SPATIO-TEMPORAL BEHAVIOR OF SOME ECOLOGICAL SYSTEMS

Abstract

In nature, the ecological communities exhibit a very complex dynamical behavior. Such complex dynamical behavior of ecological systems have attracted the attention of many mathematicians since the days of Lotka and Volterra. Ecological modelling has important role in understanding the dynamics of interaction among the species and their interaction with the environment. In this thesis, an attempt has been made to study the spatio-temporal behavior of some ecological systems. Some predator-prey, food chain and food web models have been proposed and investigated. The effect of refuge on a predator-prey system is also studied. The first chapter gives a general introduction of the spatio-temporal dynamics of ecological systems. Some mathematical preliminaries which are required for the study are discussed. In addition, a brief literature review is given. The second and third chapters deal with the analysis of the spatio-temporal be-havior of a predator-prey system under homogeneous Neumann boundary conditions. The Beddington-DeAngelis functional response with and without prey refuge is con-sidered. The predator equation is written according to the modified Leslie-Gower scheme in which the conventional carrying capacity term is being replaced by the renewable resources for the predator. The local asymptotical stability of the biologi-cally feasible equilibrium points and the global stability of the coexistence equilibrium state in the presence as well as absence of diffusion are discussed. Turing bifurcation analysis, which relies on significant differences in between predator and prey diffusion coefficients, is carried out and Turing space is obtained. In this space Turing patterns iii iv are formed. From the numerical simulation results, it is observed that in the absence of refuge the self-replicating spatial patterns are formed. Whereas in the presence of prey refuge stripe patterns, spotted patterns and the coexistence of the two is ob-served. Numerical simulations are also performed for the case when diffusivity is the same for both the species, which exclude the possibility of Turing instability; there-fore, all patterns should be ascribed to non-Turing patterns. Regular spatio-temporal oscillatory patterns, non-stationary irregular patterns and spatio-temporal chaos are obtained. The numerical simulations reveal the dependence of the patterns on the choice of initial distribution, reaction parametric values and the diffusive ability of the two species. 'File existence of non-constant positive steady states is also investigated by using Leray-Schauder theorem. The fourth chapter is devoted to a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response and diffusion under homoge-neous Neumann boundary conditions. Sufficient conditions for Hopf and Turing bifur-cations are derived. The emergence of spatial patterns is presented through theoretical analysis and numerical simulations. The results of numerical simulations reveal the formation of labyrinth patterns, and the coexistence of spotted and striped patterns. The fifth chapter investigates the spatio-temporal behavior of a three level food web system consisting of prey, a specialist and generalist predator. The boundedness of the solution of the system, the local and global stability analysis of the non-negative steady states of the corresponding temporal system as well as the spatio-temporal system are discussed. Extensive numerical simulations are carried out in a two-dimensional space. Spiral patterns and spatio-temporal chaotic patters are obtained. Sufficient conditions for the existence of non-constant positive steady state solution are given with the help of Leray-Schauder theorem. In the sixth chapter, a spatial three species food web system with two independent preys and a predator is considered. The preys are assumed to grow logistically. The predator follows the modified Leslie-Gower dynamics and feeds upon the prey species V according to Holling Type II functional response. A sufficient condition for the local stability of the constant positive steady state of the corresponding temporal system is given. In addition, the local as well as global stability of the constant positive steady state of the spatio-temporal system are discussed. Lastly, the existence and non-existence of non- constant positive steady states are investigated and sufficient conditions are obtained. The last chapter addresses the conclusions and future scope of the work. List of references and list of publications are appended at the end of the