Two populations compete for the same space. One occupies a slightly better position — closer to resources, more defensible, marginally warmer. The other has more individuals. The intuition from well-mixed competition is that numbers compensate for disadvantage: enough individuals can overwhelm a positional edge.
The compensation is exponential, not linear.
Guo and Rappel (arXiv:2603.10911, March 2026) analyze competing populations in a spatial domain with asymmetric conditions. A population with a linear spatial disadvantage — say, a growth rate that decreases linearly with distance from the boundary — requires an exponentially larger initial population to maintain dominance. A 10% positional handicap isn't compensated by 10% more individuals. It requires orders-of-magnitude more.
Worse: even transient numerical superiority is insufficient. Having more individuals at a given moment does not stabilize dominance. The spatial disadvantage reasserts itself over time because growth rates are position-dependent and the advantaged population recovers from any perturbation at a rate proportional to its positional advantage. Stable dominance from the weaker position requires a sustained, non-reciprocal interaction bias — something beyond just having more bodies.
The mechanism is spatial amplification. In well-mixed systems, each individual competes with every other individual equally. In spatial systems, local interactions dominate, and local interactions are governed by local conditions. The spatial disadvantage compounds across the domain: each local interaction is slightly unfavorable, and the compounding across many local interactions produces an exponential gap at the population level.
The structural lesson: spatial structure converts linear parameters into exponential outcomes. Position isn't just an advantage — it's a multiplier that acts on every local interaction simultaneously. The leap from linear cause to exponential effect means that small spatial asymmetries are far more consequential than they appear when measured locally.