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

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

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

Swansea University Author: Valeria Giunta Orcid Logo

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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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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 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.
Keywords: Animal movement, Energy functional, Mathematical ecology, Nonlocal advection, Partial differential equation, 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 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.
Issue: 5

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