Population models in almost periodic environments

Abstract

Abstract. We establish the basic theory of almost periodic sequences on Z + . Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models.

Almost periodic solutions of Volterra difference systems

“…is an almost periodic function (see [24]) and b (t, k) = h exp(k − t) is a convolution type function, the conditions (A1-A4) are satis ed and we guarantee the existence of an almost periodic solution by Theorem 6. However, Theorem 7 is insu cient to ensure the existence of an almost periodic solution of the system (15), since I + hA ≥ + hU B for any positive integer h (see Example 1).…”

“…For details on the almost periodic sequences, we refer the reader to the pioneering work of Diagana et al (see [24]). The basic properties of discrete almost periodic functions are given in the following theorem:…”

Almost Periodic Parameters for the Second Moment Stability of Linear Stochastic Systems

“…Motivation for applications of almost periodic parameters is not limited to the case handled in this paper, as one can see in the study of population models [18], in characterizing almost periodic solutions for unidimensional AR models [19], in determining the controllability of nonlinear systems [20], in the construction of the fundamental matrix for autonomous systems [21], in the estimation of stochastic processes with almost periodic covariance functions [22], just to cite a few examples. These illustrate some of the effective applications arising from the almost periodic system theory.…”

On the stochastic Beverton–Holt equation with survival rates

“…The second source is a recent paper by Diagana, Elaydi, and Yakubu [2], in which a fundamental theory of almost periodic sequences on Z + has been established. Furthermore, the abstract results were then applied to study the existence of almost periodic solutions to the nonautonomous Beverton-Holt difference equation (1.2) x(n + 1) = γ n x(n)…”

“…We then extend the theorem of Haskell-Sacker [5] to (1.2) and show the existence of a unique asymptotically stable invariant density (Theorem 4.2). Based upon the results of [2], we develop a theory of random mean almost periodic sequences on Z + . The developed theory is then applied to investigate the stochastic Beverton-Holt equation (1.2).…”