A Nonlocal Continuum Model for Biological Aggregation

Abstract

We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady state clumps are approached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions, as well, and is demonstrated via numerical simulations in two dimensions.

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

“…We should emphasise here that all these nonlocal models for cell invasion are variations or generalisations of nonlocal models developed over the last two decades to describe the dynamics of cell populations (Edelstein-Keshet & Ermentrout, 1990), bacterial populations (Othmer et al, 1988;Xue et al, 2011;Perthame et al, 2016) or the dynamics of self-organised animal populations (Mogilner & Edelstein-Keshet, 1999;Topaz et al, 2006;Eftimie et al, 2007;Fetecau & Eftimie, 2010;Carrillo de la Plata et al, 2015;Fetecau, 2011). While the models describing collective cell movement are usually of parabolic type, the latest models for collective bacterial movement and collective animal movement are usually of hyperbolic/transport type.…”

Emergent singular solutions of nonlocal density-magnetization equations in one dimension

“…Continuum aggregation equations have been used to model gravitational collapse and the subsequent emergence of stars [12], the localization of biological populations [13,14,15], and the self-assembly of nanoparticles [16]. These are complexes of atoms or molecules that form mesoscale structures with particle-like behavior.…”

Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach

“…The authors analyzed statistical properties of the model, including phase transition and scaling (Vicsek et al, 1995). A long-range interaction has been incorporated into the SPP model (Mikhailov and Zanette, 1999), and continuum, "hydrodynamic" versions of this model have been introduced Tu, 1995, 1998;Topaz et al, 2006). Recently, Couzin et al (2002Couzin et al ( , 2005) have introduced a model to provide insights into the mechanism of decision making in biological systems, which reproduces many important observations made in the field, and provides new insights into these phenomena.…”