A Nonlocal Continuum Model for Biological Aggregation

  title={A Nonlocal Continuum Model for Biological Aggregation},
  author={Chad M. Topaz and A. Bertozzi and Mark A. Lewis},
  journal={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… 

Nonlocal Aggregation Models: A Primer of Swarm Equilibria

This work derives a nonlocal partial differential equation describing the evolving population density of a continuum aggregation and finds exact analytical expressions for the equilibria.


We construct and investigate a new nonlocal kinetic model for the formation and movement of animal groups in two dimensions. The model generalizes to two dimensions, the one-dimensional hyperbolic

Global attractor for a nonlocal model for biological aggregation

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

A simple model for biological aggregation with asymmetric sensing

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.

Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

  • 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.

Asymptotic Dynamics of Attractive-Repulsive Swarms

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.

Analysis of stationary patterns arising from a time-discrete metapopulation model with nonlocal competition

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

A Continuum Three-Zone Model for Swarms

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

Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling.

The comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two- and three-dimensional nonlocal interaction conservation equations.



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

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

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.

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

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.

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

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

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.

Moving and staying together without a leader

Pattern formation in interacting and diffusing systems in population biology.