  We study the connection of star formation to atomic (HI) and molecular
hydrogen (H$_2$) in isolated, low metallicity dwarf galaxies with
high-resolution ($m_{\rm gas}$ = 4 M$_\odot$, $N_{\rm ngb}$ = 100) SPH
simulations. The model includes self-gravity, non-equilibrium cooling,
shielding from an interstellar radiation field, the chemistry of H$_2$
formation, H$_2$-independent star formation, supernova feedback and metal
enrichment. We find that the H$_2$ mass fraction is sensitive to the adopted
dust-to-gas ratio and the strength of the interstellar radiation field, while
the star formation rate is not. Star formation is regulated by stellar
feedback, keeping the gas out of thermal equilibrium for densities $n <$ 1
cm$^{-3}$. Because of the long chemical timescales, the H$_2$ mass remains out
of chemical equilibrium throughout the simulation. Star formation is
well-correlated with cold ( T $\leqslant$ 100 K ) gas, but this dense and cold
gas - the reservoir for star formation - is dominated by HI, not H$_2$. In
addition, a significant fraction of H$_2$ resides in a diffuse, warm phase,
which is not star-forming. The ISM is dominated by warm gas (100 K $<$ T
$\leqslant 3\times 10^4$ K) both in mass and in volume. The scale height of the
gaseous disc increases with radius while the cold gas is always confined to a
thin layer in the mid-plane. The cold gas fraction is regulated by feedback at
small radii and by the assumed radiation field at large radii. The decreasing
cold gas fractions result in a rapid increase in depletion time (up to 100
Gyrs) for total gas surface densities $\Sigma_{\rm HI+H_2} \lesssim$ 10
M$_\odot$pc$^{-2}$, in agreement with observations of dwarf galaxies in the
Kennicutt-Schmidt plane.
