Persistence and non-persistence of a mutualism system with stochastic perturbation

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March 2012, 32(3): 867-889. doi: 10.3934/dcds.2012.32.867

Persistence and non-persistence of a mutualism system with stochastic perturbation

Chunyan Ji ^1, and Daqing Jiang ^1,

1.	School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China, China

Received October 2010 Revised March 2011 Published October 2011

Abstract
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In this paper, we consider a $n$-species Lotka-Volterra mutualism system with stochastic perturbation. Sufficient criteria for persistence in mean and stationary distribution of the system are established. Besides, we show the large white noise will make the system nonpersistent. Finally, we illustrate the dynamic behavior of the system with $n=3$ and their approximations via a range of numerical experiments.

Keywords: ergodicity, $p$th moment, nonpersistent., persistent in mean, stationary distribution.

Mathematics Subject Classification: Primary: 62F10, 34F05; Secondary: 92B0.

Citation: Chunyan Ji, Daqing Jiang. Persistence and non-persistence of a mutualism system with stochastic perturbation. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 867-889. doi: 10.3934/dcds.2012.32.867

References:

[1]	L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[2]	A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
[3]	L. S. Chen and J. Chen, "Nonlinear Biological Dynamical System," Science Press, Beijing, 1993.
[4]	M. Fan and K. Wang, Positive periodic solutions of a periodic integro-differential competition system with infinite delays, Z. Angew. Math. Mech., 81 (2001), 197-203.
[5]	M. Fan and K. Wang, Periodicity in a delayed ratio-dependent pedator-prey system, J. Math. Anal. Appl., 262 (2001), 179-190. doi: 10.1006/jmaa.2001.7555.
[6]	T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.
[7]	M. E. Gilpin, A Liapunov function for competition communities, J. Theor. Biol., 44 (1974), 35-48. doi: 10.1016/S0022-5193(74)80028-7.
[8]	B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261-275. doi: 10.1086/283384.
[9]	H. B. Guo, M. Y. Li and Z. S. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14 (2006), 259-284.
[10]	R. Z. Has'minskiǐ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.
[11]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.
[12]	J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems. Mathematical Aspects of Selection," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988.
[13]	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.
[14]	C. Y. Ji, D. Q. Jiang, L. Hong and Q. S. Yang, Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation, Math. Probl. Eng., 2010, Art. ID 684926, 18 pp.
[15]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.
[16]	M. Y. Li and Z. S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.
[17]	X. R. Mao, "Stochastic Differential Equations and Applications," Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, 1997.
[18]	X. R. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.
[19]	R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, N.J., 1973.
[20]	L. R. Nie and D. C. Mei, Noise and time delay: Suppressed population explosion of the mutualism system, Europhys. Lett., 79 (2007), no. 20005, 6 pp.
[21]	G. Strang, "Linear Algebra and its Applications," Second edition, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.
[22]	M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140-178. doi: 10.1016/0040-5809(77)90040-5.
[23]	N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2^nd edition, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
[24]	D. B. West, "Introduction to Graph Theory," Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
[25]	C. H. Zeng, G. Q. Zhang and X. F. Zhou, Dynamical properties of a mutualism system in the presence of noise and time delay, Braz. J. Phys., 39 (2009), 256-259. doi: 10.1590/S0103-97332009000300001.
[26]	C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.

show all references

References:

[1]	L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[2]	A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
[3]	L. S. Chen and J. Chen, "Nonlinear Biological Dynamical System," Science Press, Beijing, 1993.
[4]	M. Fan and K. Wang, Positive periodic solutions of a periodic integro-differential competition system with infinite delays, Z. Angew. Math. Mech., 81 (2001), 197-203.
[5]	M. Fan and K. Wang, Periodicity in a delayed ratio-dependent pedator-prey system, J. Math. Anal. Appl., 262 (2001), 179-190. doi: 10.1006/jmaa.2001.7555.
[6]	T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.
[7]	M. E. Gilpin, A Liapunov function for competition communities, J. Theor. Biol., 44 (1974), 35-48. doi: 10.1016/S0022-5193(74)80028-7.
[8]	B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261-275. doi: 10.1086/283384.
[9]	H. B. Guo, M. Y. Li and Z. S. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14 (2006), 259-284.
[10]	R. Z. Has'minskiǐ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.
[11]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.
[12]	J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems. Mathematical Aspects of Selection," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988.
[13]	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.
[14]	C. Y. Ji, D. Q. Jiang, L. Hong and Q. S. Yang, Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation, Math. Probl. Eng., 2010, Art. ID 684926, 18 pp.
[15]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.
[16]	M. Y. Li and Z. S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.
[17]	X. R. Mao, "Stochastic Differential Equations and Applications," Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, 1997.
[18]	X. R. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.
[19]	R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, N.J., 1973.
[20]	L. R. Nie and D. C. Mei, Noise and time delay: Suppressed population explosion of the mutualism system, Europhys. Lett., 79 (2007), no. 20005, 6 pp.
[21]	G. Strang, "Linear Algebra and its Applications," Second edition, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.
[22]	M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140-178. doi: 10.1016/0040-5809(77)90040-5.
[23]	N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2^nd edition, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
[24]	D. B. West, "Introduction to Graph Theory," Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
[25]	C. H. Zeng, G. Q. Zhang and X. F. Zhou, Dynamical properties of a mutualism system in the presence of noise and time delay, Braz. J. Phys., 39 (2009), 256-259. doi: 10.1590/S0103-97332009000300001.
[26]	C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.

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