# American Institute of Mathematical Sciences

June  2016, 21(4): 1189-1202. doi: 10.3934/dcdsb.2016.21.1189

## 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

Received  July 2015 Revised  November 2015 Published  March 2016

This article is concerned with a mutualism ecological model with Lévy noise. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistence in mean are also established.
Citation: Mei Li, Hongjun Gao, Bingjun Wang. Analysis of a non-autonomous mutualism model driven by Levy jumps. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1189-1202. doi: 10.3934/dcdsb.2016.21.1189
##### References:
 [1] E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction,, Cambridge University Press, (2004). Google Scholar [2] D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $3^{nd}$ edition, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar [20] F. C. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2012). doi: 10.1142/p821. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] R. A. Lipster, Strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146. Google Scholar [31] X. R. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997). doi: 10.1533/9780857099402. Google Scholar [32] R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459. doi: 10.1007/978-0-387-21830-4_7. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [2] D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $3^{nd}$ edition, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar [20] F. C. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2012). doi: 10.1142/p821. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] R. A. Lipster, Strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146. Google Scholar [31] X. R. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997). doi: 10.1533/9780857099402. Google Scholar [32] R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459. doi: 10.1007/978-0-387-21830-4_7. Google Scholar [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. Google Scholar [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. Google Scholar [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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