Stability and Asymptoticity of Volterra Difference Equations: A Progress Report

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Ζ-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried out in Section 6. Finally, we state some problems

Stability and Asymptoticity of Volterra Difference Equations: A Progress Report

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Ζ-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried out in Section 6. Finally, we state some problems

Nonautonomous Difference Equations: Open Problems and Conjectures

Autonomous difference equations of the form xn+1 = ƒ (xn) may model populations of species with nonoverlaping generations such as fish, orchard pests, etc. The drawback of such models is that they do not account for environmental fluctuations or seasonal changes. Hence we are led to nonautonomous difference equations of the form xn+1 = ƒ (xn), n ∈ Ζ+. Our main focus in this note will be on periodic difference equations in which the sequence ƒn is periodic. Most of the open problems and conjectures in this part are motivated by recent work by Elaydi and Sacker [3], Elaydi and Yakubu [4] [5], and Elaydi [2]. The second part of the paper discussed the connection between a nonautonomous difference equation and its limiting equation. We present here several conjectures and open problems pertaining to the properties of omega limited sets (see Kempf [7]) and the question of lifting properties from the limiting equation to the original equation. For the convenience of the reader we introduce in Section 4 some rudiments of the theory of skew-product dynamical systems [8]

Is the World Evolving Discretely?

Difference Equations can model effectively almost all physical and artificial phenomena. Even the highly celebrated differential system of Lorenz [6], which models a fluid: =’x yx σσ + yrxxzy −+−=

General Allee Effect in Two-Species Population Biology

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator–prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a ‘phase space core’ of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar systems. Finally, we show how the presence of semistable equilibria increases the number of possible Allee effect cores

Difference Equations from Discretization of a Continuous Epidemic Model with Immigration of Infectives

A continuous-time epidemic model with immigration of infectives is introduced. Systems of difference equations obtained from the continuous-time model by using nonstandard discretization technique are presented. Comparisons between the continuous-time model and its discrete counter-part are made

Population Models with Allee Effect: A New Model

In this paper we develop several mathematical models of Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model that is our focus of study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity

Basin of Attraction of Periodic Orbits of Maps on the Real Line

We prove a conjecture by Elaydi and Yakubu which states that the basin of attraction of an attracting 2 k -cycle of the Ricker\u27s map is where E is the set of all eventually 2 r -periodic points. The result is then extended to a more general class of continuous maps on the real line

Periodic Difference Equations, Population Biology and the Cushing-Henson Conjectures

We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. We show that the periodic fluctuations in the carrying capacity always have a deleterious effect on the average population, thus answering in the affirmative the second of the conjectures. Independently Ryusuke Kon [9], [10] discovered a solution to the second conjecture and in fact proved the result for a wider class of difference equations including the Beverton-Holt equation. The work of Davydova, Diekmann and van Gils, [6] should also be noted. There they study nonlinear Leslie matrix models describing the population dynamics of an age-structured semelparous species, a species whose individuals reproduce only once and die afterwards. See also the work of N.V. Davydova, [5] where the notion of families of single year class maps is introduced