Dynamics of an almost periodic logistic integrodifferential equation

Abstract

Sufficient conditions are derived for the existence of a globally attrac- tive positive almost-periodic solution of the logistic integrodifferential equation dN(t) dt = N(t) a(t) − b(t) 1 0 K �(s)N(t − s)ds , t ≥ 0, � ∈ (0, ∞), in which a(t), b(t) are continuous positive almost-periodic functions defined on (−∞, ∞) and K� : (0, ∞) → (0, ∞) is piecewise continuous and integrable on (0, ∞), whereis a positive-valued parameter. We obtain sufficient conditions for all positive solutions to have level-crossings about the unique almost-periodic solution. Existence of a positive solution with no level-crossings about the almost-periodic solution is also discussed. It is well-known that the environments of most natural populations change with time and that such changes induce variation in the growth characteristics of populations. For instance, favourable weather conditions stimulate an increase in the body size and reproduction while unfavourable environments can lead to a decline in the birth rate and an increase in mortality. Temporal variations of an environment of a population are usually incorporated in model systems by the introduction of time-dependent parameters in governing equations. Such governing equations are nonautonomous, and studies of nonautonomous equations have not attained a satisfactory level of maturity comparable to that of autonomous equations. The reader is referred to the recent monograph of Gopalsamy (8) for an extensive discussion of multispecies dynamics in temporally uniform environments governed by autonomous differential equations with discrete and continuously distributed delays. The purpose of this article is to derive a set of algebraic sufficient conditions for the existence of a globally attractive positive almost-periodic solution of the logistic integrodifferential equation dN(t) dt = N(t) �