Mazenc Frédéric, Malisoff Michael, De Leenheer Patrick
Projet MERE INRIA-INRA, UMR Analyse des Systémes et Biométrie INRA 2, pl. Viala, 34060 Montpellier, France.
Math Biosci Eng. 2007 Apr;4(2):319-38. doi: 10.3934/mbe.2007.4.319.
We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long-term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat-tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long-term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function.
我们使用李雅普诺夫型分析方法研究了单一物种竞争单一营养物质的恒化器模型。我们设计了稀释率函数，使得恒化器的所有解都收敛到一个规定的周期解。从恒化器生物学的角度来看，这意味着无论为物种浓度和营养物质选择何种正初始水平，长期的物种浓度和底物水平都能紧密地近似规定的振荡行为。这一点很重要，因为它再现了物种和底物浓度振荡的现实生态情况。我们表明，当模型通过被驱使灭绝的其他物种进行扩充时，稳定性得以维持。我们还给出了在稀释率和初始浓度受到小扰动的情况下，恒化器跟踪方程的输入到状态稳定性结果。这意味着长期的物种浓度和底物行为具有非常理想的鲁棒性，因为当稀释率函数出现小的意外变化时，它仍然能够在小误差范围内继续近似规定的振荡。
