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Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion
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 |
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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