Nonlinear Mathematical Models: Theory and Methods

Abstract

Understanding the process that controls the connections between species and the transmission of diseases is a critical task in a complex network of natural and social systems. This thesis presents significant results related to these complex systems to better understand their dynamics, emphasizing the dynamics of infectious diseases and the connections between prey and predator. The motivation behind our study is the fundamental need to understand the complex dynamics that determine the growth and decrease in populations and the sensitive equilibrium of the system. Prey-predator interactions, with complex mathematical models control, provide insight into the complex balance of life in the natural environment. Our goal is to provide practical outcomes with broad implications for animal conservation and management while extending the boundaries of ecological knowledge by exploring the theoretical foundations of dynamical systems. In addition, the epidemic models help to strengthen international health systems as a response to the ongoing threat of infectious diseases. These models offer an opportunity to investigate the critical points that may cause epidemics or the emergence of stable, controlled states through the complex combination of parameters and variables they possess. Our goal is to contribute to managing and maintaining viral diseases by providing knowledge and methods that go beyond theoretical research and into the real world. This will ultimately assist societies throughout the world. This thesis employs computational simulations, mathematical exploration, and practical applications. Our motivation arises from the conviction that extensive investigation into the dynamics of dynamic systems may contribute to more sensible approaches to preserving biodiversity, appropriate ecosystem management, and maintaining the stability of the global community. Let us give an overview of the topics that each chapter covers. Chapter 1. Introduction and preliminaries. The main focus of this chapter is to provide readers with the fundamental concepts of dynamical systems used in this thesis. This chapter will present the basic definitions, the conditions for fixed point stability, bifurcation analysis, chaos control, and the related literature. Chapter 2. Fixed points stability, bifurcation analysis, and chaos control of a Lotka-Volterra model with two predators and their prey. The study of the population dynamics of a three-species Lotka-Volterra model is crucial in gaining a deeper 6 understanding of the delicate balance between prey and predator populations. Therefore, in this chapter, we discussed the stability of fixed points and the occurrence of Hopf bifurcation in a three-dimensional predator-prey model. Using bifurcation theory, our study provides a comprehensive analysis of the conditions for the existence of Hopf bifurcation. This is validated through detailed numerical simulations and visual representations demonstrating the potential for chaos in these systems. To mitigate the instability, we employ a hybrid control strategy that ensures the stability of the controlled model even in the presence of Hopf bifurcation. This chapter is significant in advancing the field of ecology but also has far-reaching practical implications for wildlife management and conservation efforts. Our results provide a deeper understanding of the complex dynamics of prey-predator interactions and have the potential to inform sustainable management practices and ensure the survival of these species. Chapter 3. Fixed points stability, periodic behavior, bifurcation analysis, and chaos control of a prey-predator model incorporating the Allee effect and fear effect. In this chapter, we analyze the dynamics of a two-dimensional prey-predator model that incorporates the Allee and fear effects. We conduct stability analysis of the fixed points in discrete and continuous forms and focus on the periodic behavior of the discrete-time model. In addition, we discussed the bifurcation behavior of discrete and continuous models using bifurcation theory and presented numerical examples to validate our theoretical findings. We also identified the direction of bifurcation using attractive bifurcation plots and employed a simple control technique to avoid bifurcation. This chapter contributes to a better understanding of the prey-predator system and has implications for other complex systems in various fields, including population dynamics, physical models, epidemiology, and economics. Overall, this chapter reveals additional illumination on the prey-predator model’s dynamics and increases our understanding of its dynamic behavior. Chapter 4. Fixed points stability, multi-parameter bifurcation analysis, and chaos control of a prey-predator model incorporating the Allee effect and fear effect. This chapter presents the dynamic analysis of the prey-predator model by adding the fear and Allee effects. We also present the stability, bifurcation analysis, and chaos control of the model. From the numerical examples, we conclude that the crowding effect should be minimized to maintain the stability of the model. Also, in the interior fixed point, when fear and Allee effects are taken as bifurcation parameters, backward bifurcations occur, which shows that in the presence of the crowding effect, the increase of the fear effect stabilizes the model. 7 Similarly, the more significant Allee effect stabilizes the model. While the decrease of these two effects causes an increase in growth rate, which causes bifurcation in the system due to overcrowding, the addition of the Allee effect and the fear effect should be, to a certain extent, so that the excess of both impacts controls the crowding development. A simple control method is employed to prevent bifurcation. This chapter improves our comprehension of the prey-predator system while potentially having implications for other complex systems in various fields, including population dynamics, epidemiology, and economics. Chapter 5. Fixed points stability, bifurcation analysis, and chaos control of an epidemic model with vaccination and vital dynamics. The spread of infectious diseases remains a significant threat to global health and stability. A crucial aspect of controlling and mitigating the impact of these diseases is a detailed understanding of their dynamics. This chapter thoroughly examines a discrete-time epidemic model’s stability and bifurcation characteristics, considering vaccination and vital dynamics. We may better understand the system’s behavior presented in this chapter with mathematical techniques from nonlinear dynamics. From studying the stability of fixed points, we can learn much about how the system responds to parameter changes and the circumstances needed for profound disease control. Additionally, investigating bifurcation occurrences provides a more detailed relationship between small parameter changes and qualitative changes in system behavior. The complex interactions between various parameters and their effects on the system’s dynamics are mainly illustrated in the study of one-parametric bifurcation and co-dimension two-parameter bifurcation. This chapter also shows how crucial chaos control is in modeling epidemics. Managing the chaos in the system is an essential tool for preventing the spread of infectious diseases and ensuring long-term disease control. All the chapters contain theorems related to a qualitative analysis of different models, for which we have provided proofs. The bibliography section gives all of the references used in this study. 8