Regularity, Asymptotic Solutions and Travelling Waves Analysis in a Porous Medium System to Model the Interaction between Invasive and Invaded Species

José Luis Díaz Palencia, Julián Roa González, Saeed Ur Rahman and Antonio Naranjo Redondo
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José Luis Díaz Palencia: Escuela Politécnica Superior, Universidad Francisco de Vitoria, 28223 Madrid, Spain
Julián Roa González: Department of Education, Universidad a Distancia de Madrid, 28400 Madrid, Spain
Saeed Ur Rahman: Department of Mathematics, COMSATS University Islamabad, Abbottabad 22060, Pakistan
Antonio Naranjo Redondo: Technology Programs, Schiller International University, 28002 Madrid, Spain

Mathematics, 2022, vol. 10, issue 7, 1-19

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 self-similar structure that permits showing 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 species 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 species TW is not trivial in the nonlinear diffusion case and reflects the existence of a TW trajectory governed by the invaded species runaway (in the direction of the advection) and the diffusion (acting in a finite speed front or support).

Keywords: porous medium equation; travelling waves; geometric perturbation; nonlinear diffusion; advection (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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