Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps. (English) Zbl 1285.34047
Authors’ abstract: This paper is concerned with a stochastic predator-prey system with modified Leslie-Gower and Holling-type II schemes with Lévy jumps. First, we prove there is a unique positive solution to the system with a positive initial value. Then we establish sufficient conditions for stability in mean and extinction of the system. Finally, we introduce some numerical simulations to support the main results. The results shows that the Lévy jumps can change the properties of the population systems significantly.
Reviewer: George Karakostas (Ioannina)
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 92D25 | Population dynamics (general) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
References:
| [1] | Aziz-Alaoui, M. A.; Daher Okiye, M., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075 (2003) · Zbl 1063.34044 |
| [2] | Nindjin, A. F.; Aziz-Alaoui, M. A.; Cadivel, M., Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. RWA, 7, 1104-1118 (2006) · Zbl 1104.92065 |
| [3] | Guo, H.; Song, X., An impulsive predator-prey system with modified Leslie-Gower and Holling type II schemes, Chaos Solitons Fractals, 36, 1320-1331 (2008) · Zbl 1148.34034 |
| [4] | Song, X.; Li, Y., Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect, Nonlinear Anal. RWA, 9, 64-79 (2008) · Zbl 1142.34031 |
| [5] | Yafia, R.; Adnani, F.; Alaoui, H. T., Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Nonlinear Anal. RWA, 9, 2055-2067 (2008) · Zbl 1156.34342 |
| [6] | Chen, L.; Chen, F., Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22, 1330-1334 (2009) · Zbl 1173.34333 |
| [7] | Chen, F.; Chen, L.; Xie, X., On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. RWA, 10, 2905-2908 (2009) · Zbl 1167.92032 |
| [8] | Ji, C.; Jiang, D.; Shi, N., Analysis of a predator-prey model with modified Leslie-Gower and Hollingtype II schemes with stochastic perturbation, J. Math. Anal. Appl., 359, 482-498 (2009) · Zbl 1190.34064 |
| [9] | Ji, C.; Jiang, D.; Shi, N., A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 377, 435-440 (2011) · Zbl 1216.34040 |
| [10] | Nie, L.; Teng, Z.; Hu, L.; Peng, J., Qualitative analysis of a modified Leslie-Gower and Holling-type II predator-prey model with state dependent impulsive effects, Nonlinear Anal. RWA, 11, 1364-1373 (2010) · Zbl 1228.37058 |
| [11] | Wang, Q.; Zhou, J.; Wang, Z.; Ding, M.; Zhang, H., Existence and attractivity of a periodic solution for a ratio-dependent Leslie system with feedback controls, Nonlinear Anal. RWA, 12, 24-33 (2011) · Zbl 1208.34078 |
| [12] | Guan, X.; Wang, W.; Cai, Y., Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. RWA, 12, 2385-2395 (2011) · Zbl 1225.49038 |
| [13] | Yu, S., Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Discrete Dyn. Nat. Soc. (2012) · Zbl 1248.34050 |
| [14] | Banerjee, M.; Banerjee, S., Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236, 64-76 (2012) · Zbl 1375.92077 |
| [15] | Bao, J.; Mao, X.; Yin, G.; Yuan, C., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74, 6601-6616 (2011) · Zbl 1228.93112 |
| [16] | Bao, J.; Yuan, C., Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391, 363-375 (2012) · Zbl 1316.92063 |
| [17] | Peng, S.; Zhu, X., Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116, 370-380 (2006) · Zbl 1096.60026 |
| [18] | Liu, M.; Wang, K., Survival analysis of a stochastic cooperation system in a polluted environment, J. Biol. Systems, 19, 183-204 (2011) · Zbl 1228.92074 |
| [19] | Lipster, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228 (1980) · Zbl 0435.60037 |
| [20] | Applebaum, D., Lévy Processes and Stochastics Calculus (2009), Cambridge University Press · Zbl 1200.60001 |
| [21] | Kunita, H., Itô’s stochastic calculus: its surprising power for applications, Stochastic Process. Appl., 120, 622-652 (2010) · Zbl 1202.60079 |
| [22] | Protter, P.; Talay, D., The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25, 393-423 (1997) · Zbl 0876.60030 |
| [23] | Zhu, C.; Yin, G., On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71, e1370-e1379 (2009) · Zbl 1238.34059 |
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