Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species

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

doi:10.1007/s00285-022-01824-1

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Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species

Valeria Giunta

, Thomas Hillen, Mark A. Lewis, Jonathan R. Potts

Journal of Mathematical Biology, Volume: 85, Issue: 5

Swansea University Author: Valeria Giunta

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DOI (Published version): 10.1007/s00285-022-01824-1

Abstract

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qual...

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Published in:	Journal of Mathematical Biology
ISSN:	0303-6812 1432-1416
Published:	Springer Science and Business Media LLC 2022
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URI:	https://cronfa.swan.ac.uk/Record/cronfa64701
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Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. 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spelling	v2 64701 2023-10-10 Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species 50456cce4b2c7be66f8302d418963b0c 0000-0003-1156-7136 Valeria Giunta Valeria Giunta true false 2023-10-10 SMA Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species. Journal Article Journal of Mathematical Biology 85 5 Springer Science and Business Media LLC 0303-6812 1432-1416 Animal movement, Energy functional, Mathematical ecology, Nonlocal advection, Partial differential equation, Stability 30 11 2022 2022-11-30 10.1007/s00285-022-01824-1 http://dx.doi.org/10.1007/s00285-022-01824-1 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 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 grateful for support from the Natural Science and Engineering Council of Canada (NSERC) Discovery Grant RGPIN-2017-04158. MAL gratefully acknowledges support from NSERC Discovery Grant RGPIN-2018-05210 and the Canada Research Chair program. 2023-11-28T14:09:18.4131372 2023-10-10T12:21:13.4340219 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 64701__29133__46b1b56489644930be51450e4018e7c7.pdf 64701.VOR.pdf 2023-11-28T14:07:47.9356746 Output 1186135 application/pdf Version of Record true © The Author(s) 2022. Distributed under the terms of a Creative Commons Attribution 4.0 International License (CC BY 4.0). true eng https://creativecommons.org/licenses/by/4.0/
title	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species
spellingShingle	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species Valeria Giunta
title_short	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species
title_full	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species
title_fullStr	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species
title_full_unstemmed	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species
title_sort	Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species
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
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container_title	Journal of Mathematical Biology
container_volume	85
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publishDate	2022
institution	Swansea University
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doi_str_mv	10.1007/s00285-022-01824-1
publisher	Springer Science and Business Media LLC
college_str	Faculty of Science and Engineering
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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/10.1007/s00285-022-01824-1
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description	Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.
published_date	2022-11-30T14:09:19Z
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Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species

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