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Analysis of a non-autonomous mutualism model driven by Levy jumps
1. | Institute of mathematics, Nanjing Normal University, Nanjing 210023, China, China |
2. | Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023 |
References:
[1] |
E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction,, Cambridge University Press, (2004).
|
[2] |
D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009).
doi: 10.1017/CBO9780511809781. |
[3] |
J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, J. Math. Anal. Appl., 391 (2012), 363.
doi: 10.1016/j.jmaa.2012.02.043. |
[4] |
J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Anal., 74 (2011), 6601.
doi: 10.1016/j.na.2011.06.043. |
[5] |
L. J. Chen, L. J. Chen and Z. Li, Permanence of a delayed discrete mutualism model with feedback controls,, Math. Comput. Model., 50 (2009), 1083.
doi: 10.1016/j.mcm.2009.02.015. |
[6] |
L. Chen and J. Chen, Nonlinear Biological Dynamical System,, Science Press, (1993). Google Scholar |
[7] |
F. D. Chen and M. S. You, Permanence for an integrodifferential model of mutualism,, Appl. Math. Comput., 186 (2007), 30.
doi: 10.1016/j.amc.2006.07.085. |
[8] |
N. H. Du and V. H. Sam, Dynamics of a stochastic Lotka-Volterra model perturbed by white noise,, J. Math. Anal. Appl., 324 (2006), 82.
doi: 10.1016/j.jmaa.2005.11.064. |
[9] |
A. Friedman, Stochastic Differential Equations and Their Applications,, Academic Press, (1976). Google Scholar |
[10] |
B. S. Goh, Stability in models of mutualism,, Amer. Natur., 113 (1979), 261.
doi: 10.1086/283384. |
[11] |
Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays,, J. Math. Anal. Appl., 375 (2011), 42.
doi: 10.1016/j.jmaa.2010.08.017. |
[12] |
V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems,, Math. Biosci., 111 (1992), 1.
doi: 10.1016/0025-5564(92)90078-B. |
[13] |
J. N. Holland, D. L. DeAngelis and J. L. Bronstein, Population dynamics and mutualism: Functional responses of benefits and costs,, Amer. Natur., 159 (2002), 231. Google Scholar |
[14] |
J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology., 91 (2010), 1286. Google Scholar |
[15] |
N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland, (1981).
|
[16] |
D. Q. Jiang, N. Z. Shi and X. Y. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,, J. Math. Anal. Appl., 340 (2008), 588.
doi: 10.1016/j.jmaa.2007.08.014. |
[17] |
C. Y. Ji, D. Q. Jiang and N. Z. 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. |
[18] |
C. Y. Ji and D. Q. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation,, Discrete Contin. Dyn. Syst., 32 (2012), 867.
doi: 10.3934/dcds.2012.32.867. |
[19] |
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $3^{nd}$ edition, (1991).
doi: 10.1007/978-1-4612-0949-2. |
[20] |
F. C. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2012).
doi: 10.1142/p821. |
[21] |
X. Li, A. Gray, D, Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching,, J. Math. Anal. Appl., 376 (2011), 11.
doi: 10.1016/j.jmaa.2010.10.053. |
[22] |
Q. Liu and Y. Liang, Persistence and extinction of a stochastic non-autonomous Gilpin-Ayala system driven by Lévy noise,, Commun Nonlinear Sci. Numer. Simul., 19 (2014), 3745.
doi: 10.1016/j.cnsns.2014.02.027. |
[23] |
M. Li, H. J. Gao, C. F. Shun and Y. Z. Gong, Analysis of a mutualism model with stochastic perturbations,, Int. J. Biomath., 8 (2015).
doi: 10.1142/S1793524515500722. |
[24] |
Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems,, Nonlinear. Anal., 19 (1992), 963.
doi: 10.1016/0362-546X(92)90107-P. |
[25] |
M. Liu and K. Wang, Survival analysis of a stochastic cooperation system in a polluted environment,, J. Biol. Syst., 19 (2011), 183.
doi: 10.1142/S0218339011003877. |
[26] |
M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations,, Discrete. Contin. Dyn. Syst., 33 (2013), 2495.
doi: 10.3934/dcds.2013.33.2495. |
[27] |
M. Liu and K. Wang, Analysis of a stochastic autonomous mutualism model,, J. Math. Anal. Appl., 402 (2013), 392.
doi: 10.1016/j.jmaa.2012.11.043. |
[28] |
M. Liu and K. Wang, Stochastic Lotka-Volterra systems with Lévy noise,, J. Math. Anal. Appl., 410 (2014), 750.
doi: 10.1016/j.jmaa.2013.07.078. |
[29] |
X. Y. Li and X. R. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbations,, Discrete. Contin. Dyn. Syst., 24 (2009), 523.
doi: 10.3934/dcds.2009.24.523. |
[30] |
R. A. Lipster, Strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217.
doi: 10.1080/17442508008833146. |
[31] |
X. R. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997).
doi: 10.1533/9780857099402. |
[32] |
R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.
doi: 10.1007/978-0-387-21830-4_7. |
[33] |
R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (2001). Google Scholar |
[34] |
X. Mao, S. Sabais and E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model,, J. Math. Anal. Appl., 287 (2003), 141.
doi: 10.1016/S0022-247X(03)00539-0. |
[35] |
Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment,, J. Math. Anal. Appl., 323 (2006), 938.
doi: 10.1016/j.jmaa.2005.11.009. |
[36] |
A. R. Thompson, R. M. Nisbet and R. J. Schmitt, Dynamics of mutualist populations that are demographically open,, J. Anim. Ecol., 75 (2006), 1239. Google Scholar |
[37] |
J. A. Yan, Lectures on Theory of Measure,, Science Press, (2004). Google Scholar |
show all references
References:
[1] |
E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction,, Cambridge University Press, (2004).
|
[2] |
D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009).
doi: 10.1017/CBO9780511809781. |
[3] |
J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, J. Math. Anal. Appl., 391 (2012), 363.
doi: 10.1016/j.jmaa.2012.02.043. |
[4] |
J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Anal., 74 (2011), 6601.
doi: 10.1016/j.na.2011.06.043. |
[5] |
L. J. Chen, L. J. Chen and Z. Li, Permanence of a delayed discrete mutualism model with feedback controls,, Math. Comput. Model., 50 (2009), 1083.
doi: 10.1016/j.mcm.2009.02.015. |
[6] |
L. Chen and J. Chen, Nonlinear Biological Dynamical System,, Science Press, (1993). Google Scholar |
[7] |
F. D. Chen and M. S. You, Permanence for an integrodifferential model of mutualism,, Appl. Math. Comput., 186 (2007), 30.
doi: 10.1016/j.amc.2006.07.085. |
[8] |
N. H. Du and V. H. Sam, Dynamics of a stochastic Lotka-Volterra model perturbed by white noise,, J. Math. Anal. Appl., 324 (2006), 82.
doi: 10.1016/j.jmaa.2005.11.064. |
[9] |
A. Friedman, Stochastic Differential Equations and Their Applications,, Academic Press, (1976). Google Scholar |
[10] |
B. S. Goh, Stability in models of mutualism,, Amer. Natur., 113 (1979), 261.
doi: 10.1086/283384. |
[11] |
Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays,, J. Math. Anal. Appl., 375 (2011), 42.
doi: 10.1016/j.jmaa.2010.08.017. |
[12] |
V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems,, Math. Biosci., 111 (1992), 1.
doi: 10.1016/0025-5564(92)90078-B. |
[13] |
J. N. Holland, D. L. DeAngelis and J. L. Bronstein, Population dynamics and mutualism: Functional responses of benefits and costs,, Amer. Natur., 159 (2002), 231. Google Scholar |
[14] |
J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology., 91 (2010), 1286. Google Scholar |
[15] |
N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland, (1981).
|
[16] |
D. Q. Jiang, N. Z. Shi and X. Y. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,, J. Math. Anal. Appl., 340 (2008), 588.
doi: 10.1016/j.jmaa.2007.08.014. |
[17] |
C. Y. Ji, D. Q. Jiang and N. Z. 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. |
[18] |
C. Y. Ji and D. Q. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation,, Discrete Contin. Dyn. Syst., 32 (2012), 867.
doi: 10.3934/dcds.2012.32.867. |
[19] |
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $3^{nd}$ edition, (1991).
doi: 10.1007/978-1-4612-0949-2. |
[20] |
F. C. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2012).
doi: 10.1142/p821. |
[21] |
X. Li, A. Gray, D, Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching,, J. Math. Anal. Appl., 376 (2011), 11.
doi: 10.1016/j.jmaa.2010.10.053. |
[22] |
Q. Liu and Y. Liang, Persistence and extinction of a stochastic non-autonomous Gilpin-Ayala system driven by Lévy noise,, Commun Nonlinear Sci. Numer. Simul., 19 (2014), 3745.
doi: 10.1016/j.cnsns.2014.02.027. |
[23] |
M. Li, H. J. Gao, C. F. Shun and Y. Z. Gong, Analysis of a mutualism model with stochastic perturbations,, Int. J. Biomath., 8 (2015).
doi: 10.1142/S1793524515500722. |
[24] |
Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems,, Nonlinear. Anal., 19 (1992), 963.
doi: 10.1016/0362-546X(92)90107-P. |
[25] |
M. Liu and K. Wang, Survival analysis of a stochastic cooperation system in a polluted environment,, J. Biol. Syst., 19 (2011), 183.
doi: 10.1142/S0218339011003877. |
[26] |
M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations,, Discrete. Contin. Dyn. Syst., 33 (2013), 2495.
doi: 10.3934/dcds.2013.33.2495. |
[27] |
M. Liu and K. Wang, Analysis of a stochastic autonomous mutualism model,, J. Math. Anal. Appl., 402 (2013), 392.
doi: 10.1016/j.jmaa.2012.11.043. |
[28] |
M. Liu and K. Wang, Stochastic Lotka-Volterra systems with Lévy noise,, J. Math. Anal. Appl., 410 (2014), 750.
doi: 10.1016/j.jmaa.2013.07.078. |
[29] |
X. Y. Li and X. R. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbations,, Discrete. Contin. Dyn. Syst., 24 (2009), 523.
doi: 10.3934/dcds.2009.24.523. |
[30] |
R. A. Lipster, Strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217.
doi: 10.1080/17442508008833146. |
[31] |
X. R. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997).
doi: 10.1533/9780857099402. |
[32] |
R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.
doi: 10.1007/978-0-387-21830-4_7. |
[33] |
R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (2001). Google Scholar |
[34] |
X. Mao, S. Sabais and E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model,, J. Math. Anal. Appl., 287 (2003), 141.
doi: 10.1016/S0022-247X(03)00539-0. |
[35] |
Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment,, J. Math. Anal. Appl., 323 (2006), 938.
doi: 10.1016/j.jmaa.2005.11.009. |
[36] |
A. R. Thompson, R. M. Nisbet and R. J. Schmitt, Dynamics of mutualist populations that are demographically open,, J. Anim. Ecol., 75 (2006), 1239. Google Scholar |
[37] |
J. A. Yan, Lectures on Theory of Measure,, Science Press, (2004). Google Scholar |
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