深圳大学学报理工版

研究随机噪声对微生物连续培养的影响,引入白噪声描述营养消耗率受随机噪声的干扰,建立一类具有比率型功能反应函数的随机恒化器模型. 通过构造Lyapunov函数,利用Itô公式证明系统正解的全局存在唯一性,并论证了系统的绝灭平衡点是全局随机渐近稳定的; 探讨了噪声强度大小对随机模型的解围绕相应确定性模型的平衡点振荡行为的影响.通过数值模拟验证所得理论结果的正确性.

To explore the influence of random noise on microbial continuous culture, we establish a stochastic Chemostat model with ratio-dependent functional response in which white noise is introduced to describe nutrition conversion rate influenced by random noise. By constructing stochastic Lyapunov function and using the Itô formula, we prove that there is a unique positive solution in the system with positive initial value, and the washout equilibrium is stochastically asymptotic stable. Furthermore, we investigate the impact of white noise on the behavior of the solution spirals around the positive equilibrium of deterministic system. The numerical simulations support the proposed theoretical results.

引言
2 正解的全局存在唯一性
3 绝灭平衡点的全局随机渐近稳定性
4 随机系统的解围绕确定性系统的正
5 仿真与讨论

图1 m<D, α=0.5, 确定性模型与随机模型的解曲线<br/>Fig.1 Solution curves of deterministic model and stochastic model while m<D, α=0.5

图1 m<D, α=0.5, 确定性模型与随机模型的解曲线
Fig.1 Solution curves of deterministic model and stochastic model while m<D, α=0.5

图2 m<D, α=5, 确定性模型与随机模型的解曲线<br/>Fig.2 Solution curves of deterministic model and stochastic model while m<D, α=5

图2 m<D, α=5, 确定性模型与随机模型的解曲线
Fig.2 Solution curves of deterministic model and stochastic model while m<D, α=5

图3 m>D, α=0.1, 确定性模型与随机模型的解曲线<br/>Fig.3 Solution curves of deterministic model and stochastic model while m>D, α=0.1

图3 m>D, α=0.1, 确定性模型与随机模型的解曲线
Fig.3 Solution curves of deterministic model and stochastic model while m>D, α=0.1

图4 m>D, α=0.9, 确定性模型与随机模型的解曲线<br/>Fig.4 Solution curves of deterministic model and stochastic model while m>D, α=0.9

图4 m>D, α=0.9, 确定性模型与随机模型的解曲线
Fig.4 Solution curves of deterministic model and stochastic model while m>D, α=0.9

[1] 陈兰荪, 陈键. 非线性生物动力系统[M]. 北京: 科学出版社, 1993.
[2] Smith H L, Waltman P. Theory of the Chemostat[M]. Cambridge(UK): Cambridge University Press, 1995.
[3] Williams F M. Dynamics of microbial populations, systems analysis and simulation in Ecology[M]. Cambridge,USA: Academic Press, 1971.
[4] Xu Chaoqun, Yuan Sanling, Zhang Tonghua. Asymptotic behavior of a Chemostat model with stochastic perturbation on the dilution rate[J]. Abstract & Applied Analysis, 2013(1): 233-242.
[5] Imhof L, Walcher S. Exclusion and persistence in deterministic and stochastic chemostat models[J]. Journal of Differential Equations, 2005, 217(1): 26-53.
[6] 李相龙, 许超群, 原三领. 一类随机环境中恒化器模型的动力学行为分析[J]. 生物数学学报, 2015, 30(1): 181-191.
[7] 董庆来. 具有比率型功能反应函数的随机恒化器系统的渐近性态[J]. 山东大学学报理学版, 2014, 49(3): 68-72.
[8] Chen Zhenzhen, Zhang Tonghua. Dynamics of a stochastic model for continuous flow bioreactor with Contois growth rate[J]. Journal of Mathematical Chemistry, 2013, 51(3): 1076-1091.
[9] 付桂芳, 马万彪. 由微分方程所描述的微生物连续培养动力系统(I)[J]. 微生物学通报, 2004, 31(5): 136-139.
[10] 付桂芳, 马万彪. 由微分方程所描述的微生物连续培养动力系统(II)[J]. 微生物学通报, 2004, 31(6): 128-131.
[11] 周玉平,周洁. 微生物连续发酵模型及其应用综述[J]. 微生物学通报, 2010, 37(2): 269-273.
[12] 陆征一, 周义仓. 数学生物学进展[M]. 北京: 科学出版社, 2006.
[13] 王洪礼, 高卫楼, 袁其朋. CSTR中生化反应振荡行为研究[J]. 化学反应工程与工艺, 1997, 13(3): 270-275.
[14] 王璐, 原三领, 张同华. 具有Holing II 型功能性作用函数的随机恒化器模型的渐近性态[J]. 高校应用数学学报A 辑, 2012, 27(4): 379-389.
[15] Mao Xuerong. Stochastic different equations and application[M]. Chichester, UK: Horwood Publishing, 1997.
[16] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. Siam Review, 2001, 43(3): 525-546.

备注

引言

2 正解的全局存在唯一性

3 绝灭平衡点的全局随机渐近稳定性

4 随机系统的解围绕确定性系统的正

5 仿真与讨论

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