Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate

Abstract

We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x{|}^{-2\alpha }$, $x\to \infty$. We show that if $\alpha >2$, then the system spreads at a linear speed, {while for $\alpha \in (\frac{1}{2},2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $\alpha >\frac{1}{2}$.

v2 update: A new result is added covering the case $alpha<2$ for the microscopic model. Further remarks and heuristic comments are added, including connections to other models. Many minor changes are made

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Cited by in corpus (3)

• Branching random walk with non-local competition
• The continuous-time frog model can spread arbitrarily fast
• Shape theorem for a one-dimensional growing particle system with a bounded number of occupants per site