# 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…
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## References

SHOWING 1-10 OF 61 REFERENCES

### Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups

• 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.

### A non-local model for a swarm

• 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

### Aggregation of finite-size particles with variable mobility.

• 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.

### Asymptotic Behavior for a Nonlinear Degenerate Diffusion Equation in Population Dynamics

• 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

### Self-organization in systems of self-propelled particles.

• 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.

### Stationary solutions of a spatially aggregating population model

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

### Mutual interactions, potentials, and individual distance in a social aggregation

• Psychology
Journal of mathematical biology
• 2003
It is shown quantitatively how repulsion must dominate attraction to avoid collapse of the group to a tight cluster and the existence of a well-spaced locally stable state, having a characteristic individual distance.

### Clustering due to Acceleration in the Response to Population Gradient: A Simple Self‐Organization Model

• 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.