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Spikes and localised patterns for a novel Schnakenberg model in the semi-strong interaction regime

Part of: Physiological, cellular and medical topics Partial differential equations Parabolic equations and systems Asymptotic theory

Published online by Cambridge University Press:  25 January 2021

FAHAD AL SAADI ,
ALAN CHAMPNEYS Open the ORCID record for ALAN CHAMPNEYS [Opens in a new window] ,
CHUNYI GAI  and
THEODORE KOLOKOLNIKOV
FAHAD AL SAADI
Affiliation:
Department of Engineering Mathematics, University of Bristol, BristolBS8 1UB, UK, emails: a.r.champneys@bristol.ac.uk; fa17741@bristol.ac.uk Department of Systems Engineering, Military Technological College, Muscat, Oman
ALAN CHAMPNEYS
Affiliation:
Department of Engineering Mathematics, University of Bristol, BristolBS8 1UB, UK, emails: a.r.champneys@bristol.ac.uk; fa17741@bristol.ac.uk
CHUNYI GAI
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova ScotiaB3H 4R2, Canada, emails: Chunyi.Gai@dal.ca; tkolokol@gmail.com
THEODORE KOLOKOLNIKOV
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova ScotiaB3H 4R2, Canada, emails: Chunyi.Gai@dal.ca; tkolokol@gmail.com
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Abstract

An analysis is undertaken of the formation and stability of localised patterns in a 1D Schanckenberg model, with source terms in both the activator and inhibitor fields. The aim is to illustrate the connection between semi-strong asymptotic analysis and the theory of localised pattern formation within a pinning region created by a subcritical Turing bifurcation. A two-parameter bifurcation diagram of homogeneous, periodic and localised patterns is obtained numerically. A natural asymptotic scaling for semi-strong interaction theory is found where an activator source term \[a = O(\varepsilon )\] and the inhibitor source \[b = O({\varepsilon ^2})\], with ε2 being the diffusion ratio. The theory predicts a fold of spike solutions leading to onset of localised patterns upon increase of b from zero. Non-local eigenvalue arguments show that both branches emanating from the fold are unstable, with the higher intensity branch becoming stable through a Hopf bifurcation as b increases beyond the \[O(\varepsilon )\] regime. All analytical results are found to agree with numerics. In particular, the asymptotic expression for the fold is found to be accurate beyond its region of validity, and its extension into the pinning region is found to form the low b boundary of the so-called homoclinic snaking region. Further numerical results point to both sub and supercritical Hopf bifurcation and novel spikeinsertion dynamics.

Keywords

reaction-diffusion systemspattern formationspikessnaking

MSC classification

Primary: 35B32: Bifurcation
Secondary: 35K57: Reaction-diffusion equations 92C15: Developmental biology, pattern formation 34E13: Multiple scale methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 133 - 152
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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