Canine shelter capacity is a prevalent topic, as overcrowded kennels appear to hinder physical activity and pose a potential detriment to overall canine health. In order to address this overcrowding issue, we formulate mathematical models to describe the dynamics of the dog population in a shelter across four size categories, incorporating outtake regulatory efforts through two prominent types of outtakes: adoption and partner transfers. First, we present a baseline model in which outtake regulatory efforts are fixed. We then develop an extended model with time dependent regulatory efforts, with the goal of optimizing shelter outtakes while minimizing the associated regulatory costs. The dataset used in this study was obtained from Lubbock Animal Services and consists of dog intake and outtake records spanning a period of 893 days, from July 21, 2021, to December 31, 2023. The results demonstrate that medium sized dogs constitute the largest proportion of the shelter population. Furthermore, the model provides a framework for evaluating the effectiveness of adoption and partner transfer efforts in mitigating shelter overcrowding.