Figure 4

Autonomous simulation of \(\alpha ^{*}\). The dynamic of \(N_{2}\), \(N_{3}\), and M of system (1) with two different initial values, \((N_{2}, N_{3}, M)=(70,0,5)\) and \((140,1,10)\). For both initial values, the blue curve (with \(\alpha ^{*}=\frac{1}{11}\)) tends to extinction equilibrium \(\mathcal{E}_{1}=(0,0,0)\), the black curve (with \(\alpha ^{*}=\frac{1}{7}\)) tends to hunter-free equilibrium \(\mathcal{E}_{2}=(51.06,0,0)\), and the red curve (with \(\alpha ^{*}=\frac{1}{3}\)) tends to coexistence equilibrium \(\mathcal{E}_{3}=(80.13,0.09,2.43)\)