Rarity and chaos

This paper gives counterexamples to previously developed analyses which attempted to link rarity (recurrently small densities) with irregular dynamics in one dimensional difference equation models for population growth. The problem lies in a technical definition of rarity, which turns out to be irrelevant. Nevertheless, certain numerical evidence (bifurcation diagrams for parametrized families) suggests that there are easily calculated upper and lower bounds to densities, which are closely approximated in the chaotic regime.