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Weakly nonlinear analysis of a two-species non-local advection–diffusion system
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Thomas Hillen,
Mark A. Lewis,
Jonathan R. Potts
Nonlinear Analysis: Real World Applications, Volume: 78, Start page: 104086
Swansea University Author:
Valeria Giunta
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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....
| Published in: | Nonlinear Analysis: Real World Applications |
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| ISSN: | 1468-1218 |
| Published: |
Elsevier BV
2024
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa64703 |
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| 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. 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. |
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| Keywords: |
Nonlocal interactions; Pattern formation; Amplitude equation formalism; Bifurcations; Multi-stability |
| College: |
Faculty of Science and Engineering |
| Funders: |
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. |
| Start Page: |
104086 |