This lab deals with a variation on Example 12.4. Often, contaminated soil will contain bacteria which, via metabolism or co-metabolism, is capable of eliminating the contaminant. Thus, a cost-effective method of managing and cleaning contaminated areas is to increase the level of these bacteria, in that more bacteria will result in more rapid degradation of the contaminant. The injection of nutrients needed for metabolism and colony growth has proven to be a successful technique in boosting the bacteria population. However, modeling the remediation process is nontrivial. Various processes, such as bacterial reproduction, metabolism, death, nutrient flow, and contaminant degradation, are highly coupled. Also of concern is the uncertainty in estimations of bacterial and contaminant distribution in the soil. Hence, a realistic model would need to include spatial effects and hetero-

geneities in the environment. For this reason, we instead focus on the more controlled setting found in a bioreactor, such as the ones used for drug production and sewage treatment, for example. There, simple but effective relations for bacterial growth, degradation, and production processes can be formed. Precisely this type of model was developed and used for optimal control studies in the paper by Heinricher, Lenhart, and Solomon [82]. A bioreactor with ideal mixing is considered, where a contaminant and a

bacteria known to degrade this contaminant via co-metabolism are present. The bacteria and contaminant have spatially uniform, time-varying concentrations (in g/L) x(t), z(t) respectively. The bioreactor is assumed to be rich in all nutrients needed for bacteria growth save one, whose injection into the reactor has spatially uniform, time-varying concentration u(t). Bacterial growth rate is given by Gu(t)x(t) and death rate by Dx2(t), so that

x′(t) = Gu(t)x(t)−Dx(t)2, x(0) = x0 > 0.