一类具有Lévy跳的随机三种群食物网模型

2019-10-21 09:26冀星刘桂荣
河北科技大学学报 2019年4期

冀星 刘桂荣

摘 要:為了深入研究具有双参数扰动及Lévy跳的随机三种群食物网模型的动力学性质,首先给出了模型全局正解的存在唯一性;然后通过构造Lyapunov函数,并且应用It公式和Chebyshev不等式证明了该模型的随机最终有界性;接着利用指数鞅不等式和Borel-Cantelli引理分析了种群灭绝的充分条件;最后运用数值模拟验证了相应理论结果的合理性。研究结果表明,在Lévy噪声的影响下模型是随机最终有界的,并且较大的Lévy噪声可以导致种群的灭绝。研究方法在理论证明和数值模拟方面都得到了良好的预期结果,对于探究其他随机种群模型的一些问题具有一定的借鉴意义。

关键词:定性理论;食物网模型;最终有界性;灭绝性;Lévy跳

中图分类号:O21163 文献标志码:A

文章编号:1008-1542(2019)04-0301-06

捕食者与食饵之间的相互作用是最重要的生态现象之一。近年来,三种群捕食者-食饵模型的一些动力学性质得到了许多学者的广泛研究[1-5]。

考虑到种群系统因不可避免地受到环境白噪声的影响而受到许多关注[6-12],文献[6]建立了下列随机三种群食物网模型:

3 结 论

本文研究了一类具有双参数扰动及Lévy跳的随机三种群食物网模型全局正解的存在唯一性和随机最终有界性,讨论了种群灭绝的充分条件,并运用数值模拟验证了结果的合理性。研究结果表明,在Lévy噪声的影响下模型是随机最终有界的,并且Lévy噪声可以导致种群的灭绝。因此,在考虑某些突发性环境冲击时,具有Lévy跳的随机模型有利于更好地研究种群的动力学性质。在未来的研究中,将着力于考虑该模型的一些其他的动力学性质。

参考文献/References:

[1] SEN D, GHORAI S, BANERJEE M . Complex dynamics of a three species prey-predator model with intraguild predation [J]. Ecological Complexity, 2018, 34: 9-22.

[2] PANJA P, MONDAL S K. Stability analysis of coexistence of three species prey-predator model [J]. Nonlinear Dynamics, 2015, 81(1/2): 373-382.

[3] 赵治涛, 张开蕊, 张玲. 具有食饵互惠的随机三种群捕食模型的持续与灭绝[J]. 黑龙江大学自然科学学报, 2017,34(1):23-34.

ZHAO Zhitao, ZHANG Kairui, ZHANG Ling. Persistence and extinction for a stochastic three species predation model with prey mutualism [J]. Journal of Natural Science of Heilongjiang University, 2017,34(1):23-34.

[4] CHEN Bin, WANG Minxing. Positive solutions to a three-species predator-prey model [J]. Acta Mathematica Scientia, 2008, 28(6): 1256-1266.

[5] DUBEY B, UPADHYAY R K. Persistence and extinction of one-prey and two-predators system [J]. Nonlinear Analysis Modelling & Control, 2004, 9(4): 307-329.

[6] QIU Hong, DENG Wenmin. Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system [J].Turkish Journal of Mathematics, 2017, 41: 1292-1303.

[7] DALAL N, GREENHALGH D, MAO Xuerong. A stochastic model for internal HIV dynamics [J]. Journal of Mathematical Analysis and Applications, 2008, 341(2): 1084-1101.

[8] VASILOVA M. Asymptotic behavior of a stochastic Gilpin-Ayala predator-prey system with time-dependent delay [J]. Mathematical & Computer Modelling, 2013, 57(3/4): 764-781.

[9] MAITI A, JANA M M, SAMANTA G P. Deterministic and stochastic analysis of a ratio-dependent predator-prey system with delay [J]. Nonlinear Analysis Modelling & Control, 2013, 12(3): 383-398.

[10]DU Bo. Existence, extinction and global asymptotical stability of a stochastic predator-prey model with mutual interference [J]. Journal of Applied Mathematics and Computing, 2014, 46(1/2): 79-91.

[11]刘蒙. 随机种群模型若干性质的研究[D]. 哈尔滨:哈尔滨工业大学, 2012.

LIU Meng. Analysis on Some Properties of Stochastic Population Systems [D]. Harbin: Harbin Institute of Technology, 2012.

[12]DU Bo, WANG Yamin, LIAN Xiuguo. A stochastic predator-prey model with delays [J]. Advances in Difference Equations, 2015, 2015(1): 141-148.

[13]LIU Qun, JIANG Daqing, SHI Ningzhong, et al. Stochastic mutualism model with Lévy jumps [J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 43: 78-90.

[14]臧彦超, 李俊平. 带Beddington-DeAngelis功能反应和Lévy噪聲的随机捕食-被捕食系统的渐近性质[J]. 应用数学学报, 2015, 38(2): 340-349.

ZANG Yanchao, LI Junping. A dynamics of a stochastic predator-prey system with Beddington-DeAngelis functional response and Lévy jumps [J]. Acta Mathematicae Applicatae Sinica, 2015, 38(2): 340-349.

[15]LIU Meng, ZHU Yu. Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps [J]. Nonlinear Analysis: Hybrid Systems, 2018, 30: 225-239.

[16]ZHANG Xinhong, JIANG Daqing, HAYAT T, et al. Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps [J]. Physica A: Statistical Mechanics and its Applications, 2017, 471(1): 767-777.

[17]ZHANG Qiumei, JIANG Daqing, ZHAO Yanan, et al. Asymptotic behavior of a stochastic population model with Allee effect by Lévy jumps [J]. Nonlinear Analysis: Hybrid Systems, 2017, 24: 1-12.

[18]LIU Meng , BAI Chuanzhi , DENG Meiling , et al. Analysis of stochastic two-prey one-predator model with Lévy jumps [J]. Physica A: Statistical Mechanics and Its Applications, 2016, 445: 176-188.

[19]MAO Xuerong. Stochastic Differential Equations and Applications [M]. Chichester: Horwood  Publishing Limited, 2007.

[20]LIPSTER R S H. A strong law of large numbers for local martingales [J]. Stochastics,1980, 3(4): 217-3228.

[21]ZOU Xiaoling, WANG Ke. Numerical simulations and modeling for stochastic biological systems with jumps[J]. Communications in Nonlinear Science and Numerical Simulation,2014, 19(5): 1557-1568.