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A geometric construction of traveling waves in a bioremediation model. (English) Zbl 1100.92060
Summary: Bioremediation is a promising technique for cleaning contaminated soil. We study an idealized bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a one-dimensional soil column. Using geometric singular perturbation theory, we construct traveling waves (TW) corresponding to the motion of a biologically active zone, in which the microorganisms consume both substrate and acceptor. For certain values of the parameters, traveling waves exist on a three-dimensional slow manifold within the five-dimensional phase space.
We prove persistence of the slow manifold under perturbations by controlling the nonlinearity via a change of coordinates, and we construct the wave in the transverse intersection of appropriate stable and unstable manifolds in this slow manifold. We study how the TW depends on the half-saturation constants and other parameters and investigate numerically a bifurcation in which the TW loses stability to a periodic wave.

MSC:
92D40 Ecology
37N25 Dynamical systems in biology
34E15 Singular perturbations for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
35Q80 Applications of PDE in areas other than physics (MSC2000)
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