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Stability, Relaxation and Oscillation of Biodegradation Fronts

We study the stability and oscillation of traveling fronts in a three-component, advection reaction biodegradation model. The three components are pollutant, nutrient, and bacteria concentrations. Under an explicit condition on the biomass growth and decay coefficients, we derive reduced, two-component, semi-linear hyperbolic models through a relaxation procedure, during which biomass is slaved to pollutant and nutrient concentration variables. The reduced two-component models resemble the Broadwell model of the discrete velocity gas. The traveling fronts of the reduced system are explicit and are expressed in terms of hyperbolic tangent function in the nutrient deficient regime. We perform energy estimates to prove the asymptotic stability of these fronts under explicit conditions on the coefficients in the system. In the small damping limit, we carry out Wentzel Kramers Brillouin (WKB) analysis on front perturbations and show that fronts are always stable in the two-component models. We extend the WKB analysis to derive amplitude equations for front perturbations in the original three-component model. Because of the bacteria kinetics, we find two asymptotic regimes where perturbation amplitudes grow or oscillate in time. We perform numerical simulations to illustrate the predictions of the WKB theory.