Travelling Waves in a PDE–ODE Coupled Model of Cellulolytic Biofilms with Nonlinear Diffusion

Abstract We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with diffusion coefficient that vanishes as the solution tends zero and blows up it approaches its maximum value. Stable TW such have previously been observed numerically well in biological experiments on growth cellulolytic biofilms. In this work, we provide analytical justification these observations prove existence stability results models. Using ansatz first integral, system is reduced autonomous dynamical two unknowns. Analysing corresponding phase–plane, unique shown, which possesses sharp front diffusive tail, moving constant speed. The linear space dimensions proven under suitable assumptions initial data. Finally, numerical simulations are presented affirm theoretical predictions existence, stability, parametric dependence waves.