# A Nonlocal Continuum Model for Biological Aggregation

@article{Topaz2005ANC,
title={A Nonlocal Continuum Model for Biological Aggregation},
author={Chad M. Topaz and A. Bertozzi and Mark A. Lewis},
journal={Bulletin of Mathematical Biology},
year={2005},
volume={68},
pages={1601-1623}
}
• Published 1 April 2005
• Physics
• Bulletin of Mathematical Biology
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 reached through a dynamic coarsening process. In the limit of large population size…
419 Citations
In this thesis, we study a nonlocal hyperbolic model for biological aggregations in one spatial dimension. In particular, we investigate the linear stability of the spatially homogeneous steady
We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range
• Computer Science
• 2008
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 is introduced.
• O. Aydogmus
• Mathematics
Bulletin of Mathematical Biology
• 2017
The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be
• O. Aydogmus
• Mathematics
Bulletin of mathematical biology
• 2018
A single-species, continuous time metapopulation model taking nonlocal interactions into account is proposed and it is shown that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state and as the dispersal rate decreases, amplitude of the patterns increases.
• Mathematics
SIAM J. Appl. Dyn. Syst.
• 2009
An analytical upper bound is derived for the finite blow-up time after which the solution forms one or more $\delta$-functions of the conservation equation.
• Mathematics
Discrete & Continuous Dynamical Systems - B
• 2021
The paper studies the pattern formation dynamics of a discrete in time and space model with nonlocal resource competition and dispersal. Our model is generalized from the metapopulation model
• Physics
Bulletin of mathematical biology
• 2012
We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popular
• Physics
Bulletin of Mathematical Biology
• 2011
We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popular

## References

SHOWING 1-10 OF 61 REFERENCES

• Physics
SIAM J. Appl. Math.
• 2004
A continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior, which takes the form of a conservation law in two spatial dimensions.
• Mathematics
• 1999
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
• Physics
Physical review letters
• 2005
Simulations show collapsed (clumped) states emerge from smooth initial conditions, even in one spatial dimension, in new model equations derived for dynamics of aggregation of finite-size particles.
• Mathematics
• 1983
We consider a spatially aggregating population model which provides the homogenizing process due to density-dependent diffusion and the dehomogenizing one due to a certain long-range transport. The
• Physics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2001
A continuum version of the discrete model consisting of self-propelled particles that obey simple interaction rules is developed and it is demonstrated that the agreement between the discrete and the continuum model is excellent.
where ml is a constant, 0grg c a parameter and u(x, t)O denotes the population density at position x e R and ime t0. Equations o his type, proposed by .Nagai and Mimura [4J, represent a spatially
• Computer Science
The American Naturalist
• 2004
This model is the first to achieve the formation of steady heterogeneous structures of the required shape while considering an autonomous population in a simple PDE framework, and shows the link between the acceleration and the density gradient is crucial for the appearance of clusters.
• Business
• 1998
Several hypotheses for the swarming behaviour in locusts are explored, with a goal of understanding how swarm cohesion can be maintained by the huge population of insects over long distances and long periods of time.