Two populations compete for spatial territory. One starts closer to the resource. How much larger must the farther population be to compensate for the spatial disadvantage? Guha, Ryan, and Karamched show the answer is exponential: a linear increase in distance requires an exponentially larger population to have equal probability of arriving first.
The mechanism comes from extreme value statistics. Discovery is a first-passage-time problem — the first individual from either population to reach the resource claims it. When many individuals search independently, the discovery time is determined by the fastest searcher, not the average one. The minimum of many random arrival times concentrates around its expected value exponentially fast as population size grows. A spatial disadvantage shifts the entire distribution of arrival times, and overcoming that shift by increasing population size requires exponential compensation because you're fighting the tail behavior of the minimum.
But initial discovery is not dominance. Even after one population claims the resource first, local stochastic fluctuations can reverse the advantage. Temporary superiority — arriving first — decays unless reinforced. The second finding is that sustained dominance requires a non-reciprocal interaction bias: the winning population must suppress the losing one asymmetrically, not just outcompete it symmetrically. Without this asymmetry, the absorbing state (permanent dominance) is never reached; the system fluctuates indefinitely.
The two results together form a complete picture: spatial advantage determines who arrives first (via extreme value statistics), but only structural asymmetry in the interaction determines who stays (via the absorbing state condition). Discovery and maintenance are governed by different mathematics.