具有Michaelis-Menten食物链的随机恒化器模型的渐近行为
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摘要
【目的】为了研究随机恒化器模型的渐近行为,本文考虑恒化器中一类稀释率受到白噪声干扰,具有Michaelis-Menten食物链的随机模型。首先证明模型正解的全局存在唯一性;【方法】然后通过构造Lyapunov函数,利用伊藤公式,得到模型的绝灭平衡点随机全局渐近稳定的充分条件;【结果】最后研究模型解的长期渐近行为,主要揭示在不同随机噪声条件下模型的解围绕其相应确定性模型的无捕食者平衡点和正平衡点的振荡行为。【结论】结果改进和推广现有文献的相关工作。
[Purposes]To investigate the asymptotic behaviors of a stochastic chemostat model with Michaelis-Menten food chain in which the dilution rate is disturbed by white noise.First,the global existence and uniqueness of the positive solution of the model is proved.[Methods]Then,by constructing Lyapunov function and using It's formula,the sufficient condition for the stochastic global asymptotic stability of the washout equilibrium of the model is obtained.[Findings]Finally,the long-time asymptotic behaviors of the solution of the model are studied,which mainly reveals the oscillatory behavior of the solution around the predator-free equilibrium and positive equilibrium of the corresponding deterministic model under different conditions.[Conclusions]The results improve and extend the relevant work of the existing literature.
[Purposes]To investigate the asymptotic behaviors of a stochastic chemostat model with Michaelis-Menten food chain in which the dilution rate is disturbed by white noise.First,the global existence and uniqueness of the positive solution of the model is proved.[Methods]Then,by constructing Lyapunov function and using It's formula,the sufficient condition for the stochastic global asymptotic stability of the washout equilibrium of the model is obtained.[Findings]Finally,the long-time asymptotic behaviors of the solution of the model are studied,which mainly reveals the oscillatory behavior of the solution around the predator-free equilibrium and positive equilibrium of the corresponding deterministic model under different conditions.[Conclusions]The results improve and extend the relevant work of the existing literature.
引文
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[2]WANG L,WOLKOWICZ GSK.A delayed chemostat model with general nonmonotone response functions and differential removal rates[J].Journal of Mathematical Analysis and Applications,2006,321(1):452-468.
[3]SUN S L,CHEN L S.Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration[J].Journal of Mathematical Chemistry,2007,42(4):837-847.
[4]ALI E,ASIF M,AJBAR A H.Study of chaotic behavior in predator-prey interactions in a chemostat[J].Ecological Modelling,2013,259:10-15.
[5]CHEN Z Z,ZHANG T H.Dynamics of a stochastic model for continuous flow bioreactor with contois growth rate[J].Journal of Mathematical Chemistry,2013,51(3):1076-1091.
[6]XU C Q,YUAN S L,ZHANG T H.Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate[J].Abstract&Applied Analysis.2013(1):233-242.
[7]XU C Q,YUAN S L.An analogue of break-even concentration in a simple stochastic chemostat model[J].Applied Mathematics Letters,2015,48:62-68.
[8]SUN M J,DONG Q L,WU J.Asymptotic behavior of a Lotka-Volterra food chain stochastic model in the Chemostat[J].Stochastic Analysis and Applications,2017,35(4):645-661.
[9]SUN S L,SUN Y R,ZHANG G,LIU X Z.Dynamical behavior of a stochastic two-species Monod competition chemostat model[J].Applied Mathematics and Computation,2017,298:153-170.
[10]JI C,JIANG D,SHI N.Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation[J].Journal of Mathematical Analysis and Applications,2009,359(2):482-498.