We extend the $N$ branching Brownian motions model of population invasion to higher-order asexual reproduction. Increasing reproduction order leads to qualitative changes: Invasion fronts generically cease to exist beyond binary reproduction, and in the binary case itself, their speed becomes diffusion independent. Ternary reproduction shows critical behavior, with collapse into a strongly localized “invasion bullet” in the supercritical regime, diffusive spreading in the subcritical regime, and a continuous family of fronts at criticality. These results suggest that the dominance of division and binary reproduction in nature reflects fundamental constraints on invasion dynamics.

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• Chemical waves
• Evolutionary & population dynamics
• Nonequilibrium statistical mechanics
• Random walks