Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms

By using the generalized continuation theorem, the existence of four positive periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response { x ′ ( t ) = x ( t ) [ a ( t ) − b ( t ) x ( t ) ] − c ( t ) x 2 ( t ) y ( t ) m 2 y 2 ( t ) + x 2 ( t ) − h 1 ( t ) , y ′ ( t ) = y ( t ) [ e ( t ) x 2 ( t − τ ( t ) ) m 2 y 2 ( t − τ ( t ) ) + x 2 ( t − τ ( t ) ) − d ( t ) ] − h 2 ( t ) , is established, where a ( t ) , b ( t ) , c ( t ) , e ( t ) , d ( t ) , τ ( t ) , h 1 ( t ) and h 2 ( t ) are all nonnegative periodic continuous functions with period ω > 0 , m > 0 is a constant. Our main result also improves some well-known results obtained before, especially when h 1 ( t ) = h 2 ( t ) ≡ 0 , the conditions that guarantee the existence of four positive periodic solutions reduce exactly to that of a previous conclusion.