COMPLEX DYNAMIC BEHAVIOR IN SOME NON-LINEAR CONTINUOUS SYSTEMS

Abstract

Over the years, ordinary differential equations (ODEs) have been used to develop mathematical models that represent the dynamics of predator-prey species. Comprehending and analyzing these intricate dynamics is essential for making precise predictions and developing effective management strategies for ecosystems. In this thesis, dynamics of some non-linear multi-species food chains and food-web models are developed and evaluated to investigate the dynamics with different types of interactions among the species. The work of the thesis is organized in six chapters. The chapter 1 covers the fundamental concepts of predator-prey systems as well as the mathematical techniques used to examine the existence, stability and bifurcation of these systems. In the last section, the summary of the thesis is appended. Chapter 2 focuses on the modified Hasting-Powell food chain incorporating additional food for the middle predator with Beddington-DeAngelis functional response. The four species food web consists of two prey species, intermediate predator and a top predator. For intermediate predator a modified Holling type-II functional response is considered as it predates over two prey species. It is found that the food web model is well-posed, bounded, and dissipative. The key dynamic aspects of the model are examined from the perspective of local stability. Conditions for persistence are developed and the survival of all four species is investigated. In addition to establishing local stability at various equilibrium points and feasibility, we have also quantitatively investigated the control of chaos in the system. The numerical simulations exhibits the persistence as a chaotic attractor or stable node or limit cycle. It is considered that feeding the intermediate predator with additional food might help to keep the chaos under control. With more food available, the top predator may escape extinction. In chapter 3 a three-dimensional predator-prey model with stage structure(immature and mature) in prey is considered. The model incorporates the maturation delay. It is assumed that predators are not consuming immature prey and the interaction between adult or mature prey and predator species is considered by Beddington-DeAngelis functional response. The dynamic behavior of the system is investigated using both analytical and numerical methods. A detailed analysis of the stabilities of equilibrium points is carried out. The delayed model exhibits Hopf-bifurcation. Furthermore, resulting in periodic oscillation(limit cycle) with time. Lastly, the analytical results are ix verified against the numerical results to show consistency with the theoretical analysis. Chapter 4 investigates the effects of additional food and linear harvesting of predator on the prey species. Monod-Haldane type functional response is considered for the prey predator interaction. The existence of distinct equilibrium states and their stability is analyzed. Numerical Simulation are used to elaborate the results showing that the presence of additional food plays a vital role in the existence of equilibrium points. If the quality of additional food is very high in comparison to prey, it would lead to instability of coexistence equilibrium point. Moreover, system shows bistability. The predator may extinct even in the presence of additional food. Chapter 5 examines the dynamics of a three species food chain model considering the fear effect of middle predator in the prey. The middle predator is also getting the additional food. It is assumed that the reproduction rate of prey is decreased due to the fear of predator. Prey ingestion by the predator is considered to follow the simplified Holling Type-II functional response. The local stability of equilibrium point are investigated. Numerical studies are performed to validate all of the results. Further, it was observed that chaos can also be controlled with the supply of constant additional food. Similar results is obtained for fear also. The thesis concludes with a review of findings, conclusions and future study directions in the context of predator-prey systems in Chapter 6.