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Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

Gaetana Gambino, Valeria Giunta Orcid Logo, Maria Carmela Lombardo, Gianfranco Rubino

Discrete and Continuous Dynamical Systems - B, Volume: 27, Issue: 12, Pages: 7783 - 7816

Swansea University Author: Valeria Giunta Orcid Logo

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DOI (Published version): 10.3934/dcdsb.2022063

Abstract

We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-dif...

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Published in: Discrete and Continuous Dynamical Systems - B
ISSN: 1531-3492 1553-524X
Published: American Institute of Mathematical Sciences (AIMS) 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa64702
first_indexed 2023-10-10T11:24:27Z
last_indexed 2024-11-25T14:14:33Z
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spelling 2023-11-28T13:54:40.0663876 v2 64702 2023-10-10 Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model 50456cce4b2c7be66f8302d418963b0c 0000-0003-1156-7136 Valeria Giunta Valeria Giunta true false 2023-10-10 MACS We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named cross-Turing patterns). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis. Journal Article Discrete and Continuous Dynamical Systems - B 27 12 7783 7816 American Institute of Mathematical Sciences (AIMS) 1531-3492 1553-524X Cross-diffusion, FitzHugh-Nagumo, Turing instability, out-of-phase patterns, amplitude equations 31 12 2022 2022-12-31 10.3934/dcdsb.2022063 http://dx.doi.org/10.3934/dcdsb.2022063 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2023-11-28T13:54:40.0663876 2023-10-10T12:23:09.8125675 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Gaetana Gambino 1 Valeria Giunta 0000-0003-1156-7136 2 Maria Carmela Lombardo 3 Gianfranco Rubino 4
title Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
spellingShingle Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
Valeria Giunta
title_short Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
title_full Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
title_fullStr Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
title_full_unstemmed Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
title_sort Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
author_id_str_mv 50456cce4b2c7be66f8302d418963b0c
author_id_fullname_str_mv 50456cce4b2c7be66f8302d418963b0c_***_Valeria Giunta
author Valeria Giunta
author2 Gaetana Gambino
Valeria Giunta
Maria Carmela Lombardo
Gianfranco Rubino
format Journal article
container_title Discrete and Continuous Dynamical Systems - B
container_volume 27
container_issue 12
container_start_page 7783
publishDate 2022
institution Swansea University
issn 1531-3492
1553-524X
doi_str_mv 10.3934/dcdsb.2022063
publisher American Institute of Mathematical Sciences (AIMS)
college_str Faculty of Science and Engineering
hierarchytype
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
url http://dx.doi.org/10.3934/dcdsb.2022063
document_store_str 0
active_str 0
description We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named cross-Turing patterns). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis.
published_date 2022-12-31T05:29:44Z
_version_ 1821382158269284352
score 11.04748

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