Positivity and global existence for nonlocal advection-diffusion models of interacting populations

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

doi:10.3934/math.2025949

AIMS Mathematics

2025, Volume 10, Issue 9: 21254-21272. doi: 10.3934/math.2025949

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Positivity and global existence for nonlocal advection-diffusion models of interacting populations

1.
School of Mathematics and Computer Science, University of Swansea, Computational Foundry, Crymlyn Burrows, Skewen, Swansea SA1 8DD, UK
2.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
3.
Department of Mathematics and Statistics and Department of Biology, University of Victoria, PO Box 1700 Station CSC, Victoria, BC, Canada
4.
School of Mathematical and Physical Sciences, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK

Received: 11 December 2024 Revised: 07 August 2025 Accepted: 02 September 2025 Published: 16 September 2025
MSC : 35A01, 35B09, 35B65, 35R09

We study a broad class of nonlocal advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the nonlocal presence of others. Our model allows for different nonlocal interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of nonlocal advection-diffusion equations. We demonstrate that solutions can blow up in finite time when the detection radius becomes zero, i.e. when the system is local, thus showing that nonlocality is essential for the global existence of solutions. We verify our results with numerical simulations on 2D spatial domains.
- nonlocal advection,
- positivity of PDEs,
- global existence,
- blow-up,
- numerical PDEs
Citation: Valeria Giunta, Thomas Hillen, Mark A. Lewis, Jonathan R. Potts. Positivity and global existence for nonlocal advection-diffusion models of interacting populations[J]. AIMS Mathematics, 2025, 10(9): 21254-21272. doi: 10.3934/math.2025949

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Abstract

We study a broad class of nonlocal advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the nonlocal presence of others. Our model allows for different nonlocal interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of nonlocal advection-diffusion equations. We demonstrate that solutions can blow up in finite time when the detection radius becomes zero, i.e. when the system is local, thus showing that nonlocality is essential for the global existence of solutions. We verify our results with numerical simulations on 2D spatial domains.

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