Permanence in delayed ratio-dependent predator–prey models with monotonic functional responses

In this paper, sufficient conditions for permanence of the general delayed ratio-dependent predator–prey modelx′(t)=x(t)[a(t)-b(t)x(t)]-c(t)gx(t)y(t)y(t),y′(t)=y(t)e(t)gx(t-τ)y(t-τ)-d(t),are obtained when functional response g   is monotonic, where a(t)a(t), b(t)b(t), c(t)c(t), d(t)d(t) and e(t)e(t) are all positive periodic continuous functions with period ω>0ω>0, ττ is a positive constant. We find that the conditions on existence of a positive periodic solution imply the permanence of the above system. As applications, some examples are given.