Competitive exclusion principle

Two consumers with different break-even requirements cannot both have bounded persistent abundance on a single well-mixed limiting resource when their per-capita growth is linear in that resource. For $\dot{Y}/Y=cX-d$ and $\dot{Z}/Z=eX-f$, differentiation of $\log Y/c-\log Z/e$ gives the nonzero constant $-d/c+f/e$ when their break-even levels differ. This excludes simultaneous bounded persistence without assuming convergence to an equilibrium. Additional resources, environmental variation or different interactions can invalidate the simple one-resource argument.