Global stability in a regulated logistic growth model

Abstract

We investigate global stability of the regulated logistic growth model (RLC) n'(t) = rn(t)(1-n(t-h)/K-cu(t)), u'(t) = -au(t)+bn(t-h). It was proposed by Gopalsamy and Weng [1, 2] and studied recently in [4, 5, 6, 9]. Compared with the previous results, our stability condition is of different kind and has the asymptotical form. Namely, we prove that for the fixed parameters K and mu = bcK/a (which determine the levels of steady states in the delayed logistic equation n'(t) rn(t)(1 - n(t - h)/K) and in RLG) and for every hr < root 2 the regulated logistic growth model is globally stable if we take the dissipation parameter a sufficiently large. On the other hand, studying the local stability of the positive steady state, we observe the improvement of stability for the small values of a: in this case, the inequality rh < pi(1 + mu)/2 guaranties such a stability