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

Zengji Du; Xiao Chen; Zhaosheng Feng

doi:10.3934/dcdss.2014.7.1203

Article Contents

2014, Volume 7, Issue 6: 1203-1214. Doi: 10.3934/dcdss.2014.7.1203

This issue Previous Article Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity Next Article Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain

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
2.
Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received: December 31, 2012

Revised: September 30, 2013

Published: November 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C25, 92D25.

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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), 1054-1068.doi: 10.1016/j.apm.2012.03.022.
[2]	R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.
[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), 2654-2664.doi: 10.1016/j.apm.2009.12.003.
[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), 1699-1722.doi: 10.3934/cpaa.2012.11.1699.
[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), 857-871.doi: 10.3934/dcdsb.2008.10.857.
[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), 607-614.doi: 10.3934/dcdsb.2004.4.607.
[7]	Y. Kuang, Delay Differential Equations, with Applications in Population Dynamics, Academic Press, New York, 1993.
[8]	K. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst., 32 (2012), 901-933.doi: 10.3934/dcds.2012.32.901.
[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), 644-654.doi: 10.1016/j.jmaa.2010.08.029.
[10]	A. J. Lotka, Elements of Physical Biology, Williams & Wilkins Co., Baltimore, 1925.
[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), 3654-3664.doi: 10.1016/j.nonrwa.2011.06.022.
[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), 1104-1118.doi: 10.1016/j.nonrwa.2005.10.003.
[13]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei Roma, 2 (1926), 31-113.
[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), 1061-1080.doi: 10.1017/S0308210509000134.
[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), 1705-1719.doi: 10.1216/RMJ-2008-38-5-1705.
[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), 423-433.
[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), 2130-2138.doi: 10.1016/j.cnsns.2010.08.028.
[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), 1178-1200.doi: 10.1137/080728512.
[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), 400-408.doi: 10.1016/j.jmaa.2011.05.081.