Competition in the Chemostat: A Distributed Delay Model and Its Global Asymptotic Behavior

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.