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

Finite-difference solutions of tenth-order boundary-value problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis finite difference methods are used to obtain numerical solutions for a class of high-order ordinary differential equations with applications to eigenvalue problems. Two families of numerical methods are developed for tenth-order boundary-value problems and global extrapolations on two and three grids are considered for the special problem. Special nonlinear tenth-order boundary-value problems are solved using a family of direct finite difference methods which are adapted to solve a general linear and nonlinear boundary-value problem. These methods convert the ordinary differential equation into a set of algebraic equations. If the original ordinary differential equations are linear, the finite difference equations will give linear algebraic equations. If the ordinary differential equation are nonlinear, the resulting finite difference equations will be nonlinear algebraic equations. These nonlinear equations are first linearized by Newton's method. The methods developed are of orders two, four, six, eight, ten and twelve. The error analyses are discussed. A generalized form is given to solve a class of high-order boundary-value problems by converting the differential equation to a system of first-order equations. The method based on using a Pade rational approximant to the exponential function for general boundary-value problems is applied to a tenth-order eigenvalue problem associated with instability in a Benard layer and numerical results are compared with asymtotic estimates appearing in the literature. This method may be implernented on a parallel computer. The method is extended to a twelfth-order eigenvalue problern in an appendix. The algorithms developed are tested on a variety of problems from the literature. The REDUCE package is used to obtain the parameters in the numerical methods and all computations are carried out on a Sun Workstation at Brunel University using Fortran 77 with double precision arithmetic.This study is funded by the Ministry of Education of Pakistan, Islamabad

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

Some new static anisotropic spheres

In recent years, a number of authors have found solutions to the Einstein field equations for static gravitational fields with anisotropic matter. The models generated are used to describe relativistic spheres with strong gravitational fields. It is for this reason that many investigators use a variety of techniques to attain exact solutions. The exact solutions may be used to study the physical features of charged spheroidal stars. I found a new class of exact solutions to the Einstein field equation for an anisotropic sphere with the particular choice of the anisotropic factor A= pc — pr, the difference between the radial and the tangential pressures of the fluid sphere and one of the gravitational potential Z. The condition of pressure anisotropy was reduced to a recurrence equation with variable, rational coefficients which can be solved in general. Consequently the exact solutions to the Einstein field equations corresponding to a static spherically symmetric gravitational potential was found in terms of series. I generated two linearly independent solutions by placing restriction on parameters arising in the general series. Some brief comments relating to the physical features of the model are also made

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP