Regularity, Asymptotic Solutions and Travelling Waves analysis in a
porous medium system to model the interaction between invasive and
invaded species
Abstract
This work provides an analytical approach to characterize and determine
solutions to a porous medium system of equations with views in
applications to invasive-invaded biological dynamics. Firstly, the
existence and uniqueness of solutions are proved. Afterwards, profiles
of solutions are obtained making use of the selfsimilar structure that
permits to show the existence of a diffusive front. The solutions are
then studied within the Travelling Waves (TW) domain showing the
existence of potential and exponential profiles in the stable connection
that converges to the stationary solutions in which the invasive specie
predominates. The TW profiles are shown to exist based on the geometry
perturbation theory together with an analytical-topological argument in
the phase plane. The finding of an exponential decaying rate (related
with the advection and diffusion parameters) in the invaded specie TW is
not trivial in the non-linear diffusion case and reflects the existence
of a TW trajectory governed by the invaded specie runaway (in the
direction of the advection) and the diffusion (acting along a finite
speed front or support).