In this paper, by using the analytical approach, we investigate the global behavior and bifurcation in a class of host–parasitoid models when a constant number of the hosts are safe from parasitism. We find the conditions for the existence and stability of the equilibria. We detect the existence of the Neimark–Sacker bifurcation under certain conditions. We explicitly derived the approximation of the limit curve depending on the parameters that appear in the model. We show that a locally asymptotically stable equilibrium can never be transformed into unstable by increasing a constant number of hosts that are using a refuge. Specially, we consider the effect of constant host refuge in \((S),( HV ),\) and \(( PP )\) models.The obtained results show that the constant number of hosts in refuge affects the qualitative behavior of these models in comparison to the same models without refuge. The theory is confirmed and illustrated numerically.

中文翻译：

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具有恒定宿主避难所的一类宿主-拟寄生模型中的全局行为和分支

在本文中，通过使用分析方法，当恒定数量的宿主对寄生虫无害时，我们研究了一类宿主-寄生虫模型中的整体行为和分叉。我们找到了平衡存在和稳定的条件。在某些条件下，我们检测到Neimark-Sacker分叉的存在。根据模型中出现的参数，我们明确得出了极限曲线的近似值。我们表明，通过增加使用避难所的恒定数量的宿主，局部渐近稳定的平衡永远不会转化为不稳定的平衡。特别地，我们考虑\（（S），（HV），\）和\（（PP）\）中恒定宿主避难所的影响所得结果表明，与没有避难的相同模型相比，避难所中恒定数量的宿主会影响这些模型的定性行为。该理论得到了证实和数字说明。