Permanence of periodic Holling type-IV predator–prey system with stage structure for prey

In this paper, we study the permanence of the following periodic Holling type-IV predator–prey system with stage structure for prey x ̇ 1 ( t ) = a ( t ) x 2 ( t ) − b ( t ) x 1 ( t ) − d ( t ) x 1 2 ( t ) − p ( t ) x 1 ( t ) k ( t ) + x 1 2 ( t ) y ( t ) , x ̇ 2 ( t ) = c ( t ) x 1 ( t ) − f ( t ) x 2 2 ( t ) , y ̇ ( t ) = y ( t ) ( − g ( t ) + h ( t ) x 1 ( t ) k ( t ) + x 1 2 ( t ) − q ( t ) y ( t ) ) . Under certain assumptions, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.