On competitive Lotka-Volterra model in random environments. (English) Zbl 1182.34078

The authors study asymptotic properties of a competitive Lotka-Volterra model in random environments. A continuous-time Markov chain is used to model the random environments while the population dynamics of the different species are modelled by a regime-switching diffusion. Growth rates of the population sizes are found to be bounded above, and a partial stochastic principle of competitive exclusion is obtained.


34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
Full Text: DOI


[1] Arnold, L.; Horsthemke, W.; Stucki, J. W., The influence of external real and white noise on the Lotka-Volterra model, Biomed. J., 21, 451-471 (1979) · Zbl 0433.92019
[2] Benaïm, M.; Hofbauer, J.; Sandholm, W. H., Robust permanence and impermanence for stochastic replicator dynamics, J. Biol. Dyn., 2, 180-195 (2008) · Zbl 1140.92025
[3] Chow, P.-L., Stochastic Partial Differential Equations (2007), Chapman & Hall: Chapman & Hall Boca Raton, FL · Zbl 1134.60043
[4] Cohen, J. E.; Łuczak, T.; Newman, C. M.; Zhou, Z.-M., Stochastic structure and nonlinear dynamics of food webs: Qualitative stability in a Lotka-Volterra cascade model, Proc. R. Soc. Lond. Ser. B, 240, 607-627 (1990)
[5] Christianou, M.; Kokkoris, G. D., Complexity does not affect stability in feasible model communities, J. Theoret. Biol., 273, 162-169 (2008) · Zbl 1398.92271
[6] Delgado, M.; Montenegro, M.; Suarez, A., A Lotka-Volterra symbiotic model with cross-diffusion, J. Differential Equations, 246, 2131-2149 (2009) · Zbl 1169.35008
[7] Du, N. H.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170, 399-422 (2004) · Zbl 1089.34047
[8] Du, N. H.; Sam, V. H., Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl., 324, 82-97 (2006) · Zbl 1107.92038
[9] Edelstein-Keshet, L., Mathematical Models in Biology (1988), Random House: Random House New York · Zbl 0674.92001
[10] Foster, D.; Young, P., Stochastic evolutionary game dynamics, Theoret. Popul. Biol., 38, 219-232 (1990) · Zbl 0703.92015
[11] Friedman, A., Stochastic Differential Equations and Applications, vols. I and II (1975), Academic Press: Academic Press New York · Zbl 0323.60056
[12] Fudenberg, D.; Harris, C., Evolutionary dynamics with aggregate shocks, J. Econom. Theory, 57, 420-441 (1992) · Zbl 0766.92012
[13] Goel, N. S.; Maitra, S. C.; Montroll, E. W., Nonlinear Models of Interacting Populations (1971), Academic Press: Academic Press New York
[14] Gopalsamy, K., Global asymptotic stability in Volterra population systems, J. Math. Biol., 19, 157-168 (1984) · Zbl 0535.92020
[15] Guo, H.; Chen, L., Time-limited control of a Lotka-Volterra model with impulsive harvest, Nonlinear Anal. Real World Appl., 10, 840-848 (2009) · Zbl 1167.34306
[16] Hofbauer, J.; Sigmund, K., Evolutionary Games and Population Dynamics (1998), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, United Kingdom · Zbl 0914.90287
[17] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland: North-Holland Amsterdam · Zbl 0495.60005
[18] Imhof, L. A., The long-run behavior of stochastic replicator dynamics, Ann. Appl. Probab., 15, 1019-1045 (2005) · Zbl 1081.60045
[19] Jeffries, C., Stability of predation ecosystem models, Ecology, 57, 1321-1325 (1976)
[20] Jiang, D.; Shi, N., A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303, 164-172 (2005) · Zbl 1076.34062
[21] Khasminskii, R. Z., Stochastic Stability of Differential Equations (1980), Sijthoff and Noordhoff: Sijthoff and Noordhoff Alphen aan den Rijn, Netherlands · Zbl 1259.60058
[22] Khasminskii, R. Z.; Klebaner, F. C., Long term behavior of solutions of the Lotka-Volterra systems under small random perturbations, Ann. Appl. Probab., 11, 952-963 (2001) · Zbl 1061.34513
[23] Khasminskii, R. Z.; Potsepun, N., On the replicator dynamics behavior under Stratonovich type random perturbations, Stoch. Dyn., 6, 197-211 (2006) · Zbl 1100.60030
[24] Khasminskii, R. Z.; Zhu, C.; Yin, G., Stability of regime-switching diffusions, Stochastic Process. Appl., 117, 1037-1051 (2007) · Zbl 1119.60065
[25] W. Ko, K. Ryu, Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion, Nonlinear Anal., doi:10.1016/j.na.2009.01.097; W. Ko, K. Ryu, Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion, Nonlinear Anal., doi:10.1016/j.na.2009.01.097 · Zbl 1238.35162
[26] D. Li, S. Wang, X. Zhang, D. Yang, Impulsive control of uncertain Lotka-Volterra predator-prey system chaos, Solitons Fractals, doi:10.1016/j.chaos.2008.06.021; D. Li, S. Wang, X. Zhang, D. Yang, Impulsive control of uncertain Lotka-Volterra predator-prey system chaos, Solitons Fractals, doi:10.1016/j.chaos.2008.06.021 · Zbl 1198.34092
[27] Li, X. Z.; Tang, 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
[28] Liptser, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228 (1980) · Zbl 0435.60037
[29] Lotka, A. J., Elements of Physical Biology (1925), William and Wilkins: William and Wilkins Baltimore
[30] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334, 69-84 (2007) · Zbl 1113.92052
[31] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 95-110 (2002) · Zbl 1058.60046
[32] Mao, X.; Sabanis, S.; Renshaw, R., Asymptotic behavior of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287, 141-156 (2003) · Zbl 1048.92027
[33] Mobilia, M.; Georgiev, I. T.; Täuber, U. C., Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka-Volterra models, J. Stat. Phys., 128, 447-483 (2007) · Zbl 1117.82029
[34] Yan, J.; Zhao, A.; Nieto, J. J., Existence and global attractivity of positive-periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Modelling, 40, 509-518 (2004) · Zbl 1112.34052
[35] Reichenbach, T.; Mobilia, M.; Frey, E., Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model, Phys. Rev. E, 74, 051907 (2006)
[36] Reichenbach, T.; Mobilia, M.; Frey, E., Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games, Nature, 448, 1046-1049 (2007)
[37] Reichenbach, T.; Mobilia, M.; Frey, E., Noise and correlations in a spatial population model with cyclic competition, Phys. Rev. Lett., 99, 238105 (2007)
[38] Reichenbach, T.; Mobilia, M.; Frey, E., Self-organization of mobile populations in cyclic competition, J. Theoret. Biol., 254, 368-383 (2008) · Zbl 1400.92443
[39] Roughgarden, J., Theory of Population Genetics and Evolutionary Ecology: An Introduction (1979), Macmillan: Macmillan New York
[40] Øksendal, B., Stochastic Differential Equations (2005), Springer: Springer Berlin
[41] Roubik, D. W., Experimental community studies: Time-series tests of competition between African and neotropical bees, Ecology, 64, 971-978 (1983)
[42] Skorohod, A. V., Asymptotic Methods in the Theory of Stochastic Differential Equations (1989), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0695.60055
[43] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 249-256 (1978)
[44] Stratonovich, R. L., A new representation for stochastic integrals and equations, SIAM J. Control Optim., 4, 362-371 (1966) · Zbl 0143.19002
[45] Takeuchi, Y.; Adachi, N., The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10, 401-415 (1980) · Zbl 0458.92019
[46] Takeuchi, Y.; Adachi, N.; Tokumaru, H., The stability of generalized Volterra equations, J. Math. Anal. Appl., 62, 453-473 (1978) · Zbl 0388.45011
[47] Takeuchi, Y.; Du, N. H.; Hieu, N. T.; Sato, K., Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323, 938-957 (2006) · Zbl 1113.34042
[48] Traulsen, A.; Claussen, J. C.; Hauert, C., Coevolutionary dynamics in large, but finite populations, Phys. Rev. E, 74, 1, 011901 (2006)
[49] Villadelprat, J., The period function of the generalized Lotka-Volterra centers, J. Math. Anal. Appl., 341, 834-854 (2008) · Zbl 1147.34034
[50] Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie d’animali conviventi, Mem. Acad. Lincei, 2, 31-113 (1926)
[51] Wan, L.; Zhuo, Q., Stochastic Lotka-Volterra model with infinite delay, Statist. Probab. Lett., 79, 698-706 (2009) · Zbl 1159.92321
[52] Wu, F.; Hu, S., Stochastic functional Kolmogorov-type population dynamics, J. Math. Anal. Appl., 347, 534-549 (2008) · Zbl 1158.60024
[53] Xiao, D.; Li, W., Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164, 1-15 (2000) · Zbl 0960.34022
[54] Yin, G.; Zhang, Q., Continuous-time Markov Chains and Applications: A Singular Perturbations Approach (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0896.60039
[55] Zeng, G.; Wang, F.; Nieto, J. J., Complexity of a delayed predator-prey model with impulsive harvest and holling type II functional response, Adv. Complex Syst., 11, 77-97 (2008) · Zbl 1168.34052
[56] Zhang, H.; Chen, L.; Nieto, J. J., A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. Real World Appl., 9, 1714-1726 (2008) · Zbl 1154.34394
[57] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46, 1155-1179 (2007) · Zbl 1140.93045
[58] C. Zhu, G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., doi:10.1016/j.na.2009.01.166; C. Zhu, G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., doi:10.1016/j.na.2009.01.166 · Zbl 1238.34059
[59] C. Zhu, G. Yin, Q.S. Song, Stability of random-switching systems of differential equations, Quart. Appl. Math., in press; C. Zhu, G. Yin, Q.S. Song, Stability of random-switching systems of differential equations, Quart. Appl. Math., in press · Zbl 1163.93036
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.