This paper studies how a spreading population changes when the distance that offspring travel depends on the size of the main population. Instead of assuming settlers always land a fixed distance away, the authors let the typical dispersal distance grow like L(r) = r^μ, where r is the linear size of the main colony and μ is a number between 0 and 1. They find that coupling dispersal range to population size produces qualitatively different invasion behaviors: steady linear advance, faster power-law growth, very fast exponential-like growth, or even a finite-time blow-up in the idealized model.

The authors work with two related models. The first is an analytic, mean-field model known as the coalescing colony model. In that picture a circular main colony grows at constant radial speed c and emits pointlike secondary colonies at rate λ(r) = λ0 r^θ, where θ controls how emission depends on colony size. Each newcomer is placed a distance L(r)=r^μ from the main boundary, grows at the same speed c, and is absorbed instantly if it contacts the main colony. The authors derive generalized dynamical equations (extensions of the Shigesada–Kawasaki equations) for the main-colony radius r(t) and for the size of absorbed colonies. The second model is a spatial, physical simulation in which absorbed volume is not redistributed and the main colony can develop irregular shapes.

Using the analytic equations the paper maps a phase diagram of growth regimes. A simple inequality involving the emission exponent θ, the dispersal exponent μ and the space dimension d determines the regimes. When θ + d μ ≤ d − 1 the front stays in a linear regime and the main colony advances at speed close to c. For intermediate values of θ + d μ the main colony grows faster than linear as a power of time (a power-law: r(t)∝t^β with β>1 and β set by θ and μ). For even larger values the model predicts exponential-like growth or a finite-time blow-up of the idealized equations. The special case μ = 1 behaves differently: theory predicts a nonzero, macroscopic fraction of the population remains in satellites rather than being absorbed.