# 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} }

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

- PhysicsSIAM 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.

- PhysicsPhysical 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.

- PhysicsPhysical 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

- Economics
- 1984

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…

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

- Computer ScienceThe 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.

### Pattern formation in interacting and diffusing systems in population biology.

- MathematicsAdvances in biophysics
- 1982

### Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts

- 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.