Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses

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May 2015, 14(3): 1127-1145. doi: 10.3934/cpaa.2015.14.1127

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

Received April 2014 Revised January 2015 Published March 2015

Abstract
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In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional response. We show the existence of a bounded positive invariant attracting set and establish the permanence conditions. The parameter regions for the stability and instability of the unique constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method.

Keywords: Crowley-Martin functional response, stability, Hopf bifurcation, Leslie-Gower predator-prey model, steady state bifurcation..

Mathematics Subject Classification: Primary: 35B32, 35B50, 35J65, 35K57, 37C25; Secondary: 92D2.

Citation: 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

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.
[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).
[10]	C. S. Holling, Some characteristics of simple types of predation and parasitism,, \emph{The Canad. Entomol.}, 91 (1959), 385.
[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).
[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.
[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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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.
[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).
[10]	C. S. Holling, Some characteristics of simple types of predation and parasitism,, \emph{The Canad. Entomol.}, 91 (1959), 385.
[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).
[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.
[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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