Global Hopf bifurcation of a population model with stage structure and strong Allee effect

Figure 1.  The growth rate per capita $r(y)=\frac{y^{n-1}}{1+y^m}-0.15$ in system $\dot y=\frac{y^n}{1+y^m}-0.15y=r(y)y$

Figure 2.  Figures of $\bar{y}$ and equilibria $y_1$, $y_2$ with parameters given in (18)

Figure 3.  Graphs of $S_n(\tau)$ on $\left[0, \tau^{1}\right)$ with parameters given in (18)

Figure 4.  $y_1\approx0.6986$ is unstable, and sustained oscillation occurs when $\tau\in[0, \tau^0)$, where $0 < \tau=8 < \tau^0\approx22.2$, and the initial condition is $\varphi=0.8$ for $t\in[-\tau, 0]$

Figure 5.  $y_2$ is asymptotically stable when $\tau\in[0, \tau_0)\cup\left(\tau_1, \tau^0\right)$, and the initial condition is $\varphi=1.1$ for $t\in[-\tau, 0]$

Figure 6.  $y_2\approx1.0962$ is unstable, and sustained oscillation occurs when $\tau\in(\tau_0, \tau_1)$, where $2.8\approx\tau_0 < \tau=10 < \tau_1\approx14.8$, and the initial condition is $\varphi=1.1$ for $t\in[-\tau, 0]$

Figure 7.  $\tau$, h) plane, where $h=\sqrt{2}D-\alpha |b'| e^{-\delta\tau}$

Figure 8.  Hopf bifurcation branch on the ($\tau$, d) plane, where $d=\max y(t)-\min y(t)$

Figure 9.  Stability of equilibria $0, ~y_1, ~y_2$ and periodic solutions bifurcated from $y_2$