The threshold of stochastic Gilpin-Ayala model subject to general Lévy jumps. (English) Zbl 1427.60173
Summary: This paper investigates a stochastic Gilpin-Ayala model with general Lévy jumps and stochastic perturbation to around the positive equilibrium of corresponding deterministic model. Sufficient conditions for extinction are established as well as nonpersistence in the mean, weak persistence and stochastic permanence. The threshold between weak persistence and extinction is obtained. Asymptotic behavior around the positive equilibrium of corresponding deterministic model is discussed. Our results imply the general Lévy jumps is propitious to population survival when its intensity is more than 0, and some changes profoundly if not. Numerical simulink graphics are introduced to support the analytical findings.
MSC:
| 60J76 | Jump processes on general state spaces |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34D20 | Stability of solutions to ordinary differential equations |
References:
| [1] | Gilpi, M.E., Ayala, F.G.: Global models of growth and competition. Proc. Natl. Acad. Sci. USA 70, 590-593 (1973) |
| [2] | Gilpin, M.E., Ayala, F.G.: Schoenner’ model and Drosophila competition. Theor. Popul. Biol. 9, 12-14 (1976) · doi:10.1016/0040-5809(76)90031-9 |
| [3] | May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001) · Zbl 1044.92047 |
| [4] | Gard, T.C.: Persistence in stochastic food web models. Bull. Math. Biol. 46, 357-370 (1984) · Zbl 0533.92028 · doi:10.1007/BF02462011 |
| [5] | Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in populations dynamics. Stochast. Process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0 |
| [6] | Wei, F., Wang, K.: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J. Math. Anal. Appl. 331, 516-531 (2007) · Zbl 1121.60064 · doi:10.1016/j.jmaa.2006.09.020 |
| [7] | Li, X., Mao, X.: A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching. Automatica 48, 2329-2334 (2012) · Zbl 1257.93106 · doi:10.1016/j.automatica.2012.06.045 |
| [8] | Li, W., Su, H., Wang, K.: Global stability analysis for stochastic coupled systems on networks. Automatica 47, 215-220 (2011) · Zbl 1209.93158 · doi:10.1016/j.automatica.2010.10.041 |
| [9] | Meng, Q., Jiang, H.J.: Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays. Neurocomputing 74, 362-368 (2010) · doi:10.1016/j.neucom.2010.03.029 |
| [10] | Li, X., Mao, X.: Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst. 24, 523-593 (2009) · Zbl 1161.92048 · doi:10.3934/dcds.2009.24.523 |
| [11] | Liu, M., Bai, C.: Optimal harvesting policy of a stochastic food chain population model. Appl. Math. Comput. 245, 265-270 (2014) · Zbl 1335.92081 |
| [12] | Liu, M., Bai, C., Wang, K.: Asymptotic stability of a two-group stochastic SEIR model with infinite delays. Commun. Nonlinear Sci. Numer. Simul. 19, 3444-3453 (2014) · Zbl 1470.92322 · doi:10.1016/j.cnsns.2014.02.025 |
| [13] | Lu, C., Ding, X.: Persistence and extinction in general non-autonomous logistic model with delays and stochastic perturbation. Appl. Math. Comput. 229, 1-15 (2014) · Zbl 1365.92099 · doi:10.6061/clinics/2014(01)01 |
| [14] | Lu, C., Ding, X.: Persistence and extinction of a stochastic logistic model with delays and impulsive perturbation. Acta. Math. Sci. 34, 1551-1570 (2014) · Zbl 1324.90019 · doi:10.1016/S0252-9602(14)60103-X |
| [15] | Wu, R., Zou, X., Wang, K.: Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations. Commun. Nonlinear Sci. Numer. Simul. 20, 965-974 (2015) · Zbl 1304.92117 · doi:10.1016/j.cnsns.2014.06.023 |
| [16] | 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 · doi:10.1016/j.mcm.2011.05.023 |
| [17] | Bao, J., Mao, X., Yin, G., Yuan, C.: Competitive Lotka-Volterra population dynamics with jumps. Nonlinear Anal. 74, 6601-6606 (2011) · Zbl 1228.93112 · doi:10.1016/j.na.2011.06.043 |
| [18] | Bao, J., Yuan, C.: Stochastic population dynamics driven by Lévy noise. J. Math. Anal. Appl. 391, 363-375 (2012) · Zbl 1316.92063 · doi:10.1016/j.jmaa.2012.02.043 |
| [19] | 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 · doi:10.1016/j.na.2013.02.018 |
| [20] | Zou, X., Wang, K.: Numerical simulations and modeling for stochastic biological systems with jumps. Commun. Nonlinear Sci. Numer. Simul. 5, 1557-1568 (2014) · Zbl 1457.65007 · doi:10.1016/j.cnsns.2013.09.010 |
| [21] | 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 · doi:10.1016/j.cnsns.2013.09.013 |
| [22] | Liu, Q., Jiang, D., Shi, N., Hayat, T., Alsaedi, A.: Stochastic mutualism model with Lévy jumps. Commun. Nonlinear Sci. Numer. Simul. 43, 78-90 (2017) · Zbl 1465.92139 · doi:10.1016/j.cnsns.2016.05.003 |
| [23] | Zhu, Q.: Stability analysis of stochastic delay differential equations with Lévy noise. Syst. Control Lett. 118, 62-68 (2018) · Zbl 1402.93260 · doi:10.1016/j.sysconle.2018.05.015 |
| [24] | Zhu, Q.: Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching. Int. J. Control 90, 1703-1712 (2017) · Zbl 1367.93711 · doi:10.1080/00207179.2016.1219069 |
| [25] | Zhu, Q.: Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise. J. Math. Anal. Appl. 416, 126-142 (2014) · Zbl 1309.60065 · doi:10.1016/j.jmaa.2014.02.016 |
| [26] | Lu, C., Ding, X.: Permanence and extinction of a stochastic delay logistic model with jumps. Math. Probl. Eng. 2014, Article ID 495275 (2014) · Zbl 1407.60081 |
| [27] | Applebaum, D.: Lévy Processes and Stochastics Calculus, 2nd edn. Cambridge University Press, Cambridge (2009) · Zbl 1200.60001 · doi:10.1017/CBO9780511809781 |
| [28] | Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer, Berlin (2005) · Zbl 1070.60002 |
| [29] | Wang, W., Ma, Z.: Permanence of a nonautomonous population model. Acta Math. Appl. Sin. Engl. Ser. 1, 86-95 (1998) · Zbl 0940.92020 |
| [30] | Hallam, T., Ma, Z.: Persistence in population models with demographic fluctuations. J. Math. Biol. 24, 327-339 (1986) · Zbl 0606.92022 · doi:10.1007/BF00275641 |
| [31] | Hallam, T., Ma, Z.: Effects of parameter fluctuations on community survival. Math. Biol. 86, 35-49 (1987) · Zbl 0631.92019 |
| [32] | Lipster, R.: A strong law of large numbers for local martingales. Stochastics 3, 217-228 (1980) · Zbl 0435.60037 · doi:10.1080/17442508008833146 |
| [33] | Kunita, H.: Itô’s stochastic calculus: its surprising power for applications. Stochast. Process. Appl. 120, 622-652 (2010) · Zbl 1202.60079 · doi:10.1016/j.spa.2010.01.013 |
| [34] | Wu, R., Zou, X., Wang, K.: Dynamics of logistic systems driven by Lévy noise under regime switching. Electron. J. Differ. Equ. 2014, 1-16 (2014) · Zbl 1296.60160 · doi:10.1186/1687-1847-2014-1 |
| [35] | Zou, X., Wang, K.: Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps. Nonlinear Anal. 13, 32-44 (2014) · Zbl 1325.91040 |
| [36] | Bally, V., Talay, D.: The law of the euler scheme for stochastic differential equations. Probab. Theory Rel. 104, 43-60 (1996) · Zbl 0838.60051 · doi:10.1007/BF01303802 |
| [37] | Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006) · Zbl 1126.60002 · doi:10.1142/p473 |
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