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 and Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269
References:
[1]

M. A. 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. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.

[2]

E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32 (1998), 381-408. doi: 10.1016/S0362-546X(97)00491-4.

[3]

S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.

[4]

F. Chen, L. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908. doi: 10.1016/j.nonrwa.2008.09.009.

[5]

L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330-1334. doi: 10.1016/j.aml.2009.03.005.

[6]

M. Chen and C. Wei, Existence of global solutions for a ratio-dependent food chain model with diffusion, Adv. Math. (China), 39 (2010), 679-690. doi: 1000-0917(2010)06-0679-12.

[7]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.

[8]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion. I. General existence results, Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[9]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.

[10]

H. I. Freedman, M. Agarwal and S. Devi, Analysis of stability and persistence in a ratio-dependent predator-prey resource model, Int. J. Biomath., 2 (2009), 107-118. doi: 10.1142/S1793524509000522.

[11]

S. Gakkhar and B. Singh, Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters, Chaos Solitons Fractals, 27 (2006), 1239-1255. doi: 10.1016/j.chaos.2005.04.097.

[12]

X. Guan, W. Wang and Y. Cai, Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 12 (2011), 2385-2395. doi: 10.1016/j.nonrwa.2011.02.011.

[13]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506. doi: 10.1007/s002850100079.

[14]

C. Ji, D. Jiang and N. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440. doi: 10.1016/j.jmaa.2010.11.008.

[15]

C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.

[16]

W. Ko and I. Ahn, A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: I, long time behavior and stability of equilibria, J. Math. Anal. Appl., 397 (2013), 9-28. doi: 10.1016/j.jmaa.2012.07.026.

[17]

W. Ko and I. Ahn, A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: II, stationary pattern formation, J. Math. Anal. Appl., 397 (2013), 29-45. doi: 10.1016/j.jmaa.2012.07.025.

[18]

A. Korobeinikov and W. T. Lee, Global asymptotic properties for a Leslie-Gower food chain model, Math. Biosci. Eng., 6 (2009), 585-590. doi: 10.3934/mbe.2009.6.585.

[19]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105.

[20]

P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245. doi: 10.1093/biomet/35.3-4.213.

[21]

P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31. doi: 10.1093/biomet/45.1-2.16.

[22]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219.

[23]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.

[24]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[25]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26 (1996), 1889-1903. doi: 10.1016/0362-546X(95)00058-4.

[26]

H. H. Schaefer, Topological Vector Spaces, $3^{nd}$ edition, Springer-Verlag, New York-Berlin, 1971. doi: 10.1007/978-1-4612-1468-7_3.

[27]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[28]

X. Song and Y. Li, Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect, Nonlinear Anal. Real World Appl., 9 (2008), 64-79. doi: 10.1016/j.nonrwa.2006.09.004.

[29]

K. R. Upadhyay, R. K. Naji, S. N. Raw and B. Dubey, The role of top predator interference on the dynamics of a food chain model, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 757-768. doi: 10.1016/j.cnsns.2012.08.020.

[30]

R. K. Upadhyay, S. R. K. Iyengar and V. Rai, Chaos: An ecological reality?, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 1325-1333. doi: 10.1142/S0218127498001029.

[31]

M. Wang, Nonlinear Parabolic Equation (in Chinese), Science Press, Beijing, 1993.

[32]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, 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.

[33]

G. Zhang, W. Wang and X. Wang, Coexistence states for a diffusive one-prey and two-predators model with B-D functional response, J. Math. Anal. Appl., 387 (2012), 931-948. doi: 10.1016/j.jmaa.2011.09.049.

show all references

References:
[1]

M. A. 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. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.

[2]

E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32 (1998), 381-408. doi: 10.1016/S0362-546X(97)00491-4.

[3]

S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.

[4]

F. Chen, L. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908. doi: 10.1016/j.nonrwa.2008.09.009.

[5]

L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330-1334. doi: 10.1016/j.aml.2009.03.005.

[6]

M. Chen and C. Wei, Existence of global solutions for a ratio-dependent food chain model with diffusion, Adv. Math. (China), 39 (2010), 679-690. doi: 1000-0917(2010)06-0679-12.

[7]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.

[8]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion. I. General existence results, Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[9]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.

[10]

H. I. Freedman, M. Agarwal and S. Devi, Analysis of stability and persistence in a ratio-dependent predator-prey resource model, Int. J. Biomath., 2 (2009), 107-118. doi: 10.1142/S1793524509000522.

[11]

S. Gakkhar and B. Singh, Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters, Chaos Solitons Fractals, 27 (2006), 1239-1255. doi: 10.1016/j.chaos.2005.04.097.

[12]

X. Guan, W. Wang and Y. Cai, Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 12 (2011), 2385-2395. doi: 10.1016/j.nonrwa.2011.02.011.

[13]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506. doi: 10.1007/s002850100079.

[14]

C. Ji, D. Jiang and N. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440. doi: 10.1016/j.jmaa.2010.11.008.

[15]

C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.

[16]

W. Ko and I. Ahn, A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: I, long time behavior and stability of equilibria, J. Math. Anal. Appl., 397 (2013), 9-28. doi: 10.1016/j.jmaa.2012.07.026.

[17]

W. Ko and I. Ahn, A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: II, stationary pattern formation, J. Math. Anal. Appl., 397 (2013), 29-45. doi: 10.1016/j.jmaa.2012.07.025.

[18]

A. Korobeinikov and W. T. Lee, Global asymptotic properties for a Leslie-Gower food chain model, Math. Biosci. Eng., 6 (2009), 585-590. doi: 10.3934/mbe.2009.6.585.

[19]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105.

[20]

P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245. doi: 10.1093/biomet/35.3-4.213.

[21]

P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31. doi: 10.1093/biomet/45.1-2.16.

[22]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219.

[23]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.

[24]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[25]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26 (1996), 1889-1903. doi: 10.1016/0362-546X(95)00058-4.

[26]

H. H. Schaefer, Topological Vector Spaces, $3^{nd}$ edition, Springer-Verlag, New York-Berlin, 1971. doi: 10.1007/978-1-4612-1468-7_3.

[27]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[28]

X. Song and Y. Li, Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect, Nonlinear Anal. Real World Appl., 9 (2008), 64-79. doi: 10.1016/j.nonrwa.2006.09.004.

[29]

K. R. Upadhyay, R. K. Naji, S. N. Raw and B. Dubey, The role of top predator interference on the dynamics of a food chain model, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 757-768. doi: 10.1016/j.cnsns.2012.08.020.

[30]

R. K. Upadhyay, S. R. K. Iyengar and V. Rai, Chaos: An ecological reality?, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 1325-1333. doi: 10.1142/S0218127498001029.

[31]

M. Wang, Nonlinear Parabolic Equation (in Chinese), Science Press, Beijing, 1993.

[32]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, 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.

[33]

G. Zhang, W. Wang and X. Wang, Coexistence states for a diffusive one-prey and two-predators model with B-D functional response, J. Math. Anal. Appl., 387 (2012), 931-948. doi: 10.1016/j.jmaa.2011.09.049.

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