Modelling the interaction of invasive-invaded species based on the general Bramson dynamics and with a density dependant diffusion and advection

José Luis Díaz Palencia; Abraham Otero; José Luis Díaz Palencia; Abraham Otero

doi:10.3934/mbe.2023589

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 13200-13221. doi: 10.3934/mbe.2023589

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Modelling the interaction of invasive-invaded species based on the general Bramson dynamics and with a density dependant diffusion and advection

José Luis Díaz Palencia ^{1,2
,
,},
Abraham Otero ¹

1.
Department of Information Technology, Escuela Politecnica Superior, Universidad San Pablo-CEU, CEU Universities, Campus Monteprincipe, Boadilla del Monte, Madrid 28668, Spain
2.
Department of Mathematics and Didactics, Universidad a Distancia de Madrid, UDIMA, Madrid, Spain

Academic Editor: Feliz Manuel Barrão Minhós

Received: 12 May 2023 Revised: 24 May 2023 Accepted: 29 May 2023 Published: 08 June 2023

The main goal of the presented study is to introduce a model of a pairwise invasion interaction with a nonlinear diffusion and advection. The new equation is based on the further general works introduced by Bramson (1988) to describe the invasive-invaded dynamics. This type of model is made particular with a density dependent diffusion along with an advection term. The new resulting model is then analyzed to explore the regularity, existence and uniqueness of solutions. It is well known that density dependent diffusion operators induce a propagating front with finite speed for compactly supported functions. Based on this, we introduce an analytical approach to determine the evolution of such a propagating front in the invasion dynamics. Afterward, we study the problem with travelling wave profiles and a numerical assessment. As a main finding to remark: When both species propagate with significantly different travelling wave speeds, the interaction becomes unstable, while when the species propagate with similar low speeds, the interaction stabilizes.
- invasion,
- density dependant diffusion,
- travelling waves
Citation: José Luis Díaz Palencia, Abraham Otero. Modelling the interaction of invasive-invaded species based on the general Bramson dynamics and with a density dependant diffusion and advection[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 13200-13221. doi: 10.3934/mbe.2023589

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Abstract

The main goal of the presented study is to introduce a model of a pairwise invasion interaction with a nonlinear diffusion and advection. The new equation is based on the further general works introduced by Bramson (1988) to describe the invasive-invaded dynamics. This type of model is made particular with a density dependent diffusion along with an advection term. The new resulting model is then analyzed to explore the regularity, existence and uniqueness of solutions. It is well known that density dependent diffusion operators induce a propagating front with finite speed for compactly supported functions. Based on this, we introduce an analytical approach to determine the evolution of such a propagating front in the invasion dynamics. Afterward, we study the problem with travelling wave profiles and a numerical assessment. As a main finding to remark: When both species propagate with significantly different travelling wave speeds, the interaction becomes unstable, while when the species propagate with similar low speeds, the interaction stabilizes.

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