This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.

本文是受一系列研究论文激励的，这些研究论文考虑了寄生虫在宿主-寄生虫相互作用上的外部输入。我们探索一类具有可变释放和恒定释放寄生物的宿主-寄生物模型。我们假设寄主人口的增长率是恒定的，但是除了寄生虫作用于寄主人口之外，我们不假设任何密度依赖调节。我们将获得的恒定进货结果与比例进货结果进行比较。我们观察到在特定条件下，恒定数量的寄生虫的释放最终会导致宿主种群（小虫）灭绝。在宿主种群灭绝的地方总是存在边界平衡，而寄生蜂种群稳定在恒定的种群水平上。恒定和可变的种群数量可以在独特的内部平衡点降低寄主种群的数量。另一方面，寄生虫种群水平保持不变，不依赖于放养。我们证明了Neimark-Sacker分叉的存在，并计算了封闭不变曲线的近似值。然后，我们考虑一些具有比例和恒定放养量的寄主-拟寄生虫模型，在这些模型中，我们选择已知的寄生虫概率函数。通过使用Mathematica软件包，我们提供了数值模拟来支持我们的研究。我们证明了Neimark-Sacker分叉的存在，并计算了封闭不变曲线的近似值。然后，我们考虑一些具有比例和恒定放养量的寄主-拟寄生虫模型，在这些模型中，我们选择已知的寄生虫概率函数。通过使用Mathematica软件包，我们提供了数值模拟来支持我们的研究。我们证明了Neimark-Sacker分叉的存在，并计算了封闭不变曲线的近似值。然后，我们考虑一些具有比例和恒定放养量的寄主-拟寄生物模型，在这些模型中，我们选择了已知的寄生物概率函数。通过使用Mathematica软件包，我们提供了数值模拟来支持我们的研究。