A Periodically Forced Beverton-Holt Equation

For r. 1 and K. 0 the difference equation xtþ1 rK xt; t 0; 1; 2;... K þðr 2 1Þ xt has a unique positive equilibrium K and all solutions with x0. 0 approach K as t!1: This equation (known as the Beverton–Holt equation) arises in applications to population dynamics, and in that context K is the “carrying capacity ” and r is the “inherent growth rate”. A modification of this equation that arises in the study of populations living in a periodically (seasonally) fluctuating environment replaces the constant carrying capacity K by a periodic sequence Kt of positive carrying capacities.