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 g is monotonic, where a ( t ) , b ( t ) , c ( t ) , d ( t ) and e ( t ) are all positive periodic continuous functions with period ω > 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.