American Institute of Mathematical Sciences

February  2019, 39(2): 1071-1099. doi: 10.3934/dcds.2019045

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

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China

* Corresponding author: Mingxin Wang

Received  March 2018 Revised  August 2018 Published  November 2018

Fund Project: This work was supported by NSFC Grant 11771110.

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.

Citation: Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
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References:
Graphs of $y = [d_1\mu-d_2A(\lambda ^{(1)})]^2$ and $y = 4d_1d_2[\beta\lambda ^{(1)}-A(\lambda ^{(1)})]\mu.$
$p_+(\mu)$ is decreasing and $p_{-}(\mu)$ is increasing in $(0, \mu_*]$
$p_+(\mu)$ is decreasing in $(0, \infty)$
The system (3) occurs Hopf bifurcation from $(\lambda ^{(1)}, \lambda ^{(1)})$ when $\beta = 0.1714$
Most of solutions to (3) converge to $(0, 0)$ when $\beta = 0.17157288$
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
Spatially homogeneous Hopf bifurcation of (4) when $\beta = 61.3170$ and $n = 0$
Spatially non-homogeneous Hopf bifurcation of (4) when $\beta = 78.4754$ and $n = 2$
Hopf bifurcation values of ODE problem (3)
 $0<\mub^0$ $0 $0<\mub^00
Hopf bifurcation values for $(\lambda ^{(1)}, \lambda ^{(1)})$ in PDE problem (4)
 $d_1^{-1}d_2b^0<\mub^0$ $0 $d_1^{-1}d_2b^0<\mub^00
Hopf bifurcation values for $(\lambda ^{(2)}, \lambda ^{(2)})$ in PDE problem (4)
 $0<\mu<1-b$ $1-b<\mu $0<\mu<1-b1-b<\mu
Hopf bifurcation values for $(\lambda ^{(2)}, \lambda ^{(2)})$ in PDE problem (4)
 $0<\mu $0<\mu
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
 $b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$ 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
 $b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$ 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
 $b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$ 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
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