  A class of occupancy models for detection/non-detection data is proposed to
relax the closure assumption of N$-$mixture models. We introduce a community
parameter $c$, ranging from $0$ to $1$, which characterizes a certain portion
of individuals being fixed across multiple visits. As a result, when $c$ equals
$1$, the model reduces to the N$-$mixture model; this reduced model is shown to
overestimate abundance when the closure assumption is not fully satisfied.
Additionally, by including a zero-inflated component, the proposed model can
bridge the standard occupancy model ($c=0$) and the zero-inflated N$-$mixture
model ($c=1$). We then study the behavior of the estimators for the two extreme
models as $c$ varies from $0$ to $1$. An interesting finding is that the
zero-inflated N$-$mixture model can consistently estimate the zero-inflated
probability (occupancy) as $c$ approaches $0$, but the bias can be positive,
negative, or unbiased when $c>0$ depending on other parameters. We also
demonstrate these results through simulation studies and data analysis.
