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DCDS-B

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 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
$hr < \sqrt{2}$ 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.

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