For each choice of the intrinsic growth rate r and the environmental carrying capacity K as positive, bounded, periodic functions with period p, the nonautonomous logistic equation, x ̇(t)=r(t)x(t)1- x(t) K(t), possesses an asymptotically stable, positive, periodic solution x* with period p. If K is not a constant function, but is piecewise continuous on [0, p], then the minimum and maximum values of x* are related as follows to the extrema of K: Kinf<x*min<x*max<Ksup. The following question is treated here: For each specification of the function K, which functions r come close to maximizing x*min? It is shown that if r is expressed in the form r(t)=δγ(t) with δ>0 and γ a positive p-periodic function whose average value is unity, then, for δ small enough, x*(t) is approximately equal to a number x̃(γ,K) which is independent of t; in fact, for each t, lim δ→0x*(t)=p ∫ 0 p γ(τ) K(τ)dτ-1= x ̃(γ,K). By an appropriate choice of γ, the number x̃(γ,K) can be made arbitrarily close to Ksup. The appropriate choices are those for which γ is approximately a Dirac function" with its support concentrated at times for which K is close to Ksup. © 1979."
