Table 1 The impact of the feedback gain L on the first bifurcation point \(\tau _{0}^{*}\) and critical frequency \(\omega _{0}^{*}\) for the controlled system (4.24). This table lists the values of the first bifurcation point \(\tau _{0}^{*}\) and critical frequency \(\omega _{0}^{*}\) corresponding to some feedback gain L, which are calculated from the linearized average system (4.26) of the controlled system (4.24) with the parameter values \(r=1.2\), \(k=20\), \(c=0.9\), \(d=0.3\), \(e=0.3\), \(m=0.8\), \(q=0.98\)

From: Periodic pulse control of Hopf bifurcation in a fractional-order delay predator–prey model incorporating a prey refuge