This paper studies a stochastic delay logistic model with Lévy jumps and impulsive perturbations. We show that the model has a unique global positive solution. Sufficient conditions for extinction, non-persistence in the mean, weak persistence, stochastic permanence and global asymptotic stability are established. The threshold between weak persistence and extinction is obtained. The results demonstrate that impulsive perturbations which may represent human factor play an important role in protecting the population even if it suffers sudden environmental shocks that can be discribed by Lévy jumps.
Citation: Chun Lu, Bing Li, Limei Zhou, Liwei Zhang. Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3251-3271. doi: 10.3934/mbe.2019162
This paper studies a stochastic delay logistic model with Lévy jumps and impulsive perturbations. We show that the model has a unique global positive solution. Sufficient conditions for extinction, non-persistence in the mean, weak persistence, stochastic permanence and global asymptotic stability are established. The threshold between weak persistence and extinction is obtained. The results demonstrate that impulsive perturbations which may represent human factor play an important role in protecting the population even if it suffers sudden environmental shocks that can be discribed by Lévy jumps.
| [1] | K. Golpalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic, Dordrecht, 1992. |
| [2] | K. Golpalsamy, Global asymptotic stability in Volterra's population systems, J. Math. Biol., 19 (1984), 157–168. |
| [3] | Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Valterra type systems, J. Diff. Eq.,103(1993), 221–246. |
| [4] | V. B. Kolmanovskii and V. R. Nosov, Stability of functional differential equations, Academic Press, New York, 1986. |
| [5] | B. Lisena, Global attractivity in nonautonomous logistic equations with delay, Nonlinear Anal.,9 (2008), 53–63. |
| [6] | Y.Kuang, Delay differential equations with applications in population dynamics, AcademicPress, Boston, 1993. |
| [7] | R. M. May, Stability and complexity in model ecosystems, Princeton University Press, NJ, 1973. |
| [8] | T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol.,46(1984), 357–370. |
| [9] | Q. Liu and D. Jiang, Stationary distribution and extinction of a stochastic predator-prey model with distributed delay, Appl. Math. Lett.,78(2018), 79–87. |
| [10] | W. Mao, L. Hu and X. Mao, Neutral stochastic functional differential equations with L´evy jumps under the local Lipschitz condition, Adv. Differ. Equ.,57(2017), 1–24. |
| [11] | F. Wei and K. Wang, The existence and uniquencess of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl.,331(2007), 516–531. |
| [12] | M.LiuandK.Wang, Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays, Commun. Nonlinear Sci. Numer. Simul.,17(2012), 3115–3123. |
| [13] | M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl.,375(2011), 443–457. |
| [14] | F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math.,70(2009), 641–657. |
| [15] | M. Vasilova and M. Jovanovic, Stochastic Gilpin-Ayala competition model with infinite delay, Appl. Math. Comput.,217(2011), 4944–4959. |
| [16] | C. Lu and X. Ding, Persistence and extinction in general non-autonomous logistic model with delays and stochastic perturbation, Appl. Math. Comput.,229(2014), 1–15. |
| [17] | J. Bao, X. Mao, G. Yin, et al., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal. Theory Methods Appl.,74(2011), 6601–6616. |
| [18] | M.LiuandY.Zhu, Stationarydistributionandergodicityofastochastichybridcompetitionmodel with L´evy jumps, Nonlinear Anal. Hybrid Syst.,30(2018), 225–239. |
| [19] | N.H.DuandV.H.Sam, DynamicsofastochasticLotka-Volterramodelperturbedbywhitenoise, J. Math. Anal. Appl.,324(2006), 82–97. |
| [20] | X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simul.,5(2014), 1557–1568. |
| [21] | C. Lu and X. Ding, Persistence and extinction of a stochastic Gilpin-Ayala model with jumps, Math. Meth. Appl. Sci.,38(2015), 1200–1211. |
| [22] | Q. Liu and Q. Chen, Analysis of a general stochastic non-autonomous logistic model with delays and L´evy jumps, J. Math. Anal. Appl.,433(2016), 95–120. |
| [23] | M.Liu,J.YuandP.S.Mandal, Dynamics of a stochastic delay competitive model with harvesting and Markovian switching, Appl. Math. Comput.,337(2018), 335–349. |
| [24] | M.Liu,C.DuandM.Deng, Persistence and extinction of a modified Leslie-Gower Holling-type II stochasticpredator-prey model with impulsive toxicant input in polluted environments, Nonlinear Anal.Hybrid Syst.,27(2018), 177–190. |
Figures(1)
Chun Lu, Bing Li, Limei Zhou, Liwei Zhang. Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3251-3271. doi: 10.3934/mbe.2019162
DownLoad: