Figure 2

The time trajectories of the populations x, y, and z: (a) asymptotically stable for \(\delta _{2}=0.4\) (i.e., trajectories tend to their equilibrium points); (b) limit cycle oscillation occurs at \(\delta _{2}=0.5\) (\(\delta _{2}\) crosses its threshold value \(\delta _{2}^{*}\), then the system loses stability and Hopf bifurcation exists); (c) the bifurcation diagram of model (5.2) with \(\rho =0.15\), \(\nu =0.055\), and \(\delta _{2}\in [0.38,0.60]\)