Competition in the chemostat: A distributed delay model and its global asymptotic behavior
Competition in the chemostat: A distributed delay model and its global asymptotic behavior
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DOI：
10.1137/s0036139995289842
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发表时间：
1997-10-01
影响因子：
1.9
通讯作者：
Ruan, SG
中科院分区：
文献类型：
--
作者：
Wolkowicz, GSK;Xia, HX;Ruan, SG
本文提出了一个恒化器中的两种群竞争模型，该模型利用分布时滞来模拟营养物质转化过程中的时滞，并研究了模型的全局渐近性态。该模型包括一个冲刷因子的时间延迟涉及的养分转化，因此延迟分布在物种浓度以及营养浓度（使用伽马分布）。利用线性链技巧和涨落引理，我们完全确定了模型的全局极限性态，证明了模型最多只能有一个幸存者，并给出了一个依赖于延迟核参数的预测准则.我们比较这些预测的竞争引入的定性结果，包括分布延迟模型与相应的离散延迟模型的预测，以及与相应的无延迟常微分方程模型。我们证明了离散时滞模型和相应的常微分方程模型可以作为分布时滞模型的极限情形得到。此外，假设平均延迟很小，延迟模型的预测几乎与常微分方程模型给出的预测相同。然而，当平均延迟是显着的，由延迟模型给出的关于哪个物种赢得竞争和避免灭绝的预测可以是彼此不同的或从相应的常微分方程模型的预测。通过改变延迟核中的参数，我们发现该模型似乎更有可能模仿现实。例如，计算机模拟表明，灭绝物种的平均延迟越大，物种走向灭绝的速度就越快。
In this paper, we propose a two species competition model in a chemostat that uses a distributed delay to model the lag in the process of nutrient conversion and study the global asymptotic behavior of the model. The model includes a washout factor over the time delay involved in the nutrient conversion, and hence the delay is distributed over the species concentrations as well as over the nutrient concentration (using the gamma distribution). The results are valid for a very general class of monotone growth response functions.By using the linear chain trick technique and the fluctuation lemma, we completely determine the global limiting behavior of the model, prove that there is always at most one survivor, and give a criterion to predict the outcome that is dependent upon the parameters in the delay kernel. We compare these predictions on the qualitative outcome of competition introduced by including distributed delay in the model with the predictions made by the corresponding discrete delay model, as well as with the corresponding no delay ODEs model. We show that the discrete delay model and the corresponding ODEs model can be obtained as limiting cases of the distributed delay models. Also, provided that the mean delays are small, the predictions of the delay models are almost identical with the predictions given by the ODEs model. However, when the mean delays are significant, the predictions given by the delay models concerning which species wins the competition and avoids extinction can be different from each other or from the predictions of the corresponding ODEs model. By varying the parameters in the delay kernels, we find that the model seems to have more potential to mimic reality. For example, computer simulations indicate that the larger the mean delay of the losing species, the more quickly that species proceeds toward extinction.