In this paper we study the protection zone problem to a predator-prey model subject to Beddington-DeAngelis functional responses and small prey growth rate. This is a successive work to a previous paper of the authors [X. He, S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol. 75 (2017) 239-257], where the model with large prey growth rate was considered. At first we establish the existence and multiplicity of positive steady state solutions, and then give the dynamic behavior of the evolution problem. It is proved that there may be no positive steady state, or may have at leat one, two, or even three positive steady states, depending on the parameters involved such as the growth rate, the predation rate, and the food handling time of the predators, the growth rate and the refuge ability of the preys, and the sizes of the habitat with protection zone. In addition, it is shown that the dynamics of the solutions rely on the initial state as well, e.g., though there could be multiple positive steady states, the prey will go to extinction as time tends to infinity if its initial value is small.

中文翻译：

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具有Beddington-DeAngelis功能性反应和保护区的捕食者-食饵模型的分叉分析和动态行为

在本文中，我们研究了具有Beddington-DeAngelis功能响应和小猎物成长率的捕食者－猎物模型的保护区问题。这是作者先前论文的连续工作[X. 他，SN郑，在具有Beddington-DeAngelis功能响应的扩散捕食者－食饵模型中的保护区，J。Math。生物学 75（2017）239-257]，其中考虑了具有较大猎物增长率的模型。首先我们建立了正稳态解的存在性和多重性，然后给出了演化问题的动力学行为。事实证明，可能不存在正稳态，或者至少有一个，两个甚至三个正稳态，这取决于所涉及的参数，例如生长速度，捕食速度和食物的处理时间。掠食者 猎物的生长速度和避难能力，以及具有保护区的栖息地的大小。此外，还表明解决方案的动力学也依赖于初始状态，例如，尽管可能存在多个正稳态，但是如果初始值较小，则随着时间趋于无穷大，猎物将灭绝。