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

Na Min; Mingxin Wang

doi:10.3934/dcds.2019045

Article Contents

2019, Volume 39, Issue 2: 1071-1099. Doi: 10.3934/dcds.2019045

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

Na Min^1,2, and
Mingxin Wang^1, ,

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
* Corresponding author: Mingxin Wang

Received: February 28, 2018

Revised: July 31, 2018

Published: January 2019

This work was supported by NSFC Grant 11771110

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Abstract

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.

Keywords:

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

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

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

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

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

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

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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

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

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

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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
$b_1=7-4\sqrt 3$

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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
$b_1=7-4\sqrt 3$, $h_j=\mu+(d_1+d_2)j^2/l^2$

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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
$b_1=7-4\sqrt 3$, $b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$

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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
$b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$

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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

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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

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