Permanence of Stochastic Biological Systems
DOI:
https://doi.org/10.5890/DNC.2019.06.004Abstract
Area of research related to prey-predator systems is an important topic. The concept of permanence is an important issues related to biological systems. In general permanence is considered as a combination of persistence and boundedness. Following this, this paper reviews few existing definitions of stochastic permanence. Following the existing definition, in this paper a new definition of permanence for stochastic biological systems has been proposed, which modifies the existing ones. The permanence of a general Lotka-Volterra stochastic two species cooperation, competition and predation systems is proved by new definition.References
[1] Kant, S. and Kumar, V. (2016) Dynamical Behavior of a stage structured predator-prey model, Int. J. Nonlinear Anal. Appl., 7(1), 231-241.
[2] Kant, S. and Kumar, V. (2017), Stability analysis of a predator-prey system with migrating prey and disease infection in both species, Applied Mathematical Modelling, 4,2 509-539.
[3] Kant, S. and Kumar, V. (2017), Dynamics of a prey-predator system with infection in prey, Electronic Journal of Differential Equations, 209, 1-27.
[4] Kant, S. and Kumar, V. (2015), Analysis of an eco-epedemiological model with maigrating and refuging prey, In: Agrawal P., Mohapatra R., Singh U., Srivastava H. (eds) Mathematical Analysis and its Applications, Springer Proceedings in Mathematics & Statistics, vol 143,Springer, New Delhi.
[5] Kant, S. and Kumar, V. (2015), The control for prey-predator system with time delay and refuge, In: Mohapatra R., Chowdhury D., Giri D. (eds)Mathematics and Computing, Springer Proceedings in Mathematics & Statistics, vol 139, Springer, New Delhi.
[6] Arora, C., Kumar, V., and S. Kant, (2017), Dynamics of a High Dimensional stage structured prey-predatormodel, Int. J. Appl. Comput. Math., 3(Suppl 1), 427-445. DOI: 10.1007/s40819-017-0363-z.
[7] Kant, S. and Kumar, V. (2015), Dynamical behaviour of stage structured eco-epimedemiological model, Differential Equations and Applications, 7(4), 503-215.
[8] Mao, X., Sabanis, S., and Renshaw, E. (2003), Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287, 141-156.
[9] Mao, X., Marion, G., and Renshaw, E. (2002), Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 95-110.
[10] Bao, J.,Mao, X., Yin, G., and Yuan, C. (2011), Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74, 6601-6616.
[11] Jacob, N., Wang, Y., and Yuan, C. (2009), Stochastic delay differential equations with jumps, under nonlinear growth condition, Stochastics, 81, 571-588.
[12] Rudnicki, R. (2003), Long-time behaviour of a stochastic prey-predatormodel, Stochastic Process. Appl., 108, 93-107.
[13] Nguyen, D.H. andYin, G. (2017), Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262(3), 1192-1225.
[14] Lv., J., Wang, K., and Zou, X. (2013), Remarks on stochastic permanence of population models, J. Math. Anal. Appl., 408, 561-571.
[15] Liu, M. and Fan, M. (2017), Permanence of stochastic Lotka Volterra systems, J. Nonlinear Sci., 27(2), 425-452.
[16] Jenson,W. (1987), A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25, 411-422.
[17] Jiang, D., Shi, N., and Li, X. (2008), Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340, 588-597.
[18] Li, D., Cui, J., and Song, G. (2015), Permanence and extinction for a single-species system with jump-diffusion, J. Math. Anal. Appl., 430, 438-464.
[19] Liu, M., Wang, K., and Wu, Q. (2011), Survival analysis of stochastic competitive models in a polluted environment and stochatic competitive exclusion principle, Bull. Math. Biol., 73, 1969-2012.
[20] Li, X., Gray, A., Jiang, D., and Mao, X. (2011), Suffient and necessary conditions of stochastic permanence and extinction for stochatic logistic populations under regime switching, J. Math. Anal. Appl., 376, 11-28.
[21] Rao, F. (2014), The complex dynamics of a stochastic toxic-phytoplankton-zooplanktonmodel, Adx. Differ. Equ., DOI: 10.1186/1687-1847-2014-22
[22] Rao, F., Jiang, S., Li, Y., and Liu, H. (2013), Stochastic analysis of a Hassell-Varley type predatation model, Abstr. Appl. Anal., Article ID 738342.
[23] Zhang, X. andWang, K. (2013), Asymptotic behaviour of stochastic Gilpin-Ayala mutualism model with jumps, Electon. J. Differ. Equa., 2013, 1-17.
[24] Zhang, Y., Gao, S., Fan, K., and Dai, Y. (2015), On the dynamics of a ratio-dependent predator-prey model with a specific functional response, J. Appl. Math. Comput., 48, 441-460.
[25] Qiu, H., Lv, J., and Wang, K. (2013) Two types of permanence of a stochastic mutualism model, Adv. Differ. Equ., 37, 1-17.
[26] Li, X. and Mao, X. (2009), Population dynamical behaviour of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24, 523-545.
[27] Zhu, Y. and Liu, M. (2017), Permanence and extinction in a stochastic service-resource mutualism, Applied Mathematics Letters, 69, 1-7.
[28] Zu, L., Jiang, D., and ORegan, D. (2014), Stochastic permanence, stationary distribution and extinction of a single-species nonlinear diffusion system with random perturbation, Abstract and Applied Analysis, DOI: 10.1155/2014/320460.
[29] Li, X., Jiang, D., and Mao, X. (2009), Population dynamical behaviour of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232, 427-448.
[30] Liu, M. and Wang, K. (2013), Population dynamical behaviour of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33, 2495-2522.
[31] Tran, K. and Yin, G. (2014), Stochastic competetive Lotka-Volterra ecosystems under partial observation: feedback controls for permanence and extinction, J. Frankl. Inst., 351, 4039-4064.
[32] Mandal, P.S., Abbas, S., and Banerjee, M. (2014), A comprative study of deterministic and stochastic dynamics for a non-autonomous allelopathic phytoplankton model, Appl. Math. Comput., 238, 300-318.
[33] Liu, M. and Wang, K. (2014), Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410, 750-763.
[34] Ikeda, N. and Watanabe, S. (1981), Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam.
[35] Jiang, D., Ji, C., and Li, X. (2012), Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390, 582-595.
[36] Richard, et al. (1990), Granulomatous angiitis of the nervous system, A case report of long-term survival, Surgical Neurology, 33(3), 206-212. DOI: 10.1016/0090-3019(90)90186-S
[37] Choi, et al. (2017) Long-term survival outcomes of primary breast cancer in women with or without preoperative magnetic resonence imaging: A matched cohort study, Clinical Onclology, 29(10), 653-661. DOI: 10.1016/j.clon.2017.06.015.
[38] Foliart, D.E. and et al. (2006), Magnetic field exposure and long term survival among children with leukaemia, Br J Cancer, 94(1), 161-164.
[39] Foliart, D.E. and et al. (2006), Magnetic field exposure and long term survival among children with leukaemia, Br J Cancer, 94(6), 940.
[40] Chen, X. and et al. (2012), Long term survival of mice with hepatocellular carcinoma after pulse power ablation with nanosecond pulsed electric fields, Technology in Cancer Research and Treatment, 11(1), 83-93.
[41] Sung, et al. (2012), Long term electrical survival analysis of Riata and Riata ST silicone leads: National Veternaus Affairs experience, Heart Rhythm, 9(12), 1954-1961, DOI: 10.1016/j.hrthm.2012.08.006.
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