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We show how to describe the $T \neq 0$ behavior associated with the usualBCS- Bose Einstein condensation (BEC) crossover ground state. We confine ourattention here to the BEC and near-BEC regime where analytical calculations arepossible. At finite $T$, non-condensed fermion pairs must be included, althoughthey have been generally ignored in the literature. Within this BEC regime wecompute the equations of state for the one and two channel models; these twocases correspond to whether Feshbach resonance effects are omitted or included.Differences between these two cases can be traced to differences between thenature of a Cooper pair and bosonic condensate. Our results are also comparedwith the Gross Pitaevskii equations of state for true bosons. Differences foundhere are associated with the underlying fermionic character of the system.Finally, the particle density distribution functions for a trap containingsuperfluid fermionic atoms are computed using a Thomas-Fermi approach. The oneand two channel behavior is found to be very different; we find a narrowing ofthe density profile as a result of Feshbach resonance effects. Importantly, weinfer that the ratio between bosonic and fermionic scattering lengths dependson the magnetic detuning and is generally smaller than 2. Future experimentswill be required to determine to what extent this ratio varies with magneticfields.Comment: 8 pages, 2 figure, Revtex 4, submitted to PRA; manuscript expanded,  figure adde

Particle density distributions in Fermi gas superfluids: Differences between one- and two-channel models in the Bose-Einstein-condensation limit