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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, J. Appl. Math. Compu., 43 (2013), 271-293.
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, Appl. Math. Letters, 16 (2003), 1069-1075.
doi: 10.1016/S0893-9659(03)90096-6. |
[3] |
B. I. Camara and M. Aziz-Alaoui, Dynamics of predator-prey model with diffusion, Dyn. Contin. Discrete Impuls. Syst. A, 15 (2008), 897-906. |
[4] |
B. I. Camara and M. Aziz-Alaoui, Turing and hopf patterns formation in a predator-prey model with Leslie-Gower type functional response, Dyn. Cont. Discr. Impul. Syst. B, 16 (2009), 479-488. |
[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, J. North Amer. Benth. Soc., (1989), 211-221. |
[7] |
Q. Dong, W. Ma, and M. Sun, The asymptotic behavior of a chemostat model with Crowley-Martin type functional response and time delays, J. Math. Chem., 5 (2003), 1231-1248.
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. |
[10] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canad. Entomol., 91 (1959), 385-398. |
[11] |
P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. |
[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, J. Appl. Math. Compu., 43 (2013), 479-490.
doi: 10.1007/s12190-013-0674-0. |
[13] |
R. M. May, Stability and Complexity in Model Ecosystems, volume 6, Princeton University Press, 2001. |
[14] |
C. Neuhauser, Mathematical challenges in spatial ecology, Not. AMS, 48 (2001), 1304-1314. |
[15] |
R. Peng and M. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268.
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, J. Differential Equations, 246 (2009), 2788-2812.
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, J. Appl. Math. Compu., 36 (2011), 459-472.
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, Acta Appl. Math., 114 (2011), 173-192.
doi: 10.1007/s10440-011-9607-9. |
[20] |
A. M. Turing, The chemical basis of morphogenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. |
[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, Nonlinear Analy.: Model. Cont., 15 (2010), 361-375. |
[22] |
R. K. Upadhyay and R. K. Naji, Dynamics of a three species food chain model with Crowley-Martin type functional response, Chao. Soli. Fract., 42 (2009), 1337-1346.
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, J. Differential Equations, 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[24] |
Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Scientific Press, Beijin, 1990. |
[25] |
F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.
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, J. Math. Anal. Appl., 389 (2012), 1380-1393.
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, Z. Angew. Math. Phys., 65 (2014), 1-18.
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, Nonlinear Anal.: TMA, 82 (2013), 47-65.
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, J. Math. Anal. Appl., 405 (2013), 618-630.
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, J. Appl. Math. Compu., 43 (2013), 271-293.
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, Appl. Math. Letters, 16 (2003), 1069-1075.
doi: 10.1016/S0893-9659(03)90096-6. |
[3] |
B. I. Camara and M. Aziz-Alaoui, Dynamics of predator-prey model with diffusion, Dyn. Contin. Discrete Impuls. Syst. A, 15 (2008), 897-906. |
[4] |
B. I. Camara and M. Aziz-Alaoui, Turing and hopf patterns formation in a predator-prey model with Leslie-Gower type functional response, Dyn. Cont. Discr. Impul. Syst. B, 16 (2009), 479-488. |
[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, J. North Amer. Benth. Soc., (1989), 211-221. |
[7] |
Q. Dong, W. Ma, and M. Sun, The asymptotic behavior of a chemostat model with Crowley-Martin type functional response and time delays, J. Math. Chem., 5 (2003), 1231-1248.
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. |
[10] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canad. Entomol., 91 (1959), 385-398. |
[11] |
P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. |
[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, J. Appl. Math. Compu., 43 (2013), 479-490.
doi: 10.1007/s12190-013-0674-0. |
[13] |
R. M. May, Stability and Complexity in Model Ecosystems, volume 6, Princeton University Press, 2001. |
[14] |
C. Neuhauser, Mathematical challenges in spatial ecology, Not. AMS, 48 (2001), 1304-1314. |
[15] |
R. Peng and M. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268.
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, J. Differential Equations, 246 (2009), 2788-2812.
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, J. Appl. Math. Compu., 36 (2011), 459-472.
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, Acta Appl. Math., 114 (2011), 173-192.
doi: 10.1007/s10440-011-9607-9. |
[20] |
A. M. Turing, The chemical basis of morphogenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. |
[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, Nonlinear Analy.: Model. Cont., 15 (2010), 361-375. |
[22] |
R. K. Upadhyay and R. K. Naji, Dynamics of a three species food chain model with Crowley-Martin type functional response, Chao. Soli. Fract., 42 (2009), 1337-1346.
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, J. Differential Equations, 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[24] |
Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Scientific Press, Beijin, 1990. |
[25] |
F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.
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, J. Math. Anal. Appl., 389 (2012), 1380-1393.
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, Z. Angew. Math. Phys., 65 (2014), 1-18.
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, Nonlinear Anal.: TMA, 82 (2013), 47-65.
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, J. Math. Anal. Appl., 405 (2013), 618-630.
doi: 10.1016/j.jmaa.2013.03.064. |
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