Persistence, contractivity and global stability in logistic equations with piecewise constant delays ✩

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N ∗ = 1/(a + m i=0 bi ) of the following differential equation with piecewise constant arguments:   dN(t) dt = N(t)r(t) 1 −aN(t) − m i=0 biN(n− i) , n t 0 and N(−j) = N−j 0, j= 1, 2, . . . , m, where r(t) is a nonnegative continuous function on [0,+∞), r(t) ≡ 0, m i=0 bi > 0, bi 0, i = 0, 1, 2, . . .,m, and a + m i=0 bi > 0. These new conditions depend on a, b0 and m i=1 bi , and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m = 0 and r(t) ≡ r > 0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms: dx(t) dt = x(t)r(t){1− ax(t)− g(x([t ]), x([t − 1]), . . . , x([t − m]))}, t 0, x(−k) = φ(−k) 0, 0 k m, and φ(0) > 0,where r(t) is a nonnegative continuous function on [0,+∞), r(t) ≡ 0, 1 − ax − g(x, x, . . . , x) = 0 has a unique solution x ∗ > 0 and g(x0,x1, . . . , xm) ∈ C1[(0,+∞) × (0,+∞)×· · ·×(0,+∞)].  2002 Elsevier Science (USA). All rights reserved