J Nonlinear Sci
Turing and Turing–Hopf Bifurcations for a Reaction
Diffusion Equation with Nonlocal Advection
· Xiaoming Fu
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 stability ·
Turing bifurcation · Turing–Hopf bifurcation
Mathematics Subject Classiﬁcation 35K55 · 35B32 · 35B35
Communicated by Michael Ward.
Xiaoming Fu: The research of this author is supported by China Scholarship Council.
IMB, UMR 5251, University of Bordeaux, 33400 Talence, France
IMB, UMR 5251, CNRS, 33400 Talence, France