Saber Elaydi

Lipschitz stability of nonlinear systems of differential equations

In [S], the authors introduced the notion of Lipschitz stability in differential equations. This notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation [3] and uniform stability in variation [4] on the other side. However, Lipschitz stability is new only as a nonliner phenomenon, since it coincides with uniform stability in linear systems [S]. An important feature of Lipschitz stability is that, unlike uniform stability, the linearized system inherits the property of Lipschitz stability from the original nonlinear system [5]. In this paper we pursue the study of Lipschitz stability that started in [S] using essentially the techniques of Liapunov functions. Then we give sufficient conditions for the Lipschitz stability of certain nonlinearly pertur- bed nonlinear systems. Such systems include, among other equations, certain integrodifferential and functional differential equations. Then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results [9].

Nonautonomous Beverton-Holt Equations and the Cushing-Henson Conjectures

In [3] Jim Cushing and Shandelle Henson published two conjectures (see Section 3) related to the Beverton-Holt difference equation (with growth parameter exceeding one) which said that the B-H equation with periodically varying coefficients (a) will have a globally asymptotically stable periodic solution and (b) the average of the state variable along the periodic orbit will be strictly less than the average of the carrying capacities of the individual maps. They had previously [3] proved both statements for period 2. In [4] the authors solved the first conjecture in the affirmative for arbitrary period and in a metric state space. In addition they showed that the period of the periodic “geometric cycle”, i.e. the projection of the periodic orbit onto the state space, must be a divisor of the period of the underlying system. In [5] the authors solved the second conjecture. Independently Ryusuke Kon [8], [9] 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. Also Kocic [7] has given a solution to the second conjecture. In this paper we consider the B-H equation with periodic growth parameter as well as periodic carrying capacity. We first give an estimate relating the averages of the state variable and the carrying capacities. This is done by a modification of the proof of Kocic [7]. We then refine the estimate and actually obtain an equality relating the averages, (in the case of period p = 2) thus laying to rest once and for all the p = 2 case. The general case will be treated elsewhere.

Population models with Allee effect: a new model

In this paper, we develop several population models with 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, which is the focus of our 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.

Discrete Competitive and Cooperative Models of Lotka–Volterra Type

The dynamics of discrete Lotka–Volterra system of two species is investigated. It is shown that the proposed discrete models for competitive and cooperative systems possess “dynamical consistency” with their continuous counterparts.

Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type

It is shown that uniform asymptotic stability does not imply exponential stability in linear Volterra difference equations. However, if the kernel of the equation decays exponentially. then both concepts are equivalent as in the case of ordinary difference equations.

Global Stability of Cycles: Lotka-Volterra Competition Model With Stocking

In this article, we prove that in connected metric spaces n - cycles are not globally attracting (where n S 2 ). We apply this result to a two species discrete-time Lotka-Volterra competition model with stocking. In particular, we show that an n - cycle cannot be the ultimate life-history of evolution of all population sizes. This solves Yakubus conjecture but the question on the structure of the boundary of the basins of attractions of the locally stable n - cycles is still open.

Uniform asymptoic stability in linear volterra difference equations

For linear Volterra difference equations of nonconvolution type, uniform asymptotic stability of the zero solution is characterized by the summability of the resolvent matrix. Moreover, the existence of bounded solutions of nonhomogeneous linear Volterra difference equations is studied.

Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.

Differential equations : stability and control

Reports and expands upon topics discussed at the International Conference on [title] held in Colorado Springs, Colo., June 1989. Presents recent advances in control, oscillation, and stability theories, spanning a variety of subfields and covering evolution equations, differential inclusions, functi

Asyptotics for linear difference equations I:basic theory

We present here a unified treatment of asymptotic theory of linear difference equations. This is based on anadapted theory of discrete dichotomy. The obtained results narrow the gap between Poincares Theorem and (the discrete analoue of) Levinsons Theorem.