# 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 and 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. [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), 363-375. 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), 6601-6616. 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), 1083-1089. doi: 10.1016/j.mcm.2009.02.015. [6] L. Chen and J. Chen, Nonlinear Biological Dynamical System, Science Press, Beijing, 1993. [7] F. D. Chen and M. S. You, Permanence for an integrodifferential model of mutualism, Appl. Math. Comput., 186 (2007), 30-34. 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), 82-97. doi: 10.1016/j.jmaa.2005.11.064. [9] A. Friedman, Stochastic Differential Equations and Their Applications, Academic Press, New York, 1976. [10] B. S. Goh, Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275. 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), 42-57. 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), 1-71. 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), 231-244. [14] J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology., 91 (2010), 1286-1295. [15] N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 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), 588-597. 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), 482-498. 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), 867-889. doi: 10.3934/dcds.2012.32.867. [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, $3^{nd}$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2. [20] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, $2^{nd}$ edition, Imperial college press, London, 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), 11-28. 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), 3745-3752. 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), 1550072, 18pp. doi: 10.1142/S1793524515500722. [24] Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems, Nonlinear. Anal., 19 (1992), 963-975. 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), 183-204. 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), 2495-2522. 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), 392-403. 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), 750-763. 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), 523-545. doi: 10.3934/dcds.2009.24.523. [30] R. A. Lipster, Strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146. [31] X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. doi: 10.1533/9780857099402. [32] R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. doi: 10.1007/978-0-387-21830-4_7. [33] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001. [34] X. Mao, S. Sabais and E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. 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), 938-957. 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), 1239-1251. [37] J. A. Yan, Lectures on Theory of Measure, Science Press, Beijing, 2004.

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##### 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), 363-375. 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), 6601-6616. 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), 1083-1089. doi: 10.1016/j.mcm.2009.02.015. [6] L. Chen and J. Chen, Nonlinear Biological Dynamical System, Science Press, Beijing, 1993. [7] F. D. Chen and M. S. You, Permanence for an integrodifferential model of mutualism, Appl. Math. Comput., 186 (2007), 30-34. 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), 82-97. doi: 10.1016/j.jmaa.2005.11.064. [9] A. Friedman, Stochastic Differential Equations and Their Applications, Academic Press, New York, 1976. [10] B. S. Goh, Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275. 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), 42-57. 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), 1-71. 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), 231-244. [14] J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology., 91 (2010), 1286-1295. [15] N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 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), 588-597. 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), 482-498. 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), 867-889. doi: 10.3934/dcds.2012.32.867. [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, $3^{nd}$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2. [20] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, $2^{nd}$ edition, Imperial college press, London, 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), 11-28. 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), 3745-3752. 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), 1550072, 18pp. doi: 10.1142/S1793524515500722. [24] Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems, Nonlinear. Anal., 19 (1992), 963-975. 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), 183-204. 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), 2495-2522. 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), 392-403. 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), 750-763. 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), 523-545. doi: 10.3934/dcds.2009.24.523. [30] R. A. Lipster, Strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146. [31] X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. doi: 10.1533/9780857099402. [32] R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. doi: 10.1007/978-0-387-21830-4_7. [33] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001. [34] X. Mao, S. Sabais and E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. 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), 938-957. 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), 1239-1251. [37] J. A. Yan, Lectures on Theory of Measure, Science Press, Beijing, 2004.
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