Difference Equations and Discrete Dynamical Systems

Preface

Contents

A model of competition between multiple populations in discrete time is proposed and studied. It is assumed that individuals within a single population are identical and therefore there is no structuring variable within each population. Under the assumption that population growth is modeled by Beverton-Holt functionals, it is shown that the population with maximal fitness will out compete the other population. While global stability results are provided for the case of two populations, only local stability results are obtained for the model with more than two populations.

An empirical, dynamic aggregate demand and supply (DADS) model is used to explain the policy paradox associated with the Kennedy, Reagan and Bush II administrations: that is, the possibility that tax rate decreases could in principle—and might in practice— increase tax revenue. A nonlinear elaboration is introduced to explain some aspects of monetary policy. Several factors explaining why employment lags behind the recovery of output after a recession are identified.

We study a delay differential equation with piecewise constant delays which could serve as an approximation to a corresponding delay differential equation with a finite constant delay. We consider a number of special cases.

This contribution is devoted to a discussion of the asymptotic behavior of solutions of systems of two difference equations. General principles giving a guarantee that the graph of at least one solution stays in a prescribed domain were given in previous papers of the first author. Two techniques – the so called retract type technique, and Liapunov type approach – were used separately. In the present contribution we try to connect both principles to investigate the asymptotic behavior of solutions of systems consisting of two equations. We show that under appropriate conditions containing conditions typical for the retract technique approach, and conditions typical for the Liapunov type approach, there exists at least one solution of the system considered the graph of which stays in a prescribed domain. Illustrative examples are given too. Moreover, as an application, a difference equation of Emden-Fowler type is considered.

The numerical integration of differential equations begins with the construction of appropriate discrete models. A major difficulty is that these models may have solutions not corresponding to any of those of the differential equations. Such solutions are called numerical instabilities (NI) and their elimination is of prime importance. Nonstandard finite difference (NSFD) schemes, as developed by Mickens and others, can be used to design schemes for which the elementary NI's do not occur. We show that the principle of dynamic consistency (DC) can be used to restrict the possible forms of the discrete models based on NSFD modeling rules. The existence of such schemes is illustrated using examples from heat transfer and cancer dynamics.

Some of the simplest models of population growth are one dimensional nonlinear difference equations. While such models can display wild behavior including chaos, the standard biological models have the interesting property that they display global stability if they display local stability. Various researchers have sought a simple explanation for this agreement of local and global stability. Here, we show that enveloping by a linear fractional function is sufficient for global stability. We also show that for seven standard biological models local stability implies enveloping and hence global stability. We derive two methods to demonstrate enveloping and show that these methods can easily be applied to the seven example models.

No abstract received.

Our aim in this paper is to investigate the permanence and the extreme stability of the nonlinear second-order nonautonomous difference equation of the form

A competition model of three species for one resource in a chemostat with a periodic washout rate is considered. Coexistence is indicated in [7] by numerical bifurcation analysis and in [12] by mathematical analysis. By introducing average competition functions, we obtain a necessary condition for the coexistence of a positive periodic solution and show that the condition restricts possible parameter value set to be relatively small. Further we show that the coexistence is enhanced when the period of the washout rate becomes large.

The spreading of the four main families of classical orthogonal polynomials of a discrete variable (Hahn, Meixner, Kravchuk and Charlier), which are exact solutions of the second-order hypergeometric difference equation, is studied by means of some information-theoretic measures of global (variance, Shannon entropy power) and local (Fisher information) character. The variance is calculated in a closed an compact form by means of the three-term recurrence relation of the polynomials. Then, the Cramer-Rao and Heisenberg-Shannon inequalities are used to find rigorous bounds for the other two measures.

This paper contains an approach to compute Taylor approximations of invariant manifolds associated with arbitrary fixed reference solutions of nonautonomous difference equations. Our framework is sufficiently general to include, e.g., stable and unstable manifolds of periodic orbits, or classical center-stable/-unstable manifolds corresponding to equilibria. In addition, our focus is to give applicable and quantitative results.

Finally, in the appendix we present a short manual to the Maple program IFB_Comp to calculate Taylor approximations of invariant manifolds.

A new result about the oscillation of a system of two coupled second order nonlinear difference equations is presented here. This result is complementary to those of the previous paper [4] by the authors, and leads to a çomplete characterization of oscillation for this class of systems.

Differential equations in which a very small parameter is multiplied to the highest derivative occur in many fields of science and engineering. Such differential equations form a class of “singular perturbation problems”. Classical methods fail in the numerical treatment of these problems. In particular, the standard finite difference method is not reliable. Following Mickens modelling rules [9], we design non-standard finite difference schemes. The schemes thus obtained replicate the dissipativity properties of the solution of the differential equations. The schemes are analyzed for convergence. Several numerical examples are given to support the predicted theory. More importantly, these numerical examples demonstrate uniform convergence of the non-standard schemes.

We have introduced a class of generalized autoregressive processes for which each of the autoregressive parameters is the sum of a constant and a function of a set of other random variables. It is shown that under some mild regularity conditions on these random variables the constant parts of the autoregressive parameters can be estimated from the given data set in a manner similar to those for the classical autoregressive processes. Although the sampling properties of these estimators are different from those for the classical types, we have shown that these estimators are consistent and asymptotically normally distributed.

We extend the known results of solutions of the autonomous counterpart of the difference equation in the title to the situation where any of the parameters are a period-two sequence with non-negative values and the initial conditions are positive.

Periodically forced dynamical systems are of great importance in modeling biological processes in periodically varying environments. In this paper, we survey the fundamental results of Elaydi and Yakubu, Elaydi and Sacker, Cushing and Henson, Franke and Selgrade, Franke and Yakubu on periodically forced (nonautonomous) difference equations without delay. Extensions of these results to periodically forced nonlinear difference equations with delay are posed as open problems.

No abstract received.

Difference equations are used as models for determining the dynamics of various types of processes. In many situations, the delays (or differences) can represent observation times or the time that it takes to transport informatin in the system. A basic question is: What properties of the system are preserved when subjected to small variations in the delays? The purpose of this paper is to point out some positive and negative results for linear systems, to give some applications to control problems and mention some unsolved problems for nonlinear systems.

The purpose of this paper is to provide a solution to the classical discrete linear-quadratic regulator problem under minimal assumptions. In particular, we do not assume the positive or nonnegative definiteness of the coefficients. Instead, a natural condition is imposed which is necessary for minimizing the involved discrete quadratic functional. The optimal solution is constructed from a generalized discrete Riccati equation and has a feedback form.

One of the fundamental tenets of ecology is the Competitive Exclusion Principle. According to this principle too much interspecific competition between two species results in the exclusion of one species. This Principle is supported by a wide variety of theoretical models, of which the Lotka/Volterra model based on differential equations is the most familiar. It is perhaps less well known that difference equations also played an important role in the historical development of the Competitive Exclusion Principle. The Leslie/Gower model was used in conjunction with influential competition experiments using species of Tribolium (flour beetles) carried out in the first half of the last century. This difference equation model exhibits the same dynamic scenarios as does the Lotka/Volterra model and also supports the Competitive Exclusion Principle. A recently developed competition for Tribolium species, however, exhibits a larger variety of dynamic scenarios and competitive outcomes, some of which seemingly stand in contradiction to the Principle. We discuss features of this model that differentiate it from the Leslie/Gower model. We give a simpler, lower dimensional “toy” model that illustrates some non-Lotka/Volterra dynamics.

No abstract received.

Many cells exhibit a complex behavior, characterized by brief bursts of oscillatory activity interspersed with quiescent periods during which the membrane potential changes only slowly. This behavior is called bursting. The interpretation of bursting in terms of nonlinear dynamics is one of the recent success stories of mathematical physiology and provides an excellent example of how mathematics can be used to understand complex biological dynamical systems. In the present paper we study a map, that replicates the dynamics of bursting cells, presented in [16]. Using symbolic dynamics we characterize the topological entropy of the chaotic bursts and we analyse the variation of this important numerical invariant with the parameters of the system. This procedure allows us to distinguish different chaotic scenarios.