[1]
|
W. C. Allee,
Principles of Animal Ecology, Saunders, RI, 1949.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[9]
|
B. Hassard, N. Kazarinoff and Y. H. Wan,
Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[20]
|
W. M. Ni, Diffusion, cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[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.
|
[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.
|
[23]
|
E. C. Pielou,
Mathematical Ecology, John Wiley & Sons, New York, RI, 1977.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[31]
|
S. Wiggins,
Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second edition. Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003.
|
[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.
|
[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.
|