A certain class of a host-parasitoid models, where some host are completely free from parasitism within a spatial refuge is studied. In this paper, we assume that a constant portion of host population may find a refuge and be safe from attack by parasitoids. We investigate the effect of the presence of refuge on the local stability and bifurcation of models. We give the reduction to the normal form and computation of the coefficients of the Neimark-Sacker bifurcation and the asymptotic approximation of the invariant curve. Then we apply theory to the three well-known host-parasitoid models, but now with refuge effect. In one of these models Chenciner bifurcation occurs. By using package Mathematica, we plot bifurcation diagrams, trajectories and the regions of stability and instability for each of these models.

中文翻译：

---

具有空间避难所效果的某类宿主-拟寄生物模型的稳定性。

研究了某类宿主-拟寄生虫模型，其中某些宿主在空间避难所内完全没有寄生虫。在本文中，我们假设一定数量的寄主种群可以找到庇护所，并且可以免受寄生虫的攻击。我们调查了庇护所的存在对模型的局部稳定性和分叉的影响。我们对正态形式进行归约，计算Neimark-Sacker分叉系数和不变曲线的渐近逼近。然后，我们将理论应用于三个众所周知的宿主-寄生体模型，但是现在有了庇护效果。在这些模型之一中，发生了Chenciner分叉。通过使用Mathematica软件包，我们为每个模型绘制了分叉图，轨迹以及稳定性和不稳定性区域。