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：本文针对检测/非检测数据提出了一类占有模型，以放宽N$-$混合模型的封闭性假设。我们引入一个取值范围为$0$到$1$的社区参数$c$，该参数表征了在多次访问中固定不变的个体比例。当$c$等于$1$时，该模型退化为N$-$混合模型；研究表明，当封闭性假设不完全满足时，该简化模型会高估丰度。此外，通过包含零膨胀成分，所提出的模型能够桥接标准占有模型（$c=0$）和零膨胀N$-$混合模型（$c=1$）。我们进一步研究了当$c$从$0$变化到$1$时，两种极端模型估计量的行为。一个有趣的发现是：当$c$趋近于$0$时，零膨胀N$-$混合模型能够一致地估计零膨胀概率（占有概率）；但当$c>0$时，偏差可能为正、为负或无偏，具体取决于其他参数。我们还通过模拟研究和数据分析验证了这些结果。