Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection

Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection J Nonlinear Sci https://doi.org/10.1007/s00332-018-9472-z Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection 1,2 1,2 Arnaud Ducrot · Xiaoming Fu · 1,2 Pierre Magal Received: 25 December 2017 / Accepted: 19 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this paper, we study the stability and the bifurcation properties of the positive interior equilibrium for a reaction–diffusion equation with nonlocal advection. Under rather general assumption on the nonlocal kernel, we first study the local well posedness of the problem in suitable fractional spaces and we obtain stability results for the homogeneous steady state. As a special case, we obtain that “standard” kernels such as Gaussian, Cauchy, Laplace and triangle, will lead to stability. Next we specify the model with a given step function kernel and investigate two types of bifurcations, namely Turing bifurcation and Turing–Hopf bifurcation. In fact, we prove that a single scalar equation may display these two types of bifurcations with the dominant wave number as large as we want. Moreover, similar instabilities can also be observed by using a bimodal kernel. The resulting complex spatiotemporal dynamics are illustrated by numerical simulations. Keywords Nonlocal reaction–diffusion–advection equation · Equilibria http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Nonlinear Science Springer Journals

Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Analysis; Theoretical, Mathematical and Computational Physics; Classical Mechanics; Mathematical and Computational Engineering; Economic Theory/Quantitative Economics/Mathematical Methods
ISSN
0938-8974
eISSN
1432-1467
D.O.I.
10.1007/s00332-018-9472-z
Publisher site
See Article on Publisher Site

Abstract

J Nonlinear Sci https://doi.org/10.1007/s00332-018-9472-z Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection 1,2 1,2 Arnaud Ducrot · Xiaoming Fu · 1,2 Pierre Magal Received: 25 December 2017 / Accepted: 19 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this paper, we study the stability and the bifurcation properties of the positive interior equilibrium for a reaction–diffusion equation with nonlocal advection. Under rather general assumption on the nonlocal kernel, we first study the local well posedness of the problem in suitable fractional spaces and we obtain stability results for the homogeneous steady state. As a special case, we obtain that “standard” kernels such as Gaussian, Cauchy, Laplace and triangle, will lead to stability. Next we specify the model with a given step function kernel and investigate two types of bifurcations, namely Turing bifurcation and Turing–Hopf bifurcation. In fact, we prove that a single scalar equation may display these two types of bifurcations with the dominant wave number as large as we want. Moreover, similar instabilities can also be observed by using a bimodal kernel. The resulting complex spatiotemporal dynamics are illustrated by numerical simulations. Keywords Nonlocal reaction–diffusion–advection equation · Equilibria

Journal

Journal of Nonlinear ScienceSpringer Journals

Published: May 30, 2018

References

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