Competing Species — Coexistence vs Competitive Exclusion

Gause's Competitive Exclusion Principle states that two species competing for the same resource cannot coexist indefinitely — one will outcompete the other. The Lotka-Volterra competition model: dN₁/dt = r₁N₁(1 − N₁/K₁ − α₁₂N₂/K₁), dN₂/dt = r₂N₂(1 − N₂/K₂ − α₂₁N₁/K₂). Coexistence requires inter-specific competition to be weaker than intra-specific: α₁₂α₂₁ < 1. The phase portrait shows zero-isoclines (nullclines): when they cross at a stable interior point, coexistence is possible. When they don't cross (or cross unstably), one species wins. Drag the initial condition dot in the phase portrait to explore trajectories from different starting points.