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Individual based and mean-field modeling of direct aggregation.

Martin Burger1, Jan Haškovec2, Marie-Therese Wolfram3

  • 1Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany.

Physica D. Nonlinear Phenomena
|June 14, 2014
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Biological aggregation emerges from individuals reducing movement randomness based on perceived density, not explicit attraction. This novel approach leads to pattern formation in mathematical models.

Keywords:
Degenerate parabolic equationDensity dependent random walkDirect aggregationMean field limit

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Area of Science:

  • Mathematical Biology
  • Statistical Physics
  • Dynamical Systems

Background:

  • Biological aggregation is crucial for species survival and ecosystem function.
  • Existing models often rely on explicit attractive forces between individuals.
  • Understanding emergent aggregation from local interactions is a key challenge.

Purpose of the Study:

  • To introduce and analyze novel models of biological aggregation.
  • To investigate aggregation driven solely by density-dependent stochasticity reduction.
  • To explore the mathematical properties and simulation outcomes of these models.

Main Methods:

  • Development of two models: first-order (position-based) and second-order (velocity-based) density-dependent random walks.
  • Formal derivation of mean-field limits yielding nonlocal degenerate diffusions.
  • Mathematical analysis including existence of weak solutions, steady states, and linear stability analysis.
  • Numerical simulations on both individual-based and continuum levels.

Main Results:

  • Aggregation emerges without explicit attractive forces, solely through reduced individual stochasticity.
  • The models yield nonlocal degenerate diffusion equations.
  • Existence of weak solutions and measure-valued steady states for the first-order model.
  • Identification of conditions for pattern formation through linear stability analysis.
  • Numerical simulations confirm emergent aggregation and pattern formation.

Conclusions:

  • Individual stochasticity reduction in response to local density is a sufficient mechanism for biological aggregation.
  • The derived mathematical framework (nonlocal degenerate diffusions) accurately describes emergent collective behavior.
  • These models offer a new perspective on understanding self-organization in biological systems.