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+∑j=0mbj) of the following logistic equation with general delays: (0.1)dx(t)dt=x(t)r(t)1−ax(t)−∑j=0mbjxτj(t),t⩾t0,x(t)=φ(t)⩾0,−τ⩽t⩽t0,andφ(t0)>0, where r(t) is a nonnegative continuous function on [t0,+∞), a+b−>0 or a=b−=0, and b−=∑j=0mmin(0,bj), each τj(t) is piecewise continuous on [t0,+∞), −τ⩽τj(t)⩽t for 0⩽j⩽m, and τ(t)≡min1⩽j⩽mτj(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.