This is the second part of our study on the spatially heterogeneous predator–prey model where the interaction is governed by a Crowley–Martin type functional response. In part I, we have proved that when the predator competition is strong (i.e. k is large), the model has at most one positive steady-state solution for any $\mu \in \mathbb {R}$, moreover it is globally asymptotically stable for any $\mu >0$. This part is denoted to study the effect of saturation. Our result shows that the large saturation coefficient (i.e. large m) can not only lead to the uniqueness of positive solutions, but also lead to the multiplicity of positive solutions, moreover the stability of the corresponding positive solutions is also completely obtained. This work can be regarded as a supplement of Ref. [10].

中文翻译：

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异构环境中扩散捕食者-猎物模型正解的唯一性和多重性

这是我们对空间异质捕食者 - 猎物模型研究的第二部分，其中相互作用由 Crowley-Martin 型功能响应控制。在第一部分中，我们证明了当掠食者竞争激烈时（即ķ很大），该模型对于任何一个至多有一个正稳态解$\mu \in \mathbb {R}$, 而且它对于任何一个全局渐近稳定的$\亩>0$. 这部分表示研究饱和度的影响。我们的结果表明，大的饱和系数（即大米) 不仅可以导致正解的唯一性，而且可以导致正解的多重性，而且对应的正解的稳定性也完全得到了。这项工作可以看作是参考文献的补充。[10]。