Non-linear advection-diffusion equations approximate swarming but not schooling populations. (English) Zbl 1143.92042
Summary: 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 nonlinear 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 nonlinear 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 nonlinear 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.
MSC:
| 92D50 | Animal behavior |
| 92D40 | Ecology |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
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