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

  • * Corresponding author: Junjie Wei

    * Corresponding author: Junjie Wei
This research is supported by National Natural Science Foundation of China (No. 11371111)
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  • This paper is devoted to the study of a single-species population model with stage structure and strong Allee effect. By taking $τ$ as a bifurcation parameter, we study the Hopf bifurcation and global existence of periodic solutions using Wu's theory on global Hopf bifurcation for FDEs and the Bendixson criterion for higher dimensional ODEs proposed by Li and Muldowney. Some numerical simulations are presented to illustrate our analytic results using MATLAB and DDE-BIFTOOL. In addition, interesting phenomenon can be observed such as two kinds of bistability.

    Mathematics Subject Classification: Primary: 34C23, 34D05; Secondary: 34K28.


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  • 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 $y_2$

    Table 1.  List of quantities of periodic solution bifurcating from $y_2$ under (18)

    $\delta$ $\text{Re}(c_1(0))$ $\mu_2$ $\beta_2$
    $~~\tau_0\approx 2.8$ $>0$ $-37.3115 < 0$ $>0$ $ < 0~~$
    $~~\tau_1\approx14.6$ $ < 0$ $-72.7255 < 0$ $ < 0$ $ < 0~~$
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