= 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.
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