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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

Jun Zhou 1,

1. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

Received  April 2014 Revised  January 2015 Published  March 2015

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.  Google Scholar

[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.  Google Scholar

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B. I. Camara and M. Aziz-Alaoui, Dynamics of predator-prey model with diffusion,, \emph{Dyn. Contin. Discrete Impuls. Syst. A}, 15 (2008), 897.   Google Scholar

[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.   Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons, (2004).  doi: 10.1002/0470871296.  Google Scholar

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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.  Google Scholar

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D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, volume 224,, Springer, (2001).   Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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R. M. May, Stability and Complexity in Model Ecosystems, volume 6,, Princeton University Press, (2001).   Google Scholar

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C. Neuhauser, Mathematical challenges in spatial ecology,, \emph{Not. AMS}, 48 (2001), 1304.   Google Scholar

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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.  Google Scholar

[16]

E. C. Pielou, Mathematical Ecology,, John wiley & sons, (1977).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations (in Chinese),, Scientific Press, (1990).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons, (2004).  doi: 10.1002/0470871296.  Google Scholar

[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.  Google Scholar

[8]

D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, volume 224,, Springer, (2001).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

E. C. Pielou, Mathematical Ecology,, John wiley & sons, (1977).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations (in Chinese),, Scientific Press, (1990).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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