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Global solutions for a semilinear heat equation in the exterior domain of a compact set
Persistence and nonpersistence of a mutualism system with stochastic perturbation
1.  School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China, China 
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications," WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 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 YorkLondon, 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 integrodifferential competition system with infinite delays, Z. Angew. Math. Mech., 81 (2001), 197203. 
[5] 
M. Fan and K. Wang, Periodicity in a delayed ratiodependent pedatorprey system, J. Math. Anal. Appl., 262 (2001), 179190. 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), 3548. doi: 10.1016/S00225193(74)800287. 
[8] 
B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261275. 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), 259284. 
[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 RijnGermantown, Md., 1980. 
[11] 
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525546. 
[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 predatorprey model with modified LeslieGower and Hollingtype II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482498. 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, Globalstability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 120. 
[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), 95110. doi: 10.1016/S03044149(01)001260. 
[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 YorkLondon, 1980. 
[22] 
M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140178. doi: 10.1016/00405809(77)900405. 
[23] 
N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2^{nd} edition, NorthHolland Mathematical Library, 24, NorthHolland 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), 256259. doi: 10.1590/S010397332009000300001. 
[26] 
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 11551179. doi: 10.1137/060649343. 
show all references
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications," WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 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 YorkLondon, 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 integrodifferential competition system with infinite delays, Z. Angew. Math. Mech., 81 (2001), 197203. 
[5] 
M. Fan and K. Wang, Periodicity in a delayed ratiodependent pedatorprey system, J. Math. Anal. Appl., 262 (2001), 179190. 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), 3548. doi: 10.1016/S00225193(74)800287. 
[8] 
B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261275. 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), 259284. 
[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 RijnGermantown, Md., 1980. 
[11] 
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525546. 
[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 predatorprey model with modified LeslieGower and Hollingtype II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482498. 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, Globalstability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 120. 
[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), 95110. doi: 10.1016/S03044149(01)001260. 
[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 YorkLondon, 1980. 
[22] 
M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140178. doi: 10.1016/00405809(77)900405. 
[23] 
N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2^{nd} edition, NorthHolland Mathematical Library, 24, NorthHolland 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), 256259. doi: 10.1590/S010397332009000300001. 
[26] 
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 11551179. doi: 10.1137/060649343. 
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