Competition at the front of expanding populations

Abstract

When competing species grow into new territory, the population is dominated by descendants of successful ancestors at the expansion front. Successful ancestry depends on both the reproductive advantage (fitness), as well as ability and opportunity to colonize new domains. We present a model that integrates both elements by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (KPZ equation). Macroscopic manifestations of these equations are distinct growth morphologies controlled by expansion rates, competitive abilities, or spatial anisotropy. In some cases the ability to expand in space may overcome reproductive advantage in colonizing new territory. When new traits appear with accumulating mutations, we find that variations in fitness in range expansion may be described by the Tracy--Widom distribution.

Figures (8)

• Figure 1: Geometric origin of the nonlinear term in the KPZ equation when growth is always normal to the surface. There is a corresponding drift of a neutral sector boundary proportional to the local slope.
• Figure 2: Consequences of super-diffusive boundary motion: The colors indicate different (neutral) species seeded at the bottom. Random reproduction events cause fluctuations of the sector boundaries. Coalescence of sector boundaries removes the enclosed species. The black lines trace back the ancestry of the individuals at the final front. Coalescence of these lines indicates a common ancestor.
• Figure 3: Possible morphologies of competing fronts for $s(f)=s_0f$: circular arc, composite bulge, and V-shaped dent.
• Figure 4: Phase diagram of invasion morphologies for $s(f)=s_0f$. The (lateral) invasion velocity is set either by circular front geometry for $\alpha>\alpha_c=2Ds_0/v_0$ or by the Fisher velocity $\alpha<\alpha_c$.
• Figure 5: Phase diagram of invasion morphologies with $s(f)=s_0f(f-f_0)$. For sufficiently negative $\alpha$ the mutant cannot invade the native species.