Discrete Nonlinear Planar Systems and Applications to Biological Population Models

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates

A family of tridiagonal pairs and related symmetric functions

A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order $q-$difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.Comment: 17 pages; LaTeX file with amssymb. v2: few minor changes, to appear in J.Phys.A; v3: Minor misprints, eq. (48) and orthogonality condition corrected compared to published versio

Higher-order finite-difference methods for partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients

On a differential algebraic system structure theory

International audienceThe structure at infinity and the essential structure are two control theory notions which were first defined for linear, then for the so-called affine, systems. And they were shown to be useful tools for the study of the fundamental problem of noninteracting control. They also appeared as related to the solutions of other important control problems such as disturbance decoupling. Their definitions are however entirely in terms of an algorithm, namely the so-called structure algorithm. The present work proposes new definitions with some advantages: they extend the class of systems from linear and affine systems to systems which may be described by algebraic differential equations, they are not tied to specific algorithms, and finally they provide more information on system structure. Let a system be a set of differential equations in variables which are grouped as m inputs, p outputs and n latent variables. To each input component is attached a rational integer, which, for a single input single output system defined by a single differential equation, is the difference between the order in the output and the order in the input of the differential equation defining the system. The m-tuple of these rational integers is the new structure at infinity of the system. Associated to the structure at infinity is also defined a p-tuple of rational integers representing a new notion of essential structure. The old structure at infinity is shown to be recoverable from the new one. Computations of system structure based upon the suggested definitions are quite complex. The present paper focuses on proofs of algorithms which attempt to reduce the complexity of these computations

Dynamics of a rational system of difference equations in the plane

We consider a rational system of first order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions and the stability of equilibria

On projective systems of rational difference equations

We discuss first order systems of rational difference equations which have the property that lines through the origin are mapped into lines through the origin. We call such systems projective systems of rational difference equations and we show a useful change of variables which helps us to understand the behavior in these cases. We include several examples to demonstrate the utility of this change of variables