Fig. 3

a, b The asymptotical stability of the coexistence equilibrium \(E^*(5.48181.917314.7487)\) for \((\tau _1, \tau _2, \tau _3)\) with \(\tau _1=1<\tau _1^0\). c Coexistence equilibrium \(E^*\) loses its stability at \((\tau _1, \tau _2, \tau _3)\) with \(\tau _1=4>\tau _1^0\). Stable periodic solution arising from Hopf bifurcation at \((\tau _1, \tau _2, \tau _3)\) with \(\tau _1=\tau _1^0\). d, e Stable limit cycle is observed at \((\tau _1, \tau _2, \tau _3)\) with \(\tau _1=4.2\). Here \(\tau _2=1, \tau _3=2\) and \(\tau _1^0=3.9465\)