American Institute of Mathematical Sciences

Advanced
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 and 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, 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.

[1]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[2]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[3]

Nannan Li, Wenxian Sun, Shengqiang Liu. A stage-structured predator-prey model with Crowley-Martin functional response. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022177
[4]

Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228
[5]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
[6]

Hai-Xia Li, Jian-Hua Wu, Yan-Ling Li, Chun-An Liu. Positive solutions to the unstirred chemostat model with Crowley-Martin functional response. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2951-2966. doi: 10.3934/dcdsb.2017128
[7]

Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133
[8]

Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations and Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
[9]

C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289
[10]

Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 408-425. doi: 10.3934/dcdsb.2022082
[11]

Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
[12]

Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042
[13]

Rong Zou, Shangjiang Guo. Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4189-4210. doi: 10.3934/dcdsb.2020093
[14]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2189-2219. doi: 10.3934/dcdsb.2021129
[15]

Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117
[16]

Biruk Tafesse Mulugeta, Liping Yu, Qigang Yuan, Jingli Ren. Bifurcation analysis of a predator-prey model with strong Allee effect and Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022153
[17]

Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607
[18]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
[19]

Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065
[20]

Yong Yao, Lingling Liu. Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 4787-4815. doi: 10.3934/dcdsb.2021252

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (184)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy