# 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
##### References:

show all references

##### 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
 [1] Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 [2] Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 [3] Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042 [4] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [5] Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133 [6] Safia Slimani, Paul Raynaud de Fitte, Islam Boussaada. Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5003-5039. doi: 10.3934/dcdsb.2019042 [7] C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289 [8] Jun Zhou. Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1127-1145. doi: 10.3934/cpaa.2015.14.1127 [9] Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228 [10] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [11] Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115 [12] Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203 [13] Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065 [14] Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236 [15] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [16] Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073 [17] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [18] Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011 [19] Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173 [20] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

2018 Impact Factor: 1.143