Figure 2

The asymptotical stability with delay \(\tau=13.5\). The parameters of system (5) are \(c_{1}=0.8\), \(c_{2}=0.1\), \(c_{3}=0.9\), \(c_{4}=0.6\), \(s_{1}=0.6\), \(s_{2}=0.1\), \(b=0.3\), \(d_{1}=0.9\), \(d_{2}=0.1\), \(d_{3}=0.1\), and \(\alpha= 0.98\); the initial values are \(r(t)=0.1\), \(p(t)=1\), \(u(t)=2\) (\(t \in[-\tau,0]\)). The Hopf bifurcation critical value \(\tau_{a} \approx13.5888\). It depicts the asymptotical stability of the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*}) \approx(0.3571, 0, 1.875)\) with time delay \(\tau=13.5<\tau_{a}\)