A non-local model for a swarm Export

@article{Mogilner_99, abstract = {Abstract.\ \  This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles.}, author = {Mogilner, Alexander and Edelstein-Keshet, Leah}, citeulike-article-id = {1850888}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/s002850050158}, citeulike-linkout-1 = {http://www.springerlink.com/content/0tv09y69wj2nlm2r}, day = {15}, doi = {10.1007/s002850050158}, journal = {Journal of Mathematical Biology}, keywords = {flocking\_swarms, nash\_certainty\_equivalence, nature, stochastic\_control\_large\_population\_systems}, month = {June}, number = {6}, pages = {534--570}, posted-at = {2009-07-25 21:50:42}, priority = {2}, title = {A non-local model for a swarm}, url = {http://dx.doi.org/10.1007/s002850050158}, volume = {38}, year = {1999} }

TY - JOUR ID - Mogilner_99 L3 - citeulike-article-id:1850888 N2 - Abstract.   This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles. IS - 6 JF - Journal of Mathematical Biology EP - 570 TI - A non-local model for a swarm VL - 38 SP - 534 KW - flocking_swarms KW - nash_certainty_equivalence KW - nature KW - stochastic_control_large_population_systems AU - Mogilner, Alexander AU - Edelstein-Keshet, Leah PY - 1999/06/15/ UR - http://dx.doi.org/10.1007/s002850050158 ER -