Nonlocal advection is a key process in a range of biological systems, from cells within individuals to the movement of whole organisms. Consequently, in recent years, there has been increasing attention on modeling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modeling the nonlocality. One common formalism is the aggregation-diffusion equation, a class of advection-diffusion models with nonlocal advection. This was originally used to model a single population but has recently been extended to the multispecies case to model the way organisms may alter their movement in the presence of coexistent species. Here we prove existence theorems for a class of nonlocal multispecies advection-diffusion models, with an arbitrary number of coexistent species. We prove global existence for models in $n=1$ spatial dimension and local existence for $n>1$. We describe an efficient spectral method for numerically solving these models and provide an example simulation output. Overall, this helps provide a solid mathematical foundation for studying the effect of interspecies interactions on movement and space use.


  1. advection-diffusion
  2. aggregation-diffusion
  3. existence theorems
  4. mathematical ecology
  5. nonlocal advection
  6. taxis

MSC codes

  1. 35A01
  2. 35B09
  3. 35B65
  4. 35R09
  5. 92-10
  6. 92D40

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1686 - 1708
ISSN (online): 1536-0040


Submitted: 10 June 2021
Accepted: 2 March 2022
Published online: 5 July 2022


  1. advection-diffusion
  2. aggregation-diffusion
  3. existence theorems
  4. mathematical ecology
  5. nonlocal advection
  6. taxis

MSC codes

  1. 35A01
  2. 35B09
  3. 35B65
  4. 35R09
  5. 92-10
  6. 92D40



Funding Information

Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038 : RGPIN-2017-04158
Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni https://doi.org/10.13039/100012740 : 2017YBKNCE
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/V002988/1

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