Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps. (English) Zbl 1338.60210

Summary: This paper focuses on studying the dynamics of the stochastic Gilpin-Ayala model under regime switching with jumps. The aim is to analyze what happens under the perturbations of regime switching and jumps. Some asymptotic properties are investigated and sufficient conditions for stochastic permanence, extinction, non-persistence in the mean and weak persistence are provided. The critical value among the extinction, non-persistence in the mean and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the jumps and the stationary distribution of the Markov chain.


60J75 Jump processes (MSC2010)
34D05 Asymptotic properties of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
60J27 Continuous-time Markov processes on discrete state spaces
92D25 Population dynamics (general)
Full Text: DOI


[1] Chen, F., Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays, Nonlinear Anal., 7, 1205-1222 (2006) · Zbl 1120.34062
[2] Fan, M.; Wang, K., Global periodic solutions of a generalized n-species Gilpin-Ayala competition model, Comput. Math. Appl., 40, 1141-1151 (2000) · Zbl 0954.92027
[3] Gard, T., Persistence in stochastic food web models, Bull. Math. Biol., 46, 357-370 (1984) · Zbl 0533.92028
[4] Gard, T., Stability for multispecies population models in random environments, Nonlinear Anal., 10, 1411-1419 (1986) · Zbl 0598.92017
[5] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes Appl., 97, 95-110 (2002) · Zbl 1058.60046
[6] Lian, B.; Hu, S., Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339, 419-428 (2008) · Zbl 1195.34083
[7] Jiang, D.; Shi, N., A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303, 164-172 (2005) · Zbl 1076.34062
[8] Jiang, D.; Shi, N.; Li, X., 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
[9] Liu, M.; Wang, K., Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54, 2139-2154 (2011) · Zbl 1235.60099
[10] Liu, M.; Wang, K., Asymptotic properties and simulations of a stochastic logistic model under regime switching II, Math. Comput. Model., 55, 405-418 (2012) · Zbl 1255.60129
[11] Bao, J.; Mao, X.; Yin, G.; Yuan, C., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74, 6601-6616 (2011) · Zbl 1228.93112
[12] Bao, J.; Yuan, C., Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391, 363-375 (2012) · Zbl 1316.92063
[13] Zou, R. Wu. X.; Wang, K., Dynamics of logistic system driven by Lévy noise under regime switching, Electron. J. Diff. Equ., 76, 1-16 (2014)
[14] Liu, M.; Wang, K., Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal., 85, 204-213 (2013) · Zbl 1285.34047
[15] Situ, R., Theory of Stochastic Differential Equation with Jumps and Applications (2012), Springer-Verlag: Springer-Verlag New York
[16] Applebaum, D., Lévy Process and Stochastic Calculus (2009), Cambridge University Press · Zbl 1200.60001
[17] Kunita, H., Itô stochastic calculus: its surprising power for applications, Stochastic Processes Appl., 120, 622-652 (2010) · Zbl 1202.60079
[18] Du, N.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behaviour of Lotka-Volterra competition systems: nonautonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170, 399-422 (2004) · Zbl 1089.34047
[19] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 249-256 (1978)
[20] Takeuchi, Y.; Du, N.; Hieu, N.; 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
[21] Liu, Y.; Liu, Q., A stochastic delay Gilpin-Ayala competition system under regime switching, Filomat, 27, 955-964 (2013) · Zbl 1418.34150
[22] Zou, X.; Wang, K., Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simul., 19, 1557-1568 (2014) · Zbl 1457.65007
[23] Anderson, W., Continuous-Time Markov Chains (1991), Springer
[24] Lipster, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228 (1980) · Zbl 0435.60037
[25] Mao, X.; Yuan, C., Stochastic Differential Equations with Markovian Switching (2006), Imperial College Press: Imperial College Press London · Zbl 1126.60002
[26] Liu, M.; Wang, K., Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410, 750-763 (2014) · Zbl 1327.92046
[27] Du, N.; Sam, V., Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl., 324, 82-97 (2006) · Zbl 1107.92038
[28] Luo, Q.; Mao, X., Stochastic population dynamics under regime Switching, J. Math. Anal. Appl., 334, 69-84 (2007) · Zbl 1113.92052
[29] Li, X.; Jiang, D.; Mao, X., Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232, 427-448 (2009) · Zbl 1173.60020
[30] Mao, X.; Yin, G.; Yuan, C., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273 (2007) · Zbl 1111.93082
[31] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546 (2001) · Zbl 0979.65007
[32] Yin, G.; Xi, F., Stability of regime-switching jump diffusions, SIAM J. Control Optim., 48, 4525-4549 (2010) · Zbl 1210.60089
[33] Yang, Z.; Yin, G., Stability of nonlinear regime-switching jump diffusions, Nonlinear Anal., 75, 3854-3873 (2012) · Zbl 1300.60100
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