Bifurcation and Chaos in a Discrete Fractional Order Prey-Predator System Involving Infection in Prey

Bifurcation and Chaos in a Discrete Fractional Order Prey-Predator System Involving Infection in Prey

A. George Maria Selvam (Sacred Heart College (Autonomous), India) and R. Dhineshbabu (Sacred Heart College (Autonomous), India)
Copyright: © 2020 |Pages: 25
DOI: 10.4018/978-1-7998-3741-1.ch005

Abstract

This chapter considers the dynamical behavior of a new form of fractional order three-dimensional continuous time prey-predator system and its discretized counterpart. Existence and uniqueness of solutions is obtained. The dynamic nature of the model is discussed through local stability analysis of the steady states. Qualitative behavior of the model reveals rich and complex dynamics as exhibited by the discrete-time fractional order model. Moreover, the bifurcation theory is applied to investigate the presence of Neimark-Sacker and period-doubling bifurcations at the coexistence steady state taking h as a bifurcation parameter for the discrete fractional order system. Also, the trajectories, phase diagrams, limit cycles, bifurcation diagrams, and chaotic attractors are obtained for biologically meaningful sets of parameter values for the discretized system. Finally, the analytical results are strengthened with appropriate numerical examples and they demonstrate the chaotic behavior over a range of parameters. Chaos control is achieved by the hybrid control method.
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Introduction

Epidemiology and ecology are two important research fields of mathematical biology. The first breakthrough work of Lotka (Lotka, 1925) and Volterra (Volterra, 1926) model with coupled nonlinear continuous prey predator systems are discussed in the modern mathematical biology. Moreover, the study of epidemiology and ecology is combined as eco-epidemiology. The dynamics of interacting species in prey predator model with disease spreads are analyzed in eco-epidemiology. Eco-epidemiology population growth dynamics with spread of diseases due to parasites and viruses are efficiently taken care of by complex nonlinear mathematical models. These models are also used to determine steady states, their local stability, periodic solutions, various types of bifurcation diagrams and nature of chaotic attractors (if exist). A huge number of statistical and mathematical models have been used in the study of eco-epidemiological models.

Mathematical models play a vital role in understanding the spread and control of a disease. Mathematical models are apt to model realistic situations like dynamics in population biology, bacterial or viral growth, population of an endangered species, a human population involving its age distribution or not and so on (Edelstein Keshet, 2005). Ecology, the study of relationship between environment and their species is now an enormous field considering competition and prey - predator interactions, multi-species societies, evolution of pesticide resistant strains, ecological and plant - herbivore models. There is also a lot of practical applications for single-species in the biomedical sciences.

In the past years, especially, the study of prey-predator models has been a hot topic for many researchers. Differential equations have played an important role in the investigation of prey-predator interactions and it will continue as an efficient tool in forthcoming explorations. The infectious diseases in population dynamics in a prey-predator system has been garnering great attention because natural species do not exist in isolation, the development of diseases appears due to interaction with other species for food or they are predated by other species. The authors (Prasenjet et al., 2014) studied the prey predator with infectious disease in the prey population or predator population. The nonlinear feedback controls, positive controls have been applied by many researchers to control the chaos in prey-predator system. The adaptive control methods are rapidly developing and finding its application in various fields such as electrical engineering, ecological systems, neural networks and others.

Recently, many novel forms of models have been developed to investigate the natural and social processes that enlarge over time. Now-a-days, these models are referred to as dynamical systems (Frederick, 2006). Dynamical systems are divided into two general categories, i.e. deterministic models and stochastic models respectively. Deterministic models are employed when the number of quantities involved in the process being modeled is relatively small and all the underlying scientific principles are fairly well understood. The tools of dynamical systems enable the researchers to better understand and investigate the different nonlinear characteristics, which exhibit new phenomena of the systems from various disciplines. Particularly, the tools of the dynamical models such as the applied bifurcation theories are successfully employed to discuss the qualitative behavior of nonlinear systems. This includes the enquiry in to existence of steady states and their stability and bifurcations of periodic orbits, chaotic attractors and synchronization.

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