effort is included in the model. The optimum equilibrium point is obtained. The stability of positive nonzero equilibrium point is established. List of references is appended at the end of thesis. iii