American Institute of Mathematical Sciences

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September  2015, 20(7): 2269-2290. doi: 10.3934/dcdsb.2015.20.2269

Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses

Wen-Bin Yang 1, , Yan-Ling Li 2, , Jianhua Wu 3, and  Hai-Xia Li 2,

1. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shannxi 710062
2. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China, China
3. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119

Received  May 2014 Revised  December 2014 Published  July 2015

The paper is concerned with a diffusive food chain model subject to homogeneous Robin boundary conditions, which models the trophic interactions of three levels. Using the fixed point index theory, we obtain the existence and uniqueness for coexistence states. Moreover, the existence of the global attractor and the extinction for the time-dependent model are established under certain assumptions. Some numerical simulations are done to complement the analytical results.
Keywords: numerical simulation., Leslie-Gower, Ratio-dependent, coexistence state, global attractor.
Mathematics Subject Classification: Primary: 35K57, 35J55; Secondary: 92B0.
Citation: Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269
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show all references

References:
[1]

Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[2]

Nonlinear Anal., 32 (1998), 381-408. doi: 10.1016/S0362-546X(97)00491-4.  Google Scholar

[3]

Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.  Google Scholar

[4]

Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908. doi: 10.1016/j.nonrwa.2008.09.009.  Google Scholar

[5]

Appl. Math. Lett., 22 (2009), 1330-1334. doi: 10.1016/j.aml.2009.03.005.  Google Scholar

[6]

Adv. Math. (China), 39 (2010), 679-690. doi: 1000-0917(2010)06-0679-12.  Google Scholar

[7]

Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

[8]

Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar

[9]

Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.  Google Scholar

[10]

Int. J. Biomath., 2 (2009), 107-118. doi: 10.1142/S1793524509000522.  Google Scholar

[11]

Chaos Solitons Fractals, 27 (2006), 1239-1255. doi: 10.1016/j.chaos.2005.04.097.  Google Scholar

[12]

Nonlinear Anal. Real World Appl., 12 (2011), 2385-2395. doi: 10.1016/j.nonrwa.2011.02.011.  Google Scholar

[13]

J. Math. Biol., 42 (2001), 489-506. doi: 10.1007/s002850100079.  Google Scholar

[14]

J. Math. Anal. Appl., 377 (2011), 435-440. doi: 10.1016/j.jmaa.2010.11.008.  Google Scholar

[15]

J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.  Google Scholar

[16]

J. Math. Anal. Appl., 397 (2013), 9-28. doi: 10.1016/j.jmaa.2012.07.026.  Google Scholar

[17]

J. Math. Anal. Appl., 397 (2013), 29-45. doi: 10.1016/j.jmaa.2012.07.025.  Google Scholar

[18]

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[19]

J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105.  Google Scholar

[20]

Biometrika, 35 (1948), 213-245. doi: 10.1093/biomet/35.3-4.213.  Google Scholar

[21]

Biometrika, 45 (1958), 16-31. doi: 10.1093/biomet/45.1-2.16.  Google Scholar

[22]

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[23]

Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

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Plenum Press, New York, 1992.  Google Scholar

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[26]

$3^{nd}$ edition, Springer-Verlag, New York-Berlin, 1971. doi: 10.1007/978-1-4612-1468-7_3.  Google Scholar

[27]

$2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[28]

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[29]

Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 757-768. doi: 10.1016/j.cnsns.2012.08.020.  Google Scholar

[30]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 1325-1333. doi: 10.1142/S0218127498001029.  Google Scholar

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Science Press, Beijing, 1993. Google Scholar

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in Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., 4, Elsevier/North-Holland, Amsterdam, 2008, 411-501. doi: 10.1016/S1874-5733(08)80023-X.  Google Scholar

[33]

J. Math. Anal. Appl., 387 (2012), 931-948. doi: 10.1016/j.jmaa.2011.09.049.  Google Scholar

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