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

The existence of four positive periodic solutions for a delayed ratio-dependent predator-prey model with monotonic functional responsex′(t)=x(t)a(t)-b(t)x(t)-c(t)gx(t)y(t)y(t)-h1(t),y′(t)=y(t)f(t)gx(t-τ(t))y(t-τ(t))-d(t)-h2(t),is established by using the generalized continuation theorem, where a(t),b(t),c(t),d(t),f(t),τ(t),h1(t) and h2(t) are all nonnegative periodic continuous functions with period ω>0. When h1(t)=h2(t)≡0, the conditions that guarantee the existence of four positive periodic solutions reduces exactly to that in Fan et al. (2004) [1]. And our main result also improves the corresponding results in Zhang and Hou (2010) [2] and Fan and Wang (2001) [3].