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Analysis of autonomous Lotka-Volterra competition systems with random perturbation. (English) Zbl 1258.34099
A multi-species Lotka-Volterra competition system with \(n\) interacting components is considered. Conditions for stability in time average, existence of stationary distribution, as well as extinction, are derived.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D05 Asymptotic properties of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
92D25 Population dynamics (general)
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[1] Bahar, A.; Mao, X.R., Stochastic delay Lotka-Volterra model, J. math. anal. appl., 292, 364-380, (2004) · Zbl 1043.92034
[2] Bao, J.H.; Mao, X.R.; Yin, G.; Yuan, C.G., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear anal., 74, 6601-6616, (2011) · Zbl 1228.93112
[3] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic Press New York · Zbl 0484.15016
[4] Chen, L.S.; Chen, J., Nonlinear biological dynamical system, (1993), Science Press Beijing
[5] Friedman, H.I., Deterministic mathematical models in population ecology, (1998), Dekker New York
[6] Gard, T.C., Introduction to stochastic differential equations, (1988), Dekker New York · Zbl 0682.92018
[7] Goh, B.S., Global stability in many species systems, Amer. nat., 111, 135-143, (1997)
[8] Golpalsamy, K., Globally asymptotic stability in a periodic Lotka-Volterra system, J. austral. math. soc. ser. B, 24, 160-170, (1982)
[9] Golpalsamy, K., Global asymptotic stability in volterraʼs population systems, J. math. biol., 19, 157-168, (1984)
[10] Hasminskii, R.Z., Stochastic stability of differential equations, Monographs and textbooks on mechanics of solids and fluids: mechanics and analysis, vol. 7, (1980), Sijthoff & Noordhoff Alphen aan den Rijn, The Netherlands · Zbl 0419.62037
[11] Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 525-546, (2001) · Zbl 0979.65007
[12] Hu, G.X.; Wang, K., Stability in distribution of competitive Lotka-Volterra system with Markovian switching, Appl. math. model., 35, 3189-3200, (2011) · Zbl 1228.34088
[13] Hu, Y.Z.; Wu, F.K.; Huang, C.M., Stochastic Lotka-Volterra models with multiple delays, J. math. anal. appl., 375, 42-57, (2011) · Zbl 1245.92063
[14] Ikeda Wantanabe, N., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam
[15] Ji, C.Y.; Jiang, D.Q.; Shi, N.Z.; OʼRegan, D., Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation, Math. methods appl. sci., 30, 77-89, (2007) · Zbl 1148.34040
[16] Ji, C.Y.; Jiang, D.Q.; Shi, N.Z., Analysis of a predator-prey model with modified Leslie-gower and Holling-type II schemes with stochastic perturbation, J. math. anal. appl., 359, 482-498, (2009) · Zbl 1190.34064
[17] Jiang, D.Q.; Shi, N.Z.; Zhao, Y.N., Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation, Math. comput. modelling, 42, 651-658, (2005) · Zbl 1081.92039
[18] Jiang, D.Q.; Shi, N.Z., A note on nonautonomous logistic equation with random perturbation, J. math. anal. appl., 303, 164-172, (2005) · Zbl 1076.34062
[19] Jiang, D.Q.; Zhang, B.X.; Wang, D.H.; Shi, N.Z., Existence, uniqueness and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation, Sci. China ser. A, 50, 977-986, (2007) · Zbl 1136.34324
[20] Jiang, D.Q.; Shi, N.Z.; Li, X.Y., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. anal. appl., 340, 588-597, (2008) · Zbl 1140.60032
[21] Klebaner, F.C., Introduction to stochastic calculus with applications, (1998), Imperial College Press · Zbl 0926.60002
[22] Kuang, Y.; Smith, H.L., Global stability for infinite delay Lotka-Volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077
[23] Langa, José A.; Rodríguez-Bernal, Aníbal; Suárez, Antonio, On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method, J. differential equations, 249, 414-445, (2010) · Zbl 1195.35178
[24] Li, M.Y.; Shuai, Z.S., Global-stability problem for coupled systems of differential equations on networks, J. differential equations, 248, 1-20, (2010) · Zbl 1190.34063
[25] Li, X.Y.; Mao, X.R., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete contin. dyn. syst., 24, 523-545, (2009) · Zbl 1161.92048
[26] Li, X.Z.; Tong, C.L.; Ji, X.H., The criteria for globally stable equilibrium in N-dimensional Lotka-Volterra systems, J. math. anal. appl., 240, 600-606, (1999) · Zbl 0947.34044
[27] Mao, X.R.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046
[28] Mao, X.R.; Marion, G.; Renshaw, E., Asymptotic behaviour of the stochastic Lotka-Volterra model, J. math. anal. appl., 287, 141-156, (2003) · Zbl 1048.92027
[29] May, R.M., Stability and complexity in model ecosystems, (2001), Princeton University Press New Jersey
[30] Moon, J.W., Counting labelled tree, (1970), Canadian Mathematical Congress Montreal · Zbl 0214.23204
[31] Murray, J.D., Mathematical biology, (1993), Springer-Verlag Berlin, Heidelberg · Zbl 0779.92001
[32] Polansky, P., Invariant distributions for multi-population models in random environments, Theor. popul. biol., 16, 25-34, (1979) · Zbl 0417.92019
[33] Strang, G., Linear algebra and its applications, (1988), Thomson Learning Inc.
[34] West, D.B., Introduction to graph theory, (1996), Prentice Hall Upper Saddle River · Zbl 0845.05001
[35] Xiao, D.M.; Li, W.X., Limit cycles for the competitive three-dimensional Lotka-Volterra systems, J. differential equations, 164, 1-15, (2000) · Zbl 0960.34022
[36] Zhang, L.; Teng, Z.D., N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations, Nonlinear anal. real world appl., 12, 3152-3169, (2011) · Zbl 1231.37055
[37] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. control optim., 46, 1155-1179, (2007) · Zbl 1140.93045
[38] Zhu, C.; Yin, G., On competitive Lotka-Volterra model in random environments, J. math. anal. appl., 357, 154-170, (2009) · Zbl 1182.34078
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