Permanence of a stochastic delay competition model with Lévy jumps
-
2324
Downloads
-
4586
Views
Authors
Meng Liu
- School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China.
- School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, P. R. China.
Meiling Deng
- School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China.
Zhaojuan Wang
- School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China.
Abstract
Permanence is one of the most important topics in biomathematics. The question of permanence of stochastic multi-species
models is challenging because the current approaches can not be used. In this paper, an asymptotic approach is used, and
sufficient criteria for permanence of a general n-species stochastic delay Lotka-Volterra competition model with Lévy jumps are
established. It is also shown that these criteria are sharp in some cases. The results reveal that the stochastic noises play a key
role in the permanence. This approach can be also applied to investigate the permanence of other stochastic population models
with/without time delay and/or Lévy noises.
Share and Cite
ISRP Style
Meng Liu, Meiling Deng, Zhaojuan Wang, Permanence of a stochastic delay competition model with Lévy jumps, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3245--3260
AMA Style
Liu Meng, Deng Meiling, Wang Zhaojuan, Permanence of a stochastic delay competition model with Lévy jumps. J. Nonlinear Sci. Appl. (2017); 10(6):3245--3260
Chicago/Turabian Style
Liu, Meng, Deng, Meiling, Wang, Zhaojuan. "Permanence of a stochastic delay competition model with Lévy jumps." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3245--3260
Keywords
- Permanence
- stochastic perturbations
- delay.
MSC
References
-
[1]
D. Applebaum, Lévy processes and stochastic calculus, Second edition, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2009)
-
[2]
C.-Z. Bai, Multiplicity of solutions for a class of non-local elliptic operators systems, Bull. Korean Math. Soc., 54 (2017), 715–729.
-
[3]
J.-H. Bao, X.-R. Mao, G. Yin, C.-G. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601–6616.
-
[4]
I. Barbălat, Systémes d’équations différentielles d’oscillations non linéaires, (French) Rev. Math. Pures Appl., 4 (1959), 267–270.
-
[5]
C. A. Cabrelli, U. M. Molter, The Kantorovich metric for probability measures on the circle, J. Comput. Appl. Math., 57 (1995), 345–361.
-
[6]
Y.-L. Cai, Y. Kang, M. Banerjee, W.-M. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations, 259 (2015), 7463–7502.
-
[7]
P. L. Chesson, Persistence of a Markovian population in a patchy environment, Z. Wahrsch. Verw. Gebiete, 66 (1984), 97–107.
-
[8]
P. L. Chesson, S. Ellner, Invasibility and stochastic boundedness in monotonic competition models, J. Math. Biol., 27 (1989), 117–138.
-
[9]
G. Da Prato, J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (1996)
-
[10]
K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1992)
-
[11]
D.-Q. Jiang, C.-Y. Ji, X.-Y. Li, D. O’Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582–595.
-
[12]
D.-Q. Jiang, N.-Z. Shi, X.-Y. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588–597.
-
[13]
H. Kunita, Itô’s stochastic calculus: its surprising power for applications, Stochastic Process. Appl., 120 (2010), 622– 652,.
-
[14]
D. Li, J.-A. Cui, G.-H. Song, Permanence and extinction for a single-species system with jump-diffusion, J. Math. Anal. Appl., 430 (2015), 438–464,.
-
[15]
X.-Y. Li, A. Gray, D.-Q. Jiang, X.-R. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11–28.
-
[16]
X.-Y. Li, D.-Q. Jiang, X.-R. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427–448.
-
[17]
X.-Y. Li, X.-R. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523–545.
-
[18]
M. Y. Li, Z.-S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1–20.
-
[19]
W.-X. Li, X.-Q. Zhang, C.-M. Zhang, A new method for exponential stability of coupled reaction-diffusion systems with mixed delays: combining Razumikhin method with graph theory, J. Franklin Inst., 32 (2015), 1169–1191.
-
[20]
R. S. Lipster, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217–228.
-
[21]
M. Liu, C.-Z. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597–625.
-
[22]
M. Liu, C.-Z. Bai, Dynamics of a stochastic one-prey two-predator model with Lévy jumps, Appl. Math. Comput., 284 (2016), 308–321.
-
[23]
M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1493–1508.
-
[24]
M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301–309.
-
[25]
M. Liu, C.-Z. Bai, Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete Contin. Dyn. Syst., 37 (2017), 2513–2538.
-
[26]
Q. Liu, Q.-M. Chen, Z.-H. Liu, Analysis on stochastic delay Lotka-Volterra systems driven by Lévy noise, Appl. Math. Comput., 235 (2014), 261–271.
-
[27]
M. Liu, M. Fan, Permanence of stochastic Lotka-Volterra systems, J. Nonlinear Sci., 27 (2017), 425–452.
-
[28]
M. Liu, M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math., 82 (2017), 396–423.
-
[29]
J.-L. Lv, K.Wang, Asymptotic properties of a stochastic predator-prey system with Holling II functional response, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4037–4048.
-
[30]
J.-L. Lv, K. Wang, X.-L. Zou, Remarks on stochastic permanence of population models, J. Math. Anal. Appl., 408 (2013), 561–571.
-
[31]
P. S. Mandal, S. Abbas, M. Banerjee, A comparative study of deterministic and stochastic dynamics for a non-autonomous allelopathic phytoplankton model, Appl. Math. Comput., 238 (2014), 300–318.
-
[32]
J. W. Moon, Counting labelled trees, From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress, Vancouver, (1969), Canadian Mathematical Monographs, Canadian Mathematical Congress, Montreal, Que. (1970)
-
[33]
S.-G. Peng, X.-H. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370–380.
-
[34]
H. Qiu, J.-L. Lv, K. Wang, Two types of permanence of a stochastic mutualism model, Adv. Difference Equ., 2013 (2013), 17 pages.
-
[35]
S. J. Schreiber, M. Benaïm, K. A. S. Atchadé, Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655–683.
-
[36]
H. L. Smith, H. R. Thieme, Dynamical systems and population persistence, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI (2011)
-
[37]
R.-H. Tan, Z.-J. Liu, S.-L. Guo, H.-L. Xiang, On a nonautonomous competitive system subject to stochastic and impulsive perturbations, Appl. Math. Comput., 256 (2015), 702–714.
-
[38]
K. Tran, G. Yin, Stochastic competitive Lotka-Volterra ecosystems under partial observation: feedback controls for permanence and extinction, J. Franklin Inst., 351 (2014), 4039–4064.
-
[39]
M. Vasilova, M. Jovanović, Stochastic Gilpin-Ayala competition model with infinite delay, Appl. Math. Comput., 217 (2011), 4944–4959.
-
[40]
G.-C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynam., 75 (2014), 283–287.
-
[41]
G.-C. Wu, D. Baleanu, H.-P. Xie, F.-L. Chen, Chaos synchronization of fractional chaotic maps based on the stability condition, Phys. A, 460 (2016), 374–383.
-
[42]
F.-K. Wu, Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641–657.
-
[43]
C.-M. Zhang, W.-X. Li, K. Wang, Graph-theoretic approach to stability of multi-group models with dispersal, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 259–280.
-
[44]
C.-M. Zhang, W.-X. Li, K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal. Hybrid Syst., 15 (2015), 37–51.
-
[45]
X.-H. Zhang, K. Wang, Asymptotic behavior of stochastic Gilpin-Ayala mutualism model with jumps, Electron. J. Differential Equations, 2013 (2013), 17 pages.
-
[46]
Y. Zhu, M. Liu, Permanence and extinction in a stochastic service-resource mutualism model, Appl. Math. Lett., 69 (2017), 1-7.