DYNAMICAL BEHAVIOUR OF SOME ECOLOGICAL AND EPIDEMIOLOGICAL MODELS

Abstract

This thesis is an attempt to analyze mathematical models with delay, seasonal variations and / or impulsive inputs applicable to dynamical systems in ecology, epidemiology and eco- epidemiology. The stability of dynamical models has been investigated and bifurcation analysis has been carried out. The numerical simulations are carried out to explore the possibility of chaos and complexity in nonlinear models. The efforts have been made to interpret the mathematical results. Biological relevance of the mathematical results has been explored. This thesis comprise of seven chapters. The chapter wise summary of the thesis is given below: The first chapter gives a brief introduction to the dynamics of ecological, epidemiological and eco-epidemiological systems. The related concepts are overviewed in this chapter. Brief discussion on tools / techniques used in the study is also included. It also gives the review of work done in this area. In the second chapter, the dynamical behaviour of toxin producing phytoplankton (TPP) and zooplankton has been investigated. Toxin substance as well as toxin producing phytoplankton affect the growth of zooplankton population and have a great impact on phytoplankton zooplankton interactions Mathematical models of the dynamics of viral infected phytoplankton population are rare as well. In this chapter, a prey—predator model for the toxin producing phytoplankton (TPP) and zooplankton system with the assumption that the viral disease is spreading only among the prey species, has been proposed and, though the predator feeds on both the susceptible and infected prey, the infected prey is more vulnerable to predation as is seen in nature. The dynamical behaviour of the system is investigated from the point of view of stability and bifurcation. The solution is quasi-periodic at higher values of rate of infection which is responsible for bloom. In third chapter, an SIRS epidemic model with pulse vaccination has been proposed. The model incorporates non-monotonic incidence rate of saturated mass action. The iii infection free periodic solution of impulsive system has been obtained and is found to be globally asymptotically stable when R0 <1. The supercritical bifurcation occurs at the threshold R0 =1 . The numerical simulations have been carried out to study the influence of other key parameters. The system shows complex dynamical behavior with respect to the parameter p , the fraction of susceptible that are vaccinated. In the forth chapter, deals with impulsive harvesting in prey predator system. Conditions for existence and global stability of predator free equilibrium solution have been established. The existence of nontrivial periodic solution is established when the predator free periodic solution loses its stability. Numerical simulations of the impulsive system exhibit the rich complex dynamics in the positive octant for the key parameters. The model in the fifth chapter considers the effects of seasonality in delayed prey predator system with Beddington—DeAngelis functional response. The basic problem is to understand the relationship between the magnitude of the seasonal variations and the complexity of the system. The existence of Hopf bifurcation has been established. The numerical simulations have shown that seasonal growth and delay can give rise to variety of attractors including periodic, quasi-periodic as well as chaotic oscillations. In the chapter 6 the effects of carrier dynamics on epidemics have been studied. The density of carrier population increases due to environmental factors like temperature, rain and human population related factor such as water pollution, plantation of vegetation. In this chapter, we deal with two carrier dependent epidemic models. Dynamical model for the carrier dependent infectious diseases with non-linear rate is discussed incorporating the environmental and human population related factors and an SIS model incorporating delay in spread of infection has also been investigated. In both the models the existences of carrier free iv equilibrium point and non-trivial equilibrium point have been established. In delayed model existence of Hopf bifurcation with respect to delay parameter have been explored. In undelayed model non-linear incidence rate has been taken into account and global behaviour of positive equilibrium point has been established. The last chapter is about the achievements and future scope of this work. v