title: Individual based and mean-field modeling of direct aggregation.
abstract: We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the first-order model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, together with a density-dependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density. We formally derive the corresponding mean-field limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the first-order model, in particular, we prove the existence of weak solutions and show that it allows for measure-valued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for well-posedness of the first-order model. Finally, we present results of numerical simulations for both the first- and second-order model on the individual-based and continuum levels of description.
authors.affiliation: Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany.
authors.forename: Martin
authors.initials: M
authors.lastname: Burger
authors.affiliation: King Abdullah University of Science and Technology, Thuwal, Saudi Arabia.
authors.forename: Jan
authors.initials: J
authors.lastname: Haškovec
authors.affiliation: Faculty of Mathematics, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria ; DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
authors.forename: Marie-Therese
authors.initials: MT
authors.lastname: Wolfram
keywords: Degenerate parabolic equation; Density dependent random walk; Direct aggregation; Mean field limit