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Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses
1. | School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
References:
[1] |
N. Ali and M. Jazar, Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses,, \emph{J. Appl. Math. Compu.}, 43 (2013), 271.
doi: 10.1007/s12190-013-0663-3. |
[2] |
M. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,, \emph{Appl. Math. Letters}, 16 (2003), 1069.
doi: 10.1016/S0893-9659(03)90096-6. |
[3] |
B. I. Camara and M. Aziz-Alaoui, Dynamics of predator-prey model with diffusion,, \emph{Dyn. Contin. Discrete Impuls. Syst. A}, 15 (2008), 897.
|
[4] |
B. I. Camara and M. Aziz-Alaoui, Turing and hopf patterns formation in a predator-prey model with Leslie-Gower type functional response,, \emph{Dyn. Cont. Discr. Impul. Syst. B}, 16 (2009), 479.
|
[5] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons, (2004).
doi: 10.1002/0470871296. |
[6] |
P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population,, \emph{J. North Amer. Benth. Soc.}, (1989), 211. Google Scholar |
[7] |
Q. Dong, W. Ma, and M. Sun, The asymptotic behavior of a chemostat model with Crowley-Martin type functional response and time delays,, \emph{J. Math. Chem.}, 5 (2003), 1231.
doi: 10.1007/s10910-012-0138-z. |
[8] |
D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, volume 224,, Springer, (2001).
|
[9] |
B. D. Hassard, Theory and Applications of Hopf Bifurcation, volume 41,, CUP Archive, (1981). Google Scholar |
[10] |
C. S. Holling, Some characteristics of simple types of predation and parasitism,, \emph{The Canad. Entomol.}, 91 (1959), 385. Google Scholar |
[11] |
P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species,, \emph{Biometrika}, 47 (1960), 219.
|
[12] |
X.-Q. Liu, S.-M. Zhong, B.-D. Tian, and F.-X. Zheng, Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response,, \emph{J. Appl. Math. Compu.}, 43 (2013), 479.
doi: 10.1007/s12190-013-0674-0. |
[13] |
R. M. May, Stability and Complexity in Model Ecosystems, volume 6,, Princeton University Press, (2001). Google Scholar |
[14] |
C. Neuhauser, Mathematical challenges in spatial ecology,, \emph{Not. AMS}, 48 (2001), 1304.
|
[15] |
R. Peng and M. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model,, \emph{J. Math. Anal. Appl.}, 316 (2006), 256.
doi: 10.1016/j.jmaa.2005.04.033. |
[16] |
E. C. Pielou, Mathematical Ecology,, John wiley & sons, (1977).
|
[17] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, \emph{J. Differential Equations}, 246 (2009), 2788.
doi: 10.1016/j.jde.2008.09.009. |
[18] |
X. Shi, X. Zhou, and X. Song, Analysis of a stage-structured predator-prey model with Crowley-Martin function,, \emph{J. Appl. Math. Compu.}, 36 (2011), 459.
doi: 10.1007/s12190-010-0413-8. |
[19] |
Y. Tian and P. Weng, Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes,, \emph{Acta Appl. Math.}, 114 (2011), 173.
doi: 10.1007/s10440-011-9607-9. |
[20] |
A. M. Turing, The chemical basis of morphogenesis,, \emph{Phil. Tans. R. Soc. London, 237 (1952), 37. Google Scholar |
[21] |
R. Upadhyay, S. Raw, and V. Rai, Dynamical complexities in a tri-trophic hybrid food chain model with Holling type II and Crowley-Martin functional responses,, \emph{Nonlinear Analy.: Model. Cont.}, 15 (2010), 361.
|
[22] |
R. K. Upadhyay and R. K. Naji, Dynamics of a three species food chain model with Crowley-Martin type functional response,, \emph{Chao. Soli. Fract.}, 42 (2009), 1337.
doi: 10.1016/j.chaos.2009.03.020. |
[23] |
J. Wang, J. Shi, and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong allee effect in prey,, \emph{J. Differential Equations}, 251 (2011), 1276.
doi: 10.1016/j.jde.2011.03.004. |
[24] |
Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations (in Chinese),, Scientific Press, (1990).
|
[25] |
F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, \emph{J. Differential Equations}, 246 (2009), 1944.
doi: 10.1016/j.jde.2008.10.024. |
[26] |
J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes,, \emph{J. Math. Anal. Appl.}, 389 (2012), 1380.
doi: 10.1016/j.jmaa.2012.01.013. |
[27] |
J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with bazykin functional response,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 1.
doi: 10.1007/s00033-013-0315-3. |
[28] |
J. Zhou, Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type II functional response and density-dependent diffusion,, \emph{Nonlinear Anal.: TMA}, 82 (2013), 47.
doi: 10.1016/j.na.2012.12.014. |
[29] |
J. Zhou and J. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses,, \emph{J. Math. Anal. Appl.}, 405 (2013), 618.
doi: 10.1016/j.jmaa.2013.03.064. |
show all references
References:
[1] |
N. Ali and M. Jazar, Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses,, \emph{J. Appl. Math. Compu.}, 43 (2013), 271.
doi: 10.1007/s12190-013-0663-3. |
[2] |
M. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,, \emph{Appl. Math. Letters}, 16 (2003), 1069.
doi: 10.1016/S0893-9659(03)90096-6. |
[3] |
B. I. Camara and M. Aziz-Alaoui, Dynamics of predator-prey model with diffusion,, \emph{Dyn. Contin. Discrete Impuls. Syst. A}, 15 (2008), 897.
|
[4] |
B. I. Camara and M. Aziz-Alaoui, Turing and hopf patterns formation in a predator-prey model with Leslie-Gower type functional response,, \emph{Dyn. Cont. Discr. Impul. Syst. B}, 16 (2009), 479.
|
[5] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons, (2004).
doi: 10.1002/0470871296. |
[6] |
P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population,, \emph{J. North Amer. Benth. Soc.}, (1989), 211. Google Scholar |
[7] |
Q. Dong, W. Ma, and M. Sun, The asymptotic behavior of a chemostat model with Crowley-Martin type functional response and time delays,, \emph{J. Math. Chem.}, 5 (2003), 1231.
doi: 10.1007/s10910-012-0138-z. |
[8] |
D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, volume 224,, Springer, (2001).
|
[9] |
B. D. Hassard, Theory and Applications of Hopf Bifurcation, volume 41,, CUP Archive, (1981). Google Scholar |
[10] |
C. S. Holling, Some characteristics of simple types of predation and parasitism,, \emph{The Canad. Entomol.}, 91 (1959), 385. Google Scholar |
[11] |
P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species,, \emph{Biometrika}, 47 (1960), 219.
|
[12] |
X.-Q. Liu, S.-M. Zhong, B.-D. Tian, and F.-X. Zheng, Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response,, \emph{J. Appl. Math. Compu.}, 43 (2013), 479.
doi: 10.1007/s12190-013-0674-0. |
[13] |
R. M. May, Stability and Complexity in Model Ecosystems, volume 6,, Princeton University Press, (2001). Google Scholar |
[14] |
C. Neuhauser, Mathematical challenges in spatial ecology,, \emph{Not. AMS}, 48 (2001), 1304.
|
[15] |
R. Peng and M. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model,, \emph{J. Math. Anal. Appl.}, 316 (2006), 256.
doi: 10.1016/j.jmaa.2005.04.033. |
[16] |
E. C. Pielou, Mathematical Ecology,, John wiley & sons, (1977).
|
[17] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, \emph{J. Differential Equations}, 246 (2009), 2788.
doi: 10.1016/j.jde.2008.09.009. |
[18] |
X. Shi, X. Zhou, and X. Song, Analysis of a stage-structured predator-prey model with Crowley-Martin function,, \emph{J. Appl. Math. Compu.}, 36 (2011), 459.
doi: 10.1007/s12190-010-0413-8. |
[19] |
Y. Tian and P. Weng, Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes,, \emph{Acta Appl. Math.}, 114 (2011), 173.
doi: 10.1007/s10440-011-9607-9. |
[20] |
A. M. Turing, The chemical basis of morphogenesis,, \emph{Phil. Tans. R. Soc. London, 237 (1952), 37. Google Scholar |
[21] |
R. Upadhyay, S. Raw, and V. Rai, Dynamical complexities in a tri-trophic hybrid food chain model with Holling type II and Crowley-Martin functional responses,, \emph{Nonlinear Analy.: Model. Cont.}, 15 (2010), 361.
|
[22] |
R. K. Upadhyay and R. K. Naji, Dynamics of a three species food chain model with Crowley-Martin type functional response,, \emph{Chao. Soli. Fract.}, 42 (2009), 1337.
doi: 10.1016/j.chaos.2009.03.020. |
[23] |
J. Wang, J. Shi, and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong allee effect in prey,, \emph{J. Differential Equations}, 251 (2011), 1276.
doi: 10.1016/j.jde.2011.03.004. |
[24] |
Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations (in Chinese),, Scientific Press, (1990).
|
[25] |
F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, \emph{J. Differential Equations}, 246 (2009), 1944.
doi: 10.1016/j.jde.2008.10.024. |
[26] |
J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes,, \emph{J. Math. Anal. Appl.}, 389 (2012), 1380.
doi: 10.1016/j.jmaa.2012.01.013. |
[27] |
J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with bazykin functional response,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 1.
doi: 10.1007/s00033-013-0315-3. |
[28] |
J. Zhou, Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type II functional response and density-dependent diffusion,, \emph{Nonlinear Anal.: TMA}, 82 (2013), 47.
doi: 10.1016/j.na.2012.12.014. |
[29] |
J. Zhou and J. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses,, \emph{J. Math. Anal. Appl.}, 405 (2013), 618.
doi: 10.1016/j.jmaa.2013.03.064. |
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