  Using an $sp^{3}s^{*}$ tight-binding model we demonstrate how the observed
strong bowing of the band gap and spin-orbit-splitting with increasing Bi
composition in the dilute bismide alloy GaBi$_{x}$As$_{1-x}$ can be described
in terms of a band-anticrossing interaction between the extended states of the
GaAs valence band edge and highly localised Bi-related resonant states lying
below the GaAs valence band edge. We derive a 12-band
$\textbf{k}\cdot\textbf{p}$ Hamiltonian to describe the band structure of
GaBi$_{x}$As$_{1-x}$ and show that this model is in excellent agreement with
full tight-binding calculations of the band structure in the vicinity of the
band edges, as well as with experimental measurements of the band gap and
spin-orbit-splitting across a large composition range. Based on a tight-binding
model of GaBi$_{x}$N$_{y}$As$_{1-x-y}$ we show that to a good approximation N
and Bi act independently of one another in disordered
GaBi$_{x}$N$_{y}$As$_{1-x-y}$ alloys, indicating that a simple description of
the band structure is possible. We present a 14-band
$\textbf{k}\cdot\textbf{p}$ Hamiltonian for ordered
GaBi$_{x}$N$_{y}$As$_{1-x-y}$ crystals which reproduces accurately the
essential features of full tight-binding calculations of the band structure in
the vicinity of the band edges. The $\textbf{k}\cdot\textbf{p}$ models we
present here are therefore ideally suited to the simulation of the
optoelectronic properties of these novel III-V semiconductor alloys.
