2005, 5(2): 461-468. doi: 10.3934/dcdsb.2005.5.461

Global stability in a regulated logistic growth model

1. 

Department of Mathematics, National Technical University 'KPI', Kiev, Ukraine

2. 

Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  April 2003 Revised  March 2004 Published  February 2005

We investigate global stability of the regulated logistic growth model (RLG) $n'(t)=rn(t)(1-n(t-h)/K-cu(t))$, $u'(t)=-au(t)+bn(t-h)$. 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 diff erent 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 $hr < \sqrt{2}$ 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 $rh<\pi (1+\mu)/2$ guarantees such a stability.
Citation: E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461
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