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)x2(t)y(t)m2y2(t)+x2(t)−h1(t),y′(t)=y(t)[e(t)x2(t−τ(t))m2y2(t−τ(t))+x2(t−τ(t))−d(t)]−h2(t), is established, where a(t)a(t), b(t)b(t), c(t)c(t), e(t)e(t), d(t)d(t), τ(t)τ(t), h1(t)h1(t) and h2(t)h2(t) are all nonnegative periodic continuous functions with period ω>0ω>0, m>0m>0 is a constant. Our main result also improves some well-known results obtained before, especially when h1(t)=h2(t)≡0h1(t)=h2(t)≡0, the conditions that guarantee the existence of four positive periodic solutions reduce exactly to that of a previous conclusion.