Stability, relaxation, and oscillation of biodegradation fronts

We study the stability and oscillation of traveling fronts in a three-compo nent, advection-reaction biodegradation model. The three components are pol lutant, nutrient, and bacteria concentrations. Under an explicit condition on the biomass growth and decay coefficients, we derive reduced, two-compon ent, semilinear hyperbolic models through a relaxation procedure, during wh ich biomass is slaved to pollutant and nutrient concentration variables. Th e 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-defic ient 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 t he two-component models. We extend the WKB analysis to derive amplitude equ ations for front perturbations in the original three-component model. Becau se of the bacteria kinetics, we nd two asymptotic regimes where perturbatio n amplitudes grow or oscillate in time. We perform numerical simulations to illustrate the predictions of the WKB theory.