We study the singular limit of a class of reinforced random walks on a lattice for which a complete analysis of the existence and stability of solutions is possible. We show that at a sufficiently high total density, the global minimizer of a lattice 'energy' or Lyapunov functional corresponds to aggregation at one site. At lower values of the density the stable localized solution coexists with a stable spatially-uniform solution. Similar results apply in the continuum limit, where the singular limit leads to a nonlinear diffusion equation. Numerical simulations of the lattice walk show a complicated coarsening process leading to the final aggregation.
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Movement is a fundamental process for almost all biological organisms, ranging from the single cell level to the population level, and two major classes of models are widely used to describe movement. In space-jump processes, movement is via a sequence of position jumps at random time intervals, while in velocity-jump processes movement consists of straight-line motion punctuated by random changes in velocity at random times \[1\]. Space jump processes include the familiar K. J. Painter's research has been supported by SHEFC research developmental Grant 107. D. Horstmann was supported by the Deutschen Forschungsgemeinschaft (DFG) and H. G. Othmer's research has been supported by NIH-GM29123 and NSF-DMS9805494.
- Coarsening process
- Forward-backward parabolic
- Lattice walks