Non-linear advection-diffusion equations approximate swarming but not schooling populations

Authored by Daniel Grunbaum, Karen Chan, Elizabeth Tobin, Michael T Nishizaki

Date Published: 2008

DOI: 10.1016/j.mbs.2008.06.002

Sponsors: United States National Science Foundation (NSF)

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

Abstract

Advection-diffusion equations (ADEs) are concise and tractable mathematical descriptions of population distributions used widely to address spatial problems in applied and theoretical ecology. We assessed the potential of non-linear ADEs to approximate over very large time and space scales the spatial distributions resulting from social behaviors such as swarming and schooling, in which populations are subdivided into many groups of variable size, velocity and directional persistence. We developed a simple numerical scheme to estimate coefficients in non-linear ADEs from individual-based model (IBM) simulations. Alignment responses between neighbors within groups quantitatively and qualitatively affected how populations moved. Asocial and swarming populations, and schooling populations with weak alignment tendencies, were well approximated by non-linear ADEs. For these behaviors, numerical estimates such as ours could enhance realism and efficiency in ecosystem models of social organisms. Schooling populations with strong alignment were poorly approximated, because (in contradiction to assumptions underlying the ADE approach) effective diffusion and advection were not uniquely defined functions of local density. PDE forms other than ADEs are apparently required to approximate strongly aligning populations. (C) 2008 Elsevier Inc. All rights reserved.

Tags

Migration behavior Dynamics Distributions Model Aggregation Fish Individuals Particle system