Note on permanence and global stability in delayed ratio-dependent predator–prey models with monotonic functional responses

Sufficient conditions of the permanence and global stability for the general delayed ratio-dependent predator–prey model {x′(t)=x(t)[a(t)−b(t)x(t)]−c(t)g(x(t)y(t))y(t),y′(t)=y(t)[e(t)g(x(t−τ)y(t−τ))−d(t)], are obtained when the functional response function gg is monotonic, where a(t),b(t),c(t),d(t)a(t),b(t),c(t),d(t) and e(t)e(t) are all positive periodic continuous functions with period ω>0,τω>0,τ is a positive constant. The permanence result improves Theorem 2.1 in Fan and Li (2007) [14], and the condition that guarantees the existence of positive periodic solutions for the system generalizes the corresponding result in Fan et al. (2003) [18] and Li and Wang (2006) [20]. Finally, we perform numerical simulations to support our theoretical results.