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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

Feedback gain LBifurcation point \(\tau _{0}^{*}\)Critical frequency \(\omega _{0}^{*}\)
−0.0459.4705910510.1737564826
−0.048.2158023610.1901197022
−0.0357.4134322970.2027989139
−0.036.8304689780.2134653512
−0.0256.3766323470.2228352248
−0.026.0075156970.2312894556
−0.0155.6980479070.2390574929
−0.015.4327216060.2462893820
−0.0055.2012972550.2530891673
04.9966721140.2595322466
0.13.0499796230.3549183506
0.22.2932692240.4267425452
0.31.8565800790.4895983980
0.41.5649779400.5471409136
0.51.3542492350.6009013984