Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion

<abstract><p>It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.</p></abstract>

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Regularity, Asymptotic Solutions and Travelling Waves analysis in a porous medium system to model the interaction between invasive and invaded species

10.22541/au.161736586.61686855/v1 ◽

2021 ◽

Author(s):

José Díaz ◽

Antonio Naranjo

Keyword(s):

Porous Medium ◽

Travelling Waves ◽

Stationary Solutions ◽

Uniqueness Of Solutions ◽

Linear Diffusion ◽

Diffusion Parameters ◽

Medium System ◽

Topological Argument ◽

And Diffusion ◽

Stable Connection

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).

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Existence and uniqueness of solutions for a class of non-linear diffusion equations

Journal of Mathematics of Kyoto University ◽

10.1215/kjm/1250520714 ◽

1987 ◽

Vol 27 (2) ◽

pp. 195-215 ◽

Cited By ~ 4

Author(s):

Tokuzo Shiga

Keyword(s):

Existence And Uniqueness ◽

Diffusion Equations ◽

Uniqueness Of Solutions ◽

Linear Diffusion ◽

Non Linear

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Uniqueness of solutions for a class of non-linear Volterra integral equations with convolution kernel

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100068262 ◽

1989 ◽

Vol 106 (3) ◽

pp. 547-552 ◽

Cited By ~ 29

Author(s):

P. J. Bushell ◽

W. Okrasinski

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Volterra Integral Equation ◽

Convolution Kernel ◽

Qualitative Behaviour ◽

Uniqueness Of Solutions ◽

Linear Diffusion ◽

Non Linear ◽

Linear Volterra Integral Equations ◽

Image Position

The non-linear Volterra integral equationhas been studied recently in connection with non-linear diffusion and percolation problems [4, 6, 10]. The existence, uniqueness and qualitative behaviour of non-negative, non-trivial solutions are the questions of physical interest.

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