American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

L. S. Chen and J. Chen, "Nonlinear Biological Dynamical System," Science Press, Beijing, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.  Google Scholar

[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.  Google Scholar

[8]

B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261-275. doi: 10.1086/283384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[17]

X. R. Mao, "Stochastic Differential Equations and Applications," Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, 1997.  Google Scholar

[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.  Google Scholar

[19]

R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, N.J., 1973. Google Scholar

[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.  Google Scholar

[21]

G. Strang, "Linear Algebra and its Applications," Second edition, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.  Google Scholar

[22]

M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140-178. doi: 10.1016/0040-5809(77)90040-5.  Google Scholar

[23]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2nd edition, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[24]

D. B. West, "Introduction to Graph Theory," Prentice Hall, Inc., Upper Saddle River, NJ, 1996.  Google Scholar

[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.  Google Scholar

[26]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.  Google Scholar

show all references

References:
[1]

L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[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.  Google Scholar

[3]

L. S. Chen and J. Chen, "Nonlinear Biological Dynamical System," Science Press, Beijing, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.  Google Scholar

[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.  Google Scholar

[8]

B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261-275. doi: 10.1086/283384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[17]

X. R. Mao, "Stochastic Differential Equations and Applications," Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, 1997.  Google Scholar

[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.  Google Scholar

[19]

R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, N.J., 1973. Google Scholar

[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.  Google Scholar

[21]

G. Strang, "Linear Algebra and its Applications," Second edition, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.  Google Scholar

[22]

M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140-178. doi: 10.1016/0040-5809(77)90040-5.  Google Scholar

[23]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2nd edition, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[24]

D. B. West, "Introduction to Graph Theory," Prentice Hall, Inc., Upper Saddle River, NJ, 1996.  Google Scholar

[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.  Google Scholar

[26]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.  Google Scholar

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