American Institute of Mathematical Sciences

Advanced
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

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

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.

Keywords: Leslie-Gower prey-predator model, Allee effect, Hopf bifurcation, Steady-state bifurcation.
Mathematics Subject Classification: 35B10, 35B36, 35B32, 92D25.
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, 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. Cai, C. D. Zhao, W. 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. Courchamp, T. 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. Cui, J. 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-Olivares, J. Mena-Lorca, A. 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. Li, J. 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. Wang, J. 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. Wang, J. 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. Yi, J. 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. Yi, J. 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. Cai, C. D. Zhao, W. 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. Courchamp, T. 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. Cui, J. 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-Olivares, J. Mena-Lorca, A. 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. Li, J. 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. Wang, J. 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. Wang, J. 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. Yi, J. 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. Yi, J. 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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
$b_1=7-4\sqrt 3$
Table Options

Download as excel

$0<\mu<b_0$ $b_0<\mu<b^0$ $\mu>b^0$
$0<b<b_1$
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
$b_1=7-4\sqrt 3$, $h_j=\mu+(d_1+d_2)j^2/l^2$
Table Options

Download as excel

$d_1^{-1}d_2b^0<\mu<b_0$ $\max\{d_1^{-1}d_2b^0, b_0\}<\mu<b^0$ $\mu>b^0$
$0<b<b_1$
$2r-m+1$ 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
$b_1=7-4\sqrt 3$, $b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
Table Options

Download as excel

$0<\mu<1-b$ $1-b<\mu<b_0$ $b_0<\mu<b^0$
$0<b<b_1$
$m-k$ 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
$b_2=3-2\sqrt 2$, $h_j=\mu+(d_1+d_2)j^2/l^2$
Table Options

Download as excel

$0<\mu<b_0$ $b_0<\mu<1-b$ $1-b<\mu<b^0$
$b_2<b<\frac{1}{3}$
$2r-m-k$ 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
Table Options

Download as excel

$b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$
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
Table Options

Download as excel

$b$ $\mu$ $\beta$ $d_1$ $d_2$ $l$
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, 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]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129
[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]

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
[11]

Rong Zou, Shangjiang Guo. Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4189-4210. doi: 10.3934/dcdsb.2020093
[12]

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
[13]

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
[14]

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
[15]

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
[16]

Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
[17]

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
[18]

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
[19]

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
[20]

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

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (308)
  • HTML views (192)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences