MATHEMATICAL MODELING AND OPTIMAL CONTROL TO BIOLOGICAL MODELS

Abstract

The thesis entitled “Mathematical Modeling and Optimal Control to Biological Models” comprises of six chapters followed by conclusion, future scope and a bibli ography. The abstract at the start of each chapter provides a quick overview of the research work done in that chapter. The core objective of this dissertation is to con struct and establish mathematical models of crop, pest, and natural enemies of pests, with a strong attention on pests’ detrimental effects on crops. As a consequence, This allows us to explain, recommend, propose, and provide the best pest treatment ap proach and optimal pest control technique needed to remove or reduce pest density while enhancing agricultural output. The following is the framework of the thesis: Chapter 1 introduces the motivation, biological backgrounds, a mathematical model, the relevance of functional responses, several ways for obtaining stability, the concept of optimal control, and a numerical methodology. Chapter 2 addresses the concepts, methodologies, and implementations of mathe matical models in farming. A mathematical study of two prey and one predator model has been performed in agriculture. Furthermore, an ecosystem consists of two prey and their predator; here, the prey–I, such as sugarcane crops, which require more time to develop, and the prey–II, such as vegetables, which have a shorter lifespan, are cultivated alongside sugarcane crops, with predators harming both prey–I and prey–II. The actual data of some parameters and the experimental data of other pa rameters have been used for model verification. The content of this chapter is pub lished in the form of a research paper entitled “A Prey–Predator Model Approach to Increase the Production of Crops: Mathematical Modelling and Qualitative Analysis” in International Journal of Biomathematics (World Scientific). In chapter 3, a mathematical model of an ecological perspective of prey, pest, and natural enemies of pest is addressed. The existence and stability of the steady–state xv conditions of various equilibrium points are studied. Furthermore, in the presence of the control variable, a mathematical model is formed for designing optimal pest control problems and studying the effects on crop pests. Then, the existence, char acterization, and necessary conditions of the optimal control are determined using Pontryagin’s maximum principle. Numerical simulations are then used to validate ana lytical results and to depict a better approach. The content of this chapter is published in the form of a research paper entitled “Study of a Prey–Predator Model with Pre venting Crop Pest Using Natural Enemies and Control” in American Institute of Physics, Conference Proceedings. In Chapter 4, the effectiveness of additional food has been investigated in this study of the prey–predator interaction. Providing additional food to predators has been con sidered significant to balance the biological system and ecosystem. A mathematical model of a prey-predator ecology is provided, which contains a crop, a susceptible pest, an infected pest, and a natural enemy of the pest. Further, the dynamic behav ior of the framework, the description of steady–state equilibrium behavior, and pest control are discussed. The basic reproduction number and sensitivity analysis are ad dressed to determine the most influential parameters. Furthermore, a comprehensive analysis of the optimal control strategy is performed. Pontryagin’s maximum principle is used to develop an optimal strategy for pest control. Finally, numerical simulations are carried out to support the analytical results and to explain various dynamic sys tems that are used in the model. The results of this chapter are in the form of a communicated research paper entitled “Study of Prey Predator System with Addi tional Food and Effective Pest Control Techniques in Agriculture”. Chapter 5 develops a mathematical model for describing the dynamics of the banana– nematodes and its pest detection method to help banana farmers. There are two criteria that are addressed: the mathematical model and the type of nematode pest management technique. The sensitivity analysis, local stability, global stability, and the dynamic behavior of the mathematical model are performed. Further, the math ematical model for optimal control is developed and discussed. This mathematical model describes various management strategies, such as the initial release of infect ed predators and the destruction of nematodes. Theoretical results are demonstrated and validated via numerical simulations. The content of this chapter is published in the form of a research paper entitled “A Prey Predator Model and Control of a Ne xvi matodes Pest Using Control in Banana: Mathematical Modelling and Qualitative Analysis” in International Journal of Biomathematics (World Scientific). Chapter 6 presents an interaction between the prey predator model consisting of three species: crop, pest and locust swarms. Under specific circumstances, all possi ble existence of the biological equilibrium points of the model is described. To study the dynamics of the system, the local asymptotic stability of several equilibrium points is illustrated. Different criteria are addressed for the coexistence of equilibrium solu tions. Further, we present numerical results to illustrate some biologically important circumstances. This study investigates the appropriate use of management mea sures to reduce the spread of the swarm through optimal control techniques. Two types of control variables are used: first, the application of pesticide, and second, the application of creating awareness. The content of this chapter are in the form of a communicated research paper entitled “Preventing the Spread of Locust Swarm and Pest in Agriculture: Mathematical Modeling and Qualitative Analysis”. Subsequently, the conclusion of the work carried out in the thesis is presented. We have also discussed the future scope of the current work.