A discrete mutualism model: analysis and exploration of a financial application

We perform a stability analysis on a discrete analogue of a known, continuous model of mutualism. We illustrate how the introduction of delays affects the asymptotic stability of the system’s positive nontrivial equilibrium point. In the second part of the paper we explore the insights that the model can provide when it is used in relation to interacting financial markets. We also note the limitations of such an approach

Mathematical modelling of mutualism in population ecology

This research dissertation focuses on the symbiotic interaction of mutualism, we give explanations as to what it is before mathematically modelling population dynamics of two species displaying mutualistic behaviour. Throughout the course of this dissertation, we shall be re-examining the work done in the book by Kot [16] and the paper by Joharjee and Roberts [32], whilst providing further explanations of the mathematics involved and the steps taken. We begin by constructing a model for mutualism before attempting to improve the model in order to make it more realistic. We go on to add delays to our improved model and determine the stability of its equilibrium points. We formulate models via piecewise constant arguments and via a simple Euler scheme before determining stability for both systems. A graphical comparison will then be made to explain the differences in behaviour between the two discretised systems

Computational and mathematical modelling of plant species interactions in a harsh climate

This thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction

Chaos to Permanence-Through Control Theory

Work by Cushing et al. \cite{Cushing} and Kot et al. \cite{Kot} demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species \cite{EVG}. We utilize present chaotic behavior and a control algorithm based on \cite{Vincent97,Vincent2001} to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from \cite{Harvesting}, a ratio-dependent one-prey, two-predator model from \cite{EVG} and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model \cite{Upad} and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Chaos to Permanence - Through Control Theory

Work by Cushing et al. [18] and Kot et al. [60] demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species [35]. We utilize present chaotic behavior and a control algorithm based on [66, 72] to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from [30], a ratio-dependent one-prey, two-predator model from [35] and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model [67] and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Evolution dynamics of some population models in heterogeneous environments

Spatial and/or temporal evolutions are very important topics in epidemiology and ecology. This thesis is devoted to the study of the global dynamics of some population models incorporating with environmental heterogeneities. Vector-borne diseases such as West Nile virus and malaria, pose a threat to public health worldwide. Both vector life cycle and parasite development are highly sensitive to climate factors. To understand the role of seasonality on disease spread, we start with a periodic West Nile virus transmission model with time-varying incubation periods. Apart from seasonal variations, another important feature of our environment is the spatial heterogeneity. Hence, we incorporate the movement of both vectors and hosts, temperature-dependent incubation periods, seasonal fluctuations and spatial heterogeneity into a general reaction-diffusion vector-borne disease model. By using the theory of basic reproduction number, R₀, and the theory of infinite dimensional dynamical systems, we derive R₀ and establish a threshold-type result for the global dynamics in terms of R₀ for each model. As biological invasions have significant impacts on ecology and human society, how the growth and spatial spread of invasive species interact with environment becomes an important and challenging problem. We first propose an impulsive integro-differential model to describe a single invading species with a birth pulse in the reproductive stage and a nonlocal dispersal stage. Next, we study the propagation dynamics for a class of integro-difference two-species competition models in a spatially periodic habitat

Computational and mathematical modelling of plant species interactions in a harsh climate

This thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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

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

Balance manifolds in Lotka-Volterra systems

The Lotka-Volterra equations are a dynamical system in the form of an autonomous ODE. The aim of this thesis is to explore the carrying simplex for non-competitive Lotka-Volterra systems for the case of 2- and 3-species, where it is referred to as a balance simplex. Carrying simplices were developed by M.W. Hirsch in a series of papers. They are hypersurfaces which asymptotically attract all non-zero solutions in the phase portrait. This essentially means that all the non-trivial dynamics occur on the carrying simplex, which is one dimension less than the system itself. Many of its properties have been studied by various authors, for example: E.C. Zeeman, M.L. Zeeman, S. Baigent, J. Mierczyński. The first few chapters of this thesis explores the 2-species scaled Lotka-Volterra system, where all intrinsic growth rates and intraspecific interaction rates are set to the value 1. This simplification of the model allows for an explicit, analytic form of the balance simplex to be found. This is done by transforming the system to polar co-ordinates and explicitly integrating the new system. The balance simplex for this 2-species model is precisely composed of the heteroclinic orbits connecting non-zero steady states, along with these states themselves. The later chapters of this thesis focuses on the 3-species case. The existence of the balance simplex in particular parameter cases is proven and it is shown to be piecewise analytic (when the interaction matrix containing the parameters is strictly copositive). These chapters also work towards plotting the balance simplex so it can be visualised for the 3-species system. In another chapter, more general planar Kolmogorov models are considered. Conditions sufficient for the balance simplex to exist are given, and it is again composed of heteroclinic orbits between non-zero steady states