Stability,
Relaxation and Oscillation of Biodegradation Fronts
SIAM J. APPL.
MATH, Vol. 61, No.2, pp 472 –505
Society for Industrial and Applied Mathematics
Jack X. Xin and James M. Hyman
ABSTRACT:
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.
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