December  2014, 7(6): 1203-1214. doi: 10.3934/dcdss.2014.7.1203

Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms

1. 

School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China, China

2. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  January 2013 Revised  October 2013 Published  June 2014

In this paper, we study a delayed predator-prey model with the Leslie-Gower Holling-type II functional response and harvesting terms. The existence of multiple positive periodic solutions for the system and the permanence of the predator-prey model are obtained by means of the generalized Mawhin coincidence degree theory.
Citation: 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 & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
References:
[1]

Z. Du and Y. Lv, Permanence and almost periodic solution of a Lotka-Volterra Model with mutual interference and time delays,, Appl. Math. Model., 37 (2013), 1054.  doi: 10.1016/j.apm.2012.03.022.  Google Scholar

[2]

R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations,, Springer-Verlag, (1977).   Google Scholar

[3]

D. C. He, W. T. Huang and Q. J. Xu, The dynamic complexity of an impulsive Holling II predator-prey model with mutual interference,, Appl. Math. Model, 34 (2010), 2654.  doi: 10.1016/j.apm.2009.12.003.  Google Scholar

[4]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal,, Commun. Pure Appl. Anal., 11 (2012), 1699.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[5]

S. Hsu, T. Hwang and Y. Kuang, Global dynamics of a predator-prey model with hassell-varley type functional response,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 857.  doi: 10.3934/dcdsb.2008.10.857.  Google Scholar

[6]

H. Y. Jing and Z. Y. Yang, The impact of state feedback control on a predator-prey model with functional response,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 607.  doi: 10.3934/dcdsb.2004.4.607.  Google Scholar

[7]

Y. Kuang, Delay Differential Equations, with Applications in Population Dynamics,, Academic Press, (1993).   Google Scholar

[8]

K. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates,, Discrete Contin. Dyn. Syst., 32 (2012), 901.  doi: 10.3934/dcds.2012.32.901.  Google Scholar

[9]

H. Y. Li and Y. Takeuchi, Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response,, J. Math. Anal. Appl., 374 (2011), 644.  doi: 10.1016/j.jmaa.2010.08.029.  Google Scholar

[10]

A. J. Lotka, Elements of Physical Biology,, Williams & Wilkins Co., (1925).   Google Scholar

[11]

Y. Lv and Z. Du, Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response,, Nonlinear Anal. (RWA), 12 (2011), 3654.  doi: 10.1016/j.nonrwa.2011.06.022.  Google Scholar

[12]

A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay,, Nonlinear Anal. (RWA), 7 (2006), 1104.  doi: 10.1016/j.nonrwa.2005.10.003.  Google Scholar

[13]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,, Mem. Acad. Lincei Roma, 2 (1926), 31.   Google Scholar

[14]

J. Y. Wang and Z. Feng, A non-autonomous competitive system with stage structure and distributed delays,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1061.  doi: 10.1017/S0308210509000134.  Google Scholar

[15]

L. L. Wang, Y. H. Fan and W. Gao, Periodic solutions in a delayed predator-prey model with nonmonotonic functional response,, Rocky Mountain J. Math., 38 (2008), 1705.  doi: 10.1216/RMJ-2008-38-5-1705.  Google Scholar

[16]

Z. Wang and J. Wu, Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking,, Commun. Pure Appl. Anal., 5 (2006), 423.   Google Scholar

[17]

F. Wei, Existence of multiple positive periodic solutions to a predator-prey system with harvesting terms and Holling III type functional response,, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2130.  doi: 10.1016/j.cnsns.2010.08.028.  Google Scholar

[18]

J. Xia, Z. Liu, R. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with Holling type II functional response,, SIAM J. Appl. Math., 70 (2009), 1178.  doi: 10.1137/080728512.  Google Scholar

[19]

Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solution for a predator-prey model with modified Leslie-Gower Holling-type II schemes,, J. Math. Anal. Appl., 384 (2011), 400.  doi: 10.1016/j.jmaa.2011.05.081.  Google Scholar

show all references

References:
[1]

Z. Du and Y. Lv, Permanence and almost periodic solution of a Lotka-Volterra Model with mutual interference and time delays,, Appl. Math. Model., 37 (2013), 1054.  doi: 10.1016/j.apm.2012.03.022.  Google Scholar

[2]

R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations,, Springer-Verlag, (1977).   Google Scholar

[3]

D. C. He, W. T. Huang and Q. J. Xu, The dynamic complexity of an impulsive Holling II predator-prey model with mutual interference,, Appl. Math. Model, 34 (2010), 2654.  doi: 10.1016/j.apm.2009.12.003.  Google Scholar

[4]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal,, Commun. Pure Appl. Anal., 11 (2012), 1699.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[5]

S. Hsu, T. Hwang and Y. Kuang, Global dynamics of a predator-prey model with hassell-varley type functional response,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 857.  doi: 10.3934/dcdsb.2008.10.857.  Google Scholar

[6]

H. Y. Jing and Z. Y. Yang, The impact of state feedback control on a predator-prey model with functional response,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 607.  doi: 10.3934/dcdsb.2004.4.607.  Google Scholar

[7]

Y. Kuang, Delay Differential Equations, with Applications in Population Dynamics,, Academic Press, (1993).   Google Scholar

[8]

K. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates,, Discrete Contin. Dyn. Syst., 32 (2012), 901.  doi: 10.3934/dcds.2012.32.901.  Google Scholar

[9]

H. Y. Li and Y. Takeuchi, Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response,, J. Math. Anal. Appl., 374 (2011), 644.  doi: 10.1016/j.jmaa.2010.08.029.  Google Scholar

[10]

A. J. Lotka, Elements of Physical Biology,, Williams & Wilkins Co., (1925).   Google Scholar

[11]

Y. Lv and Z. Du, Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response,, Nonlinear Anal. (RWA), 12 (2011), 3654.  doi: 10.1016/j.nonrwa.2011.06.022.  Google Scholar

[12]

A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay,, Nonlinear Anal. (RWA), 7 (2006), 1104.  doi: 10.1016/j.nonrwa.2005.10.003.  Google Scholar

[13]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,, Mem. Acad. Lincei Roma, 2 (1926), 31.   Google Scholar

[14]

J. Y. Wang and Z. Feng, A non-autonomous competitive system with stage structure and distributed delays,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1061.  doi: 10.1017/S0308210509000134.  Google Scholar

[15]

L. L. Wang, Y. H. Fan and W. Gao, Periodic solutions in a delayed predator-prey model with nonmonotonic functional response,, Rocky Mountain J. Math., 38 (2008), 1705.  doi: 10.1216/RMJ-2008-38-5-1705.  Google Scholar

[16]

Z. Wang and J. Wu, Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking,, Commun. Pure Appl. Anal., 5 (2006), 423.   Google Scholar

[17]

F. Wei, Existence of multiple positive periodic solutions to a predator-prey system with harvesting terms and Holling III type functional response,, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2130.  doi: 10.1016/j.cnsns.2010.08.028.  Google Scholar

[18]

J. Xia, Z. Liu, R. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with Holling type II functional response,, SIAM J. Appl. Math., 70 (2009), 1178.  doi: 10.1137/080728512.  Google Scholar

[19]

Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solution for a predator-prey model with modified Leslie-Gower Holling-type II schemes,, J. Math. Anal. Appl., 384 (2011), 400.  doi: 10.1016/j.jmaa.2011.05.081.  Google Scholar

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