Elsevier

Physica D: Nonlinear Phenomena

Volume 260, 1 October 2013, Pages 145-158
Physica D: Nonlinear Phenomena

Individual based and mean-field modeling of direct aggregation

https://doi.org/10.1016/j.physd.2012.11.003Get rights and content
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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.

Highlights

► We introduce two stochastic individual-based models of biological aggregation. ► The individual particle stochasticity depends on the perceived average population density. ► Both models exhibit formation of aggregates resulting from random fluctuations in the population density. ► We derive the corresponding mean field description and perform its mathematical analysis. ► Extensive numerical results are presented.

Keywords

Direct aggregation
Density dependent random walk
Degenerate parabolic equation
Mean field limit

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