• Mathematical Modelling and it's Applications in Biology, Ecology and Population Study

      Forrest-Owen, Owain (University of Chester, 2016-09-12)
      This thesis explores the topic of mathematical modelling involving the simulation of population growth associated with mathematical biology and more specifically ecology. Chapter 1 studies how populations are modelled by looking at single equation models as well as systems of equation models of continuous and discrete nature. We also consider interacting populations including predator-prey, competition and mutualism and symbiosis relationships. In Chapters 2 and 3, we review stability properties for both continuous and discrete cases including differential and difference equations respectively. For each case, we examine linear examples involving equilibrium solutions and stability theory, and non-linear examples by implementing eigenvalue, linearisation and Lyapunov methods. Chapter 4 is a study of the research paper - A Model of a Three Species Ecosystem with Mutualism Between The Predators by K. S. Reddy and N. C. Pattabhiramacharyulu [32]. Here, we study the basic definitions and assumptions of the model, examine different cases for equilibrium solutions, prove global stability of the system and implement numerical examples for the model before reviewing existence and uniqueness and permanence properties. In Chapter 5, we construct a discrete scheme of the model from Chapter 4. We do this in two ways, by using Euler's method to create one autonomous time-invariant form of the system, and utilising the method of piecewise constant arguments implemented in [6] to establish another autonomous time-invariant form of the system. For both discretisations, we study equilibrium solutions, stability, numerical examples and existence and uniqueness, and permanence properties. Finally, we conclude the findings of the thesis, summarising what we have discovered, stating new questions that arise from the investigation and examine how this work could be taken further and built upon in future.