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Pattern formation and nonlocal logistic growth.

Nadav M Shnerb1

  • 1Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 13, 2004
PubMed
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Spontaneous breakdown of translational invariance occurs in a 1D logistic growth model with nonlocal interactions. Domain walls manifest as soliton solutions, confirmed by numerical simulations.

Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Statistical Mechanics

Background:

  • The study investigates logistic growth models, which are fundamental in population dynamics and pattern formation.
  • Nonlocal interactions introduce complexity beyond nearest-neighbor effects, influencing emergent behaviors.

Purpose of the Study:

  • To analyze the spontaneous breakdown of translational invariance in a one-dimensional logistic growth process with nonlocal interactions.
  • To identify bifurcation regimes for both short- and long-range interactions.
  • To characterize domain walls as soliton solutions in specific interaction cases.

Main Methods:

  • Analytical treatment of the logistic growth equation with nonlocal interaction terms.
  • Bifurcation analysis to determine parameter regions for symmetry breaking.

Related Experiment Videos

  • Derivation of reduced dynamics and identification of soliton solutions for nearest-neighbor interactions.
  • Numerical simulations to validate analytical findings.

    • Main Results:

    • Demonstration of spontaneous breakdown of translational invariance within specific parameter ranges.
    • Identification of distinct bifurcation regimes dependent on interaction range (short vs. long).
    • Characterization of domain walls as soliton solutions for nearest-neighbor interactions.
    • Confirmation of analytical predictions through numerical simulations.

    Conclusions:

    • Nonlocal interactions can lead to spontaneous symmetry breaking in simple growth models.
    • The emergent domain walls exhibit soliton-like behavior, providing insights into pattern formation.
    • The interplay between interaction range and system parameters dictates the observed dynamics.