Perimeter Problem
Geographic expansion follows identical math to biological territory scaling.
Doubling territory area requires 1.41× longer boundary but intrusions increase 2× - making large territories disproportionately expensive to defend.
Territory scaling follows cruel mathematics. As territory area increases, boundary length increases slower than area but faster than defense capacity. For circular territory (optimal shape for defense): Area = πr², Perimeter = 2πr. Perimeter grows with √Area. Doubling territory area (2× resources) requires 1.41× longer boundary, but defensive capacity doesn't scale (same animal, same time budget). Result: Larger territories are harder to defend per unit area.
Wolf pack territories demonstrate this scaling challenge: A 4-wolf pack defending 50 mi² with 25-mile boundary faces 47 intrusions/year at 2.1 hours/day defense cost. An 8-wolf pack defending 120 mi² with 39-mile boundary (+56%) faces 134 intrusions/year (+185%) at 4.8 hours/day (+129%). The logarithmic scaling problem: Doubling pack size allows 2.3× larger territory, but boundary increases 1.56× and intrusions increase 2.85×. Defense cost per wolf increases 32%.
Business Application of Perimeter Problem
Geographic expansion follows identical math to biological territory scaling. Opening second location doubles territory but increases competitive boundary by only 41%. Opening 10th location increases boundary 216%. Defensive costs (local competition, brand enforcement, supply chain) scale faster than linear. This explains why Walmart's overlap zone defensive costs exploded to $1.2B annually.