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:
[1]

W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949. Google Scholar

[2]

M. A. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[3]

Y. L. CaiC. D. ZhaoW. M. Wang and J. F. Wang, Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Lett., 39 (2015), 2092-2106.  doi: 10.1016/j.apm.2014.09.038.  Google Scholar

[4]

F. CourchampT. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405-410.  doi: 10.1016/S0169-5347(99)01683-3.  Google Scholar

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R. H. CuiJ. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015.  Google Scholar

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L. L. Du and M. X. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485.  doi: 10.1016/j.jmaa.2010.02.002.  Google Scholar

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Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Diff. Equat., 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[8]

E. González-OlivaresJ. Mena-LorcaA. Rojas-Palma and J. D. Flores, Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model, 35 (2011), 366-381.  doi: 10.1016/j.apm.2010.07.001.  Google Scholar

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B. Hassard, N. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

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A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.  Google Scholar

[11]

P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245.  doi: 10.1093/biomet/35.3-4.213.  Google Scholar

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P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31.  doi: 10.1093/biomet/45.1-2.16.  Google Scholar

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S. B. LiJ. H. Wu and H. Nie, Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model, Comput. Math. Appl., 70 (2015), 3043-3056.  doi: 10.1016/j.camwa.2015.10.017.  Google Scholar

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Y. Li, Hopf bifurcations in general systems of Brusselator type, Nonlinear Anal.: Real World Appl., 28 (2016), 32-47.  doi: 10.1016/j.nonrwa.2015.09.004.  Google Scholar

[15]

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[16]

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N. Min and X. M. Wang, Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1721-1737.  doi: 10.3934/dcdsb.2018073.  Google Scholar

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W. J. Ni and M. X. Wang, Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.  doi: 10.3934/dcdsb.2017172.  Google Scholar

[19]

W. J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.  doi: 10.1016/j.jde.2016.06.022.  Google Scholar

[20]

W. M. Ni, Diffusion, cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[21]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[22]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Diff. Equat., 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.  Google Scholar

[23]

E. C. Pielou, Mathematical Ecology, John Wiley & Sons, New York, RI, 1977.  Google Scholar

[24]

Y. W. Qi and Y. Zhu, Global stability of Lesile-type predator-prey model, Meth. Appl. Anal., 23 (2016), 259-268.  doi: 10.4310/MAA.2016.v23.n3.a3.  Google Scholar

[25]

P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.  Google Scholar

[26]

J. F. WangJ. P. Shi and J. J. W, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[27]

J. F. WangJ. J. Wei and J. P. Shi, Global bifurcation analysis and pattern formation inhomogeneous diffusive predator-prey systems, J. Diff. Equat., 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.  Google Scholar

[28]

M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196 (2004), 172-192.  doi: 10.1016/j.physd.2004.05.007.  Google Scholar

[29]

M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Cont. Dyn. Syst. A, 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.  Google Scholar

[30]

Y. X. Wang and W. T. Li, Spatial patterns of the Holling-Tanner predator-prey model with nonlinear diffusion effects, Appl. Anal., 92 (2013), 2168-2181.  doi: 10.1080/00036811.2012.724402.  Google Scholar

[31]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second edition. Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003.  Google Scholar

[32]

F. Q. YiJ. J. Wei and J. P. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal.: Real World Appl., 9 (2008), 1038-1051.  doi: 10.1016/j.nonrwa.2007.02.005.  Google Scholar

[33]

F. Q. YiJ. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

show all references

References:
[1]

W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949. Google Scholar

[2]

M. A. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[3]

Y. L. CaiC. D. ZhaoW. M. Wang and J. F. Wang, Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Lett., 39 (2015), 2092-2106.  doi: 10.1016/j.apm.2014.09.038.  Google Scholar

[4]

F. CourchampT. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405-410.  doi: 10.1016/S0169-5347(99)01683-3.  Google Scholar

[5]

R. H. CuiJ. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015.  Google Scholar

[6]

L. L. Du and M. X. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485.  doi: 10.1016/j.jmaa.2010.02.002.  Google Scholar

[7]

Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Diff. Equat., 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[8]

E. González-OlivaresJ. Mena-LorcaA. Rojas-Palma and J. D. Flores, Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model, 35 (2011), 366-381.  doi: 10.1016/j.apm.2010.07.001.  Google Scholar

[9]

B. Hassard, N. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

[10]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.  Google Scholar

[11]

P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245.  doi: 10.1093/biomet/35.3-4.213.  Google Scholar

[12]

P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31.  doi: 10.1093/biomet/45.1-2.16.  Google Scholar

[13]

S. B. LiJ. H. Wu and H. Nie, Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model, Comput. Math. Appl., 70 (2015), 3043-3056.  doi: 10.1016/j.camwa.2015.10.017.  Google Scholar

[14]

Y. Li, Hopf bifurcations in general systems of Brusselator type, Nonlinear Anal.: Real World Appl., 28 (2016), 32-47.  doi: 10.1016/j.nonrwa.2015.09.004.  Google Scholar

[15]

Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal.: Real World Appl., 14 (2013), 1806-1816.  doi: 10.1016/j.nonrwa.2012.11.012.  Google Scholar

[16]

N. Min and X. M. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.  doi: 10.1016/j.camwa.2016.07.028.  Google Scholar

[17]

N. Min and X. M. Wang, Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1721-1737.  doi: 10.3934/dcdsb.2018073.  Google Scholar

[18]

W. J. Ni and M. X. Wang, Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.  doi: 10.3934/dcdsb.2017172.  Google Scholar

[19]

W. J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.  doi: 10.1016/j.jde.2016.06.022.  Google Scholar

[20]

W. M. Ni, Diffusion, cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[21]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[22]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Diff. Equat., 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.  Google Scholar

[23]

E. C. Pielou, Mathematical Ecology, John Wiley & Sons, New York, RI, 1977.  Google Scholar

[24]

Y. W. Qi and Y. Zhu, Global stability of Lesile-type predator-prey model, Meth. Appl. Anal., 23 (2016), 259-268.  doi: 10.4310/MAA.2016.v23.n3.a3.  Google Scholar

[25]

P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.  Google Scholar

[26]

J. F. WangJ. P. Shi and J. J. W, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[27]

J. F. WangJ. J. Wei and J. P. Shi, Global bifurcation analysis and pattern formation inhomogeneous diffusive predator-prey systems, J. Diff. Equat., 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.  Google Scholar

[28]

M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196 (2004), 172-192.  doi: 10.1016/j.physd.2004.05.007.  Google Scholar

[29]

M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Cont. Dyn. Syst. A, 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.  Google Scholar

[30]

Y. X. Wang and W. T. Li, Spatial patterns of the Holling-Tanner predator-prey model with nonlinear diffusion effects, Appl. Anal., 92 (2013), 2168-2181.  doi: 10.1080/00036811.2012.724402.  Google Scholar

[31]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second edition. Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003.  Google Scholar

[32]

F. Q. YiJ. J. Wei and J. P. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal.: Real World Appl., 9 (2008), 1038-1051.  doi: 10.1016/j.nonrwa.2007.02.005.  Google Scholar

[33]

F. Q. YiJ. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

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
$b_1=7-4\sqrt 3$
$0<\mu<b_0$ $b_0<\mu<b^0$ $\mu>b^0$
$0<b<b_1$
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
$b_1=7-4\sqrt 3$
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
$b_1=7-4\sqrt 3$, $h_j=\mu+(d_1+d_2)j^2/l^2$
$d_1^{-1}d_2b^0<\mu<b_0$ $\max\{d_1^{-1}d_2b^0, b_0\}<\mu<b^0$ $\mu>b^0$
$0<b<b_1$
$2r-m+1$ Hopf bifurcation values

$2r+2$ Hopf bifurcation values
Null
$b_1<b<1$
$m+1$ Hopf bifurcation values
Null Null
$b_1=7-4\sqrt 3$, $h_j=\mu+(d_1+d_2)j^2/l^2$
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
$b_1=7-4\sqrt 3$, $b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
$0<\mu<1-b$ $1-b<\mu<b_0$ $b_0<\mu<b^0$
$0<b<b_1$
$m-k$ 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
$b_1=7-4\sqrt 3$, $b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
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
$b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
$0<\mu<b_0$ $b_0<\mu<1-b$ $1-b<\mu<b^0$
$b_2<b<\frac{1}{3}$
$2r-m-k$ 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
$b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
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
$b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$
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
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
$b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$
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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