Permanence, contractivity and global stability in logistic equations with general delays

We obtain new conditions of the permanence and “contractivity” of solutions and the global asymptotic stability for the positive equilibrium x ∗ = 1/(a + m j=0 bj ) of the following logistic equation with general delays:   dx(t) dt = x(t)r(t) 1 − ax(t) − m j=0 bj x τj (t ) , t t0, x(t) = φ(t) 0, −τ t t0, and φ(t0) > 0, (0.1) where r(t) is a nonnegative continuous function on [t0,+∞), a + b − > 0 or a = b − = 0, and b − = m j=0 min(0,bj ), each τj (t ) is piecewise continuous on [t0,+∞), −τ τj (t ) t for 0 j m, and τ (t )≡ min1 j m τj (t )→+∞as t →+∞. The results improve that of J.W.-H. So and J.S. Yu [Hokkaido Math. J. 24 (1995) 269–286]. For a logistic equation with nonlinear delay terms, a similar result is obtained.  2004 Elsevier Inc. All rights reserved.