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Weakly nonlinear analysis of a two-species non-local advection–diffusion system

Valeria Giunta Orcid Logo, Thomas Hillen, Mark A. Lewis, Jonathan R. Potts

Nonlinear Analysis: Real World Applications, Volume: 78, Start page: 104086

Swansea University Author: Valeria Giunta Orcid Logo

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DOI (Published version): 10.1016/j.nonrwa.2024.104086

Abstract

Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions....

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Published in: Nonlinear Analysis: Real World Applications
ISSN: 1468-1218
Published: Elsevier BV 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa64703
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spelling 2024-03-01T15:28:33.4802469 v2 64703 2023-10-10 Weakly nonlinear analysis of a two-species non-local advection–diffusion system 50456cce4b2c7be66f8302d418963b0c 0000-0003-1156-7136 Valeria Giunta Valeria Giunta true false 2023-10-10 MACS Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg–Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg–Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions. Journal Article Nonlinear Analysis: Real World Applications 78 104086 Elsevier BV 1468-1218 Nonlocal interactions; Pattern formation; Amplitude equation formalism; Bifurcations; Multi-stability 23 2 2024 2024-02-23 10.1016/j.nonrwa.2024.104086 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) JRP and VG acknowledge support of Engineering and Physical Sciences Research Council (EPSRC) grant EP/V002988/1 awarded to JRP. VG is also grateful for support from the National Group of Mathematical Physics (GNFM-INdAM). TH is supported through a discovery grant of the Natural Science and Engineering Research Council of Canada (NSERC), RGPIN-2017-04158. MAL gratefully acknowledges support from NSERC Discovery Grant RGPIN-2018-05210 and from the Gilbert and Betty Kennedy Chair in Mathematical Biology. 2024-03-01T15:28:33.4802469 2023-10-10T12:25:17.6855997 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Valeria Giunta 0000-0003-1156-7136 1 Thomas Hillen 2 Mark A. Lewis 3 Jonathan R. Potts 4 64703__29610__8f4f1a915d7144af92da3055534b78a7.pdf 64703_VoR.pdf 2024-03-01T15:26:58.3159687 Output 1510262 application/pdf Version of Record true ©2024 TheAuthors. This is an open access article under the CC BY license true eng http://creativecommons.org/licenses/by/4.0/
title Weakly nonlinear analysis of a two-species non-local advection–diffusion system
spellingShingle Weakly nonlinear analysis of a two-species non-local advection–diffusion system
Valeria Giunta
title_short Weakly nonlinear analysis of a two-species non-local advection–diffusion system
title_full Weakly nonlinear analysis of a two-species non-local advection–diffusion system
title_fullStr Weakly nonlinear analysis of a two-species non-local advection–diffusion system
title_full_unstemmed Weakly nonlinear analysis of a two-species non-local advection–diffusion system
title_sort Weakly nonlinear analysis of a two-species non-local advection–diffusion system
author_id_str_mv 50456cce4b2c7be66f8302d418963b0c
author_id_fullname_str_mv 50456cce4b2c7be66f8302d418963b0c_***_Valeria Giunta
author Valeria Giunta
author2 Valeria Giunta
Thomas Hillen
Mark A. Lewis
Jonathan R. Potts
format Journal article
container_title Nonlinear Analysis: Real World Applications
container_volume 78
container_start_page 104086
publishDate 2024
institution Swansea University
issn 1468-1218
doi_str_mv 10.1016/j.nonrwa.2024.104086
publisher Elsevier BV
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
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description Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg–Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg–Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.
published_date 2024-02-23T08:25:16Z
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