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Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey

  • * Corresponding author: Mingxin Wang

    * Corresponding author: Mingxin Wang

This work was supported by NSFC Grant 11771110

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  • It is well known that the Leslie-Gower prey-predator model (without Allee effect) has a unique globally asymptotically stable positive equilibrium point, thus there is no Hopf bifurcation branching from positive equilibrium point. In this paper we study the Leslie-Gower prey-predator model with strong Allee effect in prey, and perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the steady-state bifurcation of PDE model. Moreover, by the center manifold theory and the normal form method, the direction and stability of Hopf bifurcation solutions are established. Finally, some numerical simulations are presented. Apparently, Allee effect changes the topology structure of the original Leslie-Gower model.

    Mathematics Subject Classification: 35B10, 35B36, 35B32, 92D25.

    Citation:

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  • Figure 1.  Graphs of $y = [d_1\mu-d_2A(\lambda ^{(1)})]^2$ and $y = 4d_1d_2[\beta\lambda ^{(1)}-A(\lambda ^{(1)})]\mu.$

    Figure 2.  $p_+(\mu)$ is decreasing and $p_{-}(\mu)$ is increasing in $(0, \mu_*]$

    Figure 3.  $p_+(\mu)$ is decreasing in $(0, \infty)$

    Figure 4.  The system (3) occurs Hopf bifurcation from $(\lambda ^{(1)}, \lambda ^{(1)})$ when $\beta = 0.1714$

    Figure 5.  Most of solutions to (3) converge to $(0, 0)$ when $\beta = 0.17157288$

    Figure 6.  The system (3) has two positive equilibrium points $(\lambda ^{(1)}, \lambda ^{(1)})$ and $(\lambda ^{(2)}, \lambda ^{(2)})$. The former is stable and the later unstable

    Figure 7.  Spatially homogeneous Hopf bifurcation of (4) when $\beta = 61.3170$ and $n = 0$

    Figure 8.  Spatially non-homogeneous Hopf bifurcation of (4) when $\beta = 78.4754$ and $n = 2$

    Table 1.  Hopf bifurcation values of ODE problem (3)

    $0<\mu<b_0$ $b_0<\mu<b^0$ $\mu>b^0$
    $0<b<b_1$
    One Hopf bifurcation value $\lambda ^{(1)}_{0, +}$

    Two Hopf bifurcation values $\lambda ^{(1)}_{0, -}$, $\lambda ^{(1)}_{0, +}$
    Null
    $b_1<b<1$
    One Hopf bifurcation value $\lambda ^{(1)}_{0, +}$
    Null Null
    $b_1=7-4\sqrt 3$
     | Show Table
    DownLoad: CSV

    Table 2.  Hopf bifurcation values for $(\lambda ^{(1)}, \lambda ^{(1)})$ in PDE problem (4)

    $d_1^{-1}d_2b^0<\mu<b_0$ $\max\{d_1^{-1}d_2b^0, b_0\}<\mu<b^0$ $\mu>b^0$
    $0<b<b_1$
    $2r-m+1$ Hopf bifurcation values

    $2r+2$ Hopf bifurcation values
    Null
    $b_1<b<1$
    $m+1$ Hopf bifurcation values
    Null Null
    $b_1=7-4\sqrt 3$, $h_j=\mu+(d_1+d_2)j^2/l^2$
     | Show Table
    DownLoad: CSV

    Table 3.  Hopf bifurcation values for $(\lambda ^{(2)}, \lambda ^{(2)})$ in PDE problem (4)

    $0<\mu<1-b$ $1-b<\mu<b_0$ $b_0<\mu<b^0$
    $0<b<b_1$
    $m-k$ Hopf
    bifurcation values

    $m$ Hopf
    bifurcation values
    Null
    $b_1<b<b_2$
    $2r-m-k$ Hopf
    bifurcation values

    $2r-m$ Hopf
    bifurcation values

    $2r$ Hopf
    bifurcation values
    $b_1=7-4\sqrt 3$, $b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
     | Show Table
    DownLoad: CSV

    Table 4.  Hopf bifurcation values for $(\lambda ^{(2)}, \lambda ^{(2)})$ in PDE problem (4)

    $0<\mu<b_0$ $b_0<\mu<1-b$ $1-b<\mu<b^0$
    $b_2<b<\frac{1}{3}$
    $2r-m-k$ Hopf
    bifurcation values

    $2r-k$ Hopf
    bifurcation values

    $2r$ Hopf
    bifurcation values
    $\frac{1}{3}<b<1$
    $k-m$ Hopf
    bifurcation values

    $k$ Hopf
    bifurcation values
    Null
    $b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
     | Show Table
    DownLoad: CSV

    Table 5.  Parameters' values of Hopf bifurcation for $(\lambda ^{(2)}, \lambda ^{(2)})$

    $b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$
    1 0.03 0.1 22.44329 1 0.1 1
    2 0.05 0.1 11.46339 1 0.1 1
    3 0.06 0.1 9.485507 1 0.1 1
    4 0.06 0.1 6.305220 1 0.1 1
     | Show Table
    DownLoad: CSV

    Table 6.  Parameters' values for steady-state bifurcation

    $b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$
    1 0.25 0.292 0.972 0.5 3 0.531
    2 0.062 2.431 8.667 0.5 2 1.283
    3 0.25 1 0.667 1 1 1
    4 0.062 1 10 1 1 2
     | Show Table
    DownLoad: CSV
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