Communications in Mathematical Sciences

A simple model for biological aggregation with asymmetric sensing

P. A. Milewski and X. Yang

Full-text: Open access


In this paper we introduce a simple continuum model for swarming of organisms in which there is a nonlocal aggregation term with an asymmetric sensing kernel countered by a nonlinear diffusion. The model can be thought of as capturing the dynamics of organisms that aggregate responding to visual cues and whose field of vision is predominantly in the direction of motion. The model has a variety of traveling solutions, including compact swarms where the speed of the swarm increases with the total number of organisms in the swarm (up to a largest swarm), traveling fronts and periodic waves. The dynamics of fully time-dependent solutions include complicated inelastic swarm interactions and the spontaneous breakup of small populations into compact swarms. We also comment on a kinetic formulation that leads to our equations and introduce a bi-directional model where the organisms are allowed to change directions.

Article information

Commun. Math. Sci., Volume 6, Number 2 (2008), 397-416.

First available in Project Euclid: 1 July 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D25: Population dynamics (general) 92D50: Animal behavior 35Q80: PDEs in connection with classical thermodynamics and heat transfer 35F25: Initial value problems for nonlinear first-order equations

swarming chemotaxis traveling waves


Milewski, P. A.; Yang, X. A simple model for biological aggregation with asymmetric sensing. Commun. Math. Sci. 6 (2008), no. 2, 397--416.

Export citation