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Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Part of: Parabolic equations and systems Functional-differential and differential-difference equations

Published online by Cambridge University Press:  07 May 2019

Elena Trofimchuk ,
Manuel Pinto  and
Sergei Trofimchuk
Elena Trofimchuk
Affiliation:
Department of Differential Equations, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine (trofimch@imath.kiev.ua)
Manuel Pinto
Affiliation:
Facultad de Ciencias, Universidad de Chile, Santiago, Chile (pintoj.uchile@gmail.com)
Sergei Trofimchuk
Affiliation:
Instituto de Matemática y Fisica, Universidad de Talca, Talca, Chile (trofimch@inst-mat.utalca.cl)
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Abstract

We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of ‘sufficiently small parameters’, our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar et al., we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.

Keywords

food-limited modelmonotone wavenonlocal interactionexistenceuniqueness

MSC classification

Primary: 34K60: Qualitative investigation and simulation of models
Secondary: 35K57: Reaction-diffusion equations 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2462 - 2483
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Copyright
Copyright © Royal Society of Edinburgh 2019

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