Analysis of stochastic two-prey one-predator model with Lévy jumps. (English) Zbl 1400.92437
Summary: Taking white noises and Lévy noises into account, a two-prey one-predator model in random environments is proposed and investigated. Under some simple assumptions, the critical value between persistence in the mean and extinction for each population is obtained. Then sufficient conditions for stability in distribution of the model are established. Finally, some numerical examples are introduced to validate the analytical findings.
MSC:
| 92D25 | Population dynamics (general) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60J75 | Jump processes (MSC2010) |
References:
| [1] | Freedman, H.; Waltman, P., Mathematical analysis of some three-species food-chain models, Math. Biosci., 33, 257-276 (1977) · Zbl 0363.92022 |
| [2] | Takeuchi, Y.; Adachi, N., Existence of bifurcation of stable equilibrium in two-prey, one-predator communities, Bull. Math. Biol., 45, 877-900 (1983) · Zbl 0524.92025 |
| [3] | Hutson, V.; Vickers, G., A criterion for permanent co-existence of species, with an application to a two-prey one-predator system, Math. Biosci., 63, 253-269 (1983) · Zbl 0524.92023 |
| [4] | Freedman, H.; Waltman, P., Persistence in models of three interacting predator-prey populations, Math. Biosci., 68, 213-231 (1984) · Zbl 0534.92026 |
| [5] | Feng, W., Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179, 592-609 (1993) · Zbl 0846.35067 |
| [6] | Ahmad, S.; Stamova, I. M., Almost necessary and sufficient conditions for survival of species, Nonlinear Anal., 5, 219-229 (2004) · Zbl 1080.34035 |
| [7] | Ton, T. V., Survival of three species in a nonautonomous Lotka-Volterra system, J. Math. Anal. Appl., 362, 427-437 (2010) · Zbl 1184.34057 |
| [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] | Luo, Q.; Mao, X., Stochastic population dynamics under regime switching II, J. Math. Anal. Appl., 355, 577-593 (2009) · Zbl 1162.92032 |
| [10] | Zhu, C.; Yin, G., On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71, e1370-e1379 (2009) · Zbl 1238.34059 |
| [11] | Li, X.; Mao, X., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24, 523-545 (2009) · Zbl 1161.92048 |
| [12] | Liu, M.; Wang, K., Dynamics of a two-prey one-predator system in random environments, J. Nonlinear Sci., 23, 751-775 (2013) · Zbl 1279.92088 |
| [13] | Cai, Y.; Kang, Y.; Banerjee, M.; Wang, W., A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations (2015) · Zbl 1330.35464 |
| [14] | Zhang, C.; Li, W.; Wang, K., Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Trans. Neural Netw. Learn. Syst., 26, 1698-1709 (2015) |
| [15] | Zhang, C.; Li, W.; Wang, K., Graph-theoretic approach to stability of multi-group models with dispersal, Discrete Contin. Dyn. Syst. Ser. B, 20, 259-280 (2015) · Zbl 1311.34105 |
| [16] | Liu, M.; Mandal, P. S., Dynamical behavior of an one-prey two-predator model with random perturbations, Commun. Nonlinear Sci. Numer. Simul., 28, 123-137 (2015) · Zbl 1510.92169 |
| [17] | Liu, M.; Bai, C., Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25, 277-289 (2015) · Zbl 1329.60186 |
| [18] | Bao, J.; Mao, X.; Yin, G.; Yuan, C., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74, 6601-6616 (2011) · Zbl 1228.93112 |
| [19] | Bao, J.; Yuan, C., Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391, 363-375 (2012) · Zbl 1316.92063 |
| [20] | Liu, M.; Wang, K., Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410, 750-763 (2014) · Zbl 1327.92046 |
| [21] | Liu, M.; Wang, K., Dynamics of a Leslie-Gower Holling-type II predator-prey system with Levy jumps, Nonlinear Anal., 85, 204-213 (2013) · Zbl 1285.34047 |
| [22] | Liu, Q.; Liu, Y., Persistence and extinction of a stochastic non-autonomous Gilpin-Ayala system driven by Lévy noise, Commun. Nonlinear Sci. Numer. Simul., 19, 3745-3752 (2014) · Zbl 1470.34153 |
| [23] | Zhang, X.; Wang, K., Stability analysis of a stochastic Gilpin-Ayala model driven by Lévy noise, Commun. Nonlinear Sci. Numer. Simul., 19, 1391-1399 (2014) · Zbl 1457.34096 |
| [24] | Zhang, X.; Li, W.; Wang, K., Dynamics of a stochastic Holling II one-predator two-prey system with jumps, Physica A, 421, 571-582 (2015) · Zbl 1395.37059 |
| [25] | Liu, Q.; Chen, Q., Dynamics of stochastic delay Lotka-Volterra systems with impulsive toxicant input and Lévy noise in polluted environments, Appl. Math. Comput., 256, 52-67 (2015) · Zbl 1338.91107 |
| [26] | Barbalat, I., Systems dequations differentielles d’osci d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4, 267-270 (1959) · Zbl 0090.06601 |
| [27] | Mao, X., Stochastic Differential Equations and Applications (1997), Horwood Publishing: Horwood Publishing Chichester · Zbl 0874.60050 |
| [28] | Protter, P.; Talay, D., The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25, 393-423 (1997) · Zbl 0876.60030 |
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