Stochastically ultimate boundedness and global attraction of positive solution for a stochastic competitive system. (English) Zbl 1463.92051

Summary: A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.


92D25 Population dynamics (general)
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] Fan, M.; Wang, K.; Jiang, D., Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments, Mathematical Biosciences, 160, 1, 47-61 (1999) · Zbl 0964.34059
[2] Liu, Z.; Fan, M.; Chen, L., Globally asymptotic stability in two periodic delayed competitive systems, Applied Mathematics and Computation, 197, 1, 271-287 (2008) · Zbl 1148.34046
[3] Liu, B.; Chen, L., The periodic competitive Lotka-Volterra model with impulsive effect, Mathematical Medicine and Biology, 21, 2, 129-145 (2004) · Zbl 1055.92056
[4] Song, Y.; Han, M.; Peng, Y., Stability and hopf bifurcations in a competitive Lotka-Volterra system with two delays, Chaos, Solitons &Fractals, 22, 5, 1139-1148 (2004) · Zbl 1067.34075
[5] Tang, X. H.; Zou, X., Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedback, Journal of Differential Equations, 192, 2, 502-535 (2003) · Zbl 1035.34085
[6] Jin, Z.; Zhien, M.; Maoan, H., The existence of periodic solutions of the n-species Lotka-Volterra competition systems with impulsive, Chaos, Solitons & Fractals, 22, 1, 181-188 (2004) · Zbl 1058.92046
[7] Ahmad, S., Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations, Journal of Mathematical Analysis and Applications, 127, 2, 377-387 (1987) · Zbl 0648.34037
[8] Gopalsamy, K., Stability and Oscillations in Delay Differential Equation of Population Dynamics (1992), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0752.34039
[9] Tan, R.; Liu, W.; Wang, Q.; Liu, Z., Uniformly asymptotic stability of almost periodic solutions for a competitive system with impulsive perturbations, Advances in Difference Equations, 2014, article 2 (2014) · Zbl 1417.34120
[10] May, R. M., Stability and Complexity in Model Ecosystems (2001), New York, NY, USA: Princeton University Press, New York, NY, USA
[11] Li, Y.; Gao, H., Existence, uniqueness and global asymptotic stability of positive solutions of a predator-prey system with Holling II functional response with random perturbation, Nonlinear Analysis. Theory, Methods & Applications, 68, 6, 1694-1705 (2008) · Zbl 1143.34033
[12] Li, X.; Mao, X., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete and Continuous Dynamical Systems A, 24, 2, 523-545 (2009) · Zbl 1161.92048
[13] Lv, J.; Wang, K., Asymptotic properties of a stochastic predator-prey system with Holling II functional response, Communications in Nonlinear Science and Numerical Simulation, 16, 10, 4037-4048 (2011) · Zbl 1218.92072
[14] Jiang, D.; Ji, C.; Li, X.; O’Regan, D., Analysis of autonomous Lotka-Volterra competition systems with random perturbation, Journal of Mathematical Analysis and Applications, 390, 2, 582-595 (2012) · Zbl 1258.34099
[15] Liu, M.; Wang, K., Dynamics and simulations of a logistic model with impulsive perturbations in a random environment, Mathematics and Computers in Simulation, 92, 53-75 (2013)
[16] Mao, X.; Yuan, C.; Zou, J., Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304, 1, 296-320 (2005) · Zbl 1062.92055
[17] Jiang, D.; Yu, J.; Ji, C.; Shi, N., Asymptotic behavior of global positive solution to a stochastic SIR model, Mathematical and Computer Modelling, 54, 1-2, 221-232 (2011) · Zbl 1225.60114
[18] Liu, M.; Wang, K., Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation, Applied Mathematical Modelling, 36, 11, 5344-5353 (2012) · Zbl 1254.34074
[19] Li, X.; Gray, A.; Jiang, D.; Mao, X., Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376, 1, 11-28 (2011) · Zbl 1205.92058
[20] Xing, Z.; Peng, J., Boundedness, persistence and extinction of a stochastic non-autonomous logistic system with time delays, Applied Mathematical Modelling, 36, 8, 3379-3386 (2012) · Zbl 1252.60058
[21] Mao, X., Stochastic Differential Equations and Applications (2008), Chichester, UK: Horwood Publishing, Chichester, UK
[22] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge University Press
[23] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Berlin , Germany: Springer, Berlin , Germany · Zbl 0734.60060
[24] Barbalat, I., Systems d’equations differentielle d’oscillations nonlineaires, Revue Roumaine de Mathématiques Pures et Appliquées, 4, 2, 267-270 (1959) · Zbl 0090.06601
[25] Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, Society for Industrial and Applied Mathematics, 43, 3, 525-546 (2001) · Zbl 0979.65007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.