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.

翻译：提出了一种针对检测/非检测数据的占据模型类，以放宽 N$-$混合体系的封闭性假设。引入了一个社群参数 $c$，取值范围为 $0$ 至 $1$，用于表征多次访问中某些个体固定的一定比例。因此，当 $c$ 等于 $1$ 时，模型简化为 N$-$混合体系，该简化模型在封闭性假设未完全满足时会高估种群数量。另外，通过包括零膨胀成分，所提出的模型可以连接标准占据模型（$c=0$）和零膨胀 N$-$混合体系模型（$c=1$）。我们随后研究了两个极端模型的估计器随 $c$ 变化的行为。有意思的发现是，零膨胀 N$-$混合体系模型在 $c$ 接近 $0$ 时可以一致估计到零膨胀概率（占据率），但在 $c>0$ 时，偏差可以是正、负或无偏，具体取决于其他参数。我们还通过模拟研究和数据分析展示了这些结果。