# American Institute of Mathematical Sciences

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

 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.
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
##### 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. doi: 10.1016/S0893-9659(03)90096-6. Google Scholar [2] E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems,, Nonlinear Anal., 32 (1998), 381. doi: 10.1016/S0362-546X(97)00491-4. Google Scholar [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. doi: 10.1016/S0362-546X(01)00116-X. Google Scholar [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. doi: 10.1016/j.nonrwa.2008.09.009. Google Scholar [5] L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls,, Appl. Math. Lett., 22 (2009), 1330. doi: 10.1016/j.aml.2009.03.005. Google Scholar [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. doi: 1000-0917(2010)06-0679-12. Google Scholar [7] E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729. doi: 10.1090/S0002-9947-1984-0743741-4. Google Scholar [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. doi: 10.1016/0362-546X(94)E0063-M. Google Scholar [9] Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model,, Trans. Amer. Math. Soc., 349 (1997), 2443. doi: 10.1090/S0002-9947-97-01842-4. Google Scholar [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. doi: 10.1142/S1793524509000522. Google Scholar [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. doi: 10.1016/j.chaos.2005.04.097. Google Scholar [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. doi: 10.1016/j.nonrwa.2011.02.011. Google Scholar [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. doi: 10.1007/s002850100079. Google Scholar [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. doi: 10.1016/j.jmaa.2010.11.008. Google Scholar [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. doi: 10.1016/j.jmaa.2009.05.039. Google Scholar [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. doi: 10.1016/j.jmaa.2012.07.026. Google Scholar [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. doi: 10.1016/j.jmaa.2012.07.025. Google Scholar [18] A. Korobeinikov and W. T. Lee, Global asymptotic properties for a Leslie-Gower food chain model,, Math. Biosci. Eng., 6 (2009), 585. doi: 10.3934/mbe.2009.6.585. Google Scholar [19] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, J. Math. Biol., 36 (1998), 389. doi: 10.1007/s002850050105. Google Scholar [20] P. H. Leslie, Some further notes on the use of matrices in population mathematics,, Biometrika, 35 (1948), 213. doi: 10.1093/biomet/35.3-4.213. Google Scholar [21] P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods,, Biometrika, 45 (1958), 16. doi: 10.1093/biomet/45.1-2.16. Google Scholar [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. doi: 10.1093/biomet/47.3-4.219. Google Scholar [23] L. Li, Coexistence theorems of steady states for predator-prey interacting systems,, Trans. Amer. Math. Soc., 305 (1988), 143. doi: 10.1090/S0002-9947-1988-0920151-1. Google Scholar [24] C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar [25] C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems,, Nonlinear Anal., 26 (1996), 1889. doi: 10.1016/0362-546X(95)00058-4. Google Scholar [26] H. H. Schaefer, Topological Vector Spaces,, $3^{nd}$ edition, (1971). doi: 10.1007/978-1-4612-1468-7_3. Google Scholar [27] J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (1994). doi: 10.1007/978-1-4612-0873-0. Google Scholar [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. doi: 10.1016/j.nonrwa.2006.09.004. Google Scholar [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. doi: 10.1016/j.cnsns.2012.08.020. Google Scholar [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. doi: 10.1142/S0218127498001029. Google Scholar [31] M. Wang, Nonlinear Parabolic Equation (in Chinese),, Science Press, (1993). Google Scholar [32] Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion,, in Handbook of Differential Equations: Stationary Partial Differential Equations, (2008), 411. doi: 10.1016/S1874-5733(08)80023-X. Google Scholar [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. doi: 10.1016/j.jmaa.2011.09.049. Google Scholar

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##### 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. doi: 10.1016/S0893-9659(03)90096-6. Google Scholar [2] E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems,, Nonlinear Anal., 32 (1998), 381. doi: 10.1016/S0362-546X(97)00491-4. Google Scholar [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. doi: 10.1016/S0362-546X(01)00116-X. Google Scholar [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. doi: 10.1016/j.nonrwa.2008.09.009. Google Scholar [5] L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls,, Appl. Math. Lett., 22 (2009), 1330. doi: 10.1016/j.aml.2009.03.005. Google Scholar [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. doi: 1000-0917(2010)06-0679-12. Google Scholar [7] E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729. doi: 10.1090/S0002-9947-1984-0743741-4. Google Scholar [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. doi: 10.1016/0362-546X(94)E0063-M. Google Scholar [9] Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model,, Trans. Amer. Math. Soc., 349 (1997), 2443. doi: 10.1090/S0002-9947-97-01842-4. Google Scholar [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. doi: 10.1142/S1793524509000522. Google Scholar [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. doi: 10.1016/j.chaos.2005.04.097. Google Scholar [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. doi: 10.1016/j.nonrwa.2011.02.011. Google Scholar [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. doi: 10.1007/s002850100079. Google Scholar [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. doi: 10.1016/j.jmaa.2010.11.008. Google Scholar [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. doi: 10.1016/j.jmaa.2009.05.039. Google Scholar [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. doi: 10.1016/j.jmaa.2012.07.026. Google Scholar [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. doi: 10.1016/j.jmaa.2012.07.025. Google Scholar [18] A. Korobeinikov and W. T. Lee, Global asymptotic properties for a Leslie-Gower food chain model,, Math. Biosci. Eng., 6 (2009), 585. doi: 10.3934/mbe.2009.6.585. Google Scholar [19] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, J. Math. Biol., 36 (1998), 389. doi: 10.1007/s002850050105. Google Scholar [20] P. H. Leslie, Some further notes on the use of matrices in population mathematics,, Biometrika, 35 (1948), 213. doi: 10.1093/biomet/35.3-4.213. Google Scholar [21] P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods,, Biometrika, 45 (1958), 16. doi: 10.1093/biomet/45.1-2.16. Google Scholar [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. doi: 10.1093/biomet/47.3-4.219. Google Scholar [23] L. Li, Coexistence theorems of steady states for predator-prey interacting systems,, Trans. Amer. Math. Soc., 305 (1988), 143. doi: 10.1090/S0002-9947-1988-0920151-1. Google Scholar [24] C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar [25] C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems,, Nonlinear Anal., 26 (1996), 1889. doi: 10.1016/0362-546X(95)00058-4. Google Scholar [26] H. H. Schaefer, Topological Vector Spaces,, $3^{nd}$ edition, (1971). doi: 10.1007/978-1-4612-1468-7_3. Google Scholar [27] J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (1994). doi: 10.1007/978-1-4612-0873-0. Google Scholar [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. doi: 10.1016/j.nonrwa.2006.09.004. Google Scholar [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. doi: 10.1016/j.cnsns.2012.08.020. Google Scholar [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. doi: 10.1142/S0218127498001029. Google Scholar [31] M. Wang, Nonlinear Parabolic Equation (in Chinese),, Science Press, (1993). Google Scholar [32] Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion,, in Handbook of Differential Equations: Stationary Partial Differential Equations, (2008), 411. doi: 10.1016/S1874-5733(08)80023-X. Google Scholar [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. doi: 10.1016/j.jmaa.2011.09.049. Google Scholar
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