Discrete-time models are the traditional approach for capturing population dynamics of insects living in the temperate regions of the world. We revisit classical discrete-time models of host–parasitoid population dynamics and provide novel results on the stability of the population dynamics. Discrete-time host–parasitoid models are characterized by update functions that connect the population densities from one year to the next, and a host escape response — the fraction of hosts escaping parasitism each year. For a general class of models we show that the stability can be simply characterized in terms of two quantities: the rate at which the host equilibrium changes with the host’s growth rate, and the sensitivity of the host’s escape response to the host density. Interestingly, stability is more likely to arise when the escape response is a decreasing function of the host density rather than an increasing function. Moreover, if the host’s escape response only depends on the parasitoid population density then the stability condition is further simplified to the host equilibrium density being an increasing function of the host’s reproduction rate. We interpret several mechanisms known for stabilizing host–parasitoid population dynamics in the context of these generalized stability conditions. Next, we introduce a hybrid approach for obtaining the update functions by solving ordinary differential equations that mechanistically capture the ecological interactions between the host and the parasitoid. This hybrid approach is used to study the suppression of host density by a parasitoid. Our analysis shows that when the parasitoid attacks the host at a constant rate, then the host density cannot be suppressed beyond a certain point without making the population dynamics unstable. In contrast, when the parasitoid’s attack rate increases with increasing host density (Type III functional response), then the host population density can be suppressed to arbitrarily low levels while maintaining system stability. These results have important implications for biological control where parasitoids are introduced to eliminate a pest that is the host species for the parasitoid.

离散时间模型是捕捉生活在世界温带地区昆虫种群动态的传统方法。我们重新审视了宿主 - 寄生蜂种群动态的经典离散时间模型，并提供了关于种群动态稳定性的新结果。离散时间宿主 - 寄生蜂模型的特点是更新函数将一年到下一年的种群密度连接起来，以及宿主逃逸反应——每年逃避寄生的宿主的比例。对于一般类型的模型，我们表明稳定性可以简单地用两个量来表征：宿主平衡随宿主生长速率变化的速率，以及宿主逃逸响应对宿主密度的敏感性。有趣的是，当逃逸响应是宿主密度的递减函数而不是递增函数时，稳定性更有可能出现。此外，如果宿主的逃逸反应仅取决于寄生蜂种群密度，则稳定性条件进一步简化为宿主平衡密度是宿主繁殖率的增函数。我们在这些广义稳定性条件的背景下解释了几种已知的稳定宿主 - 寄生蜂种群动态的机制。接下来，我们介绍了一种通过求解常微分方程来获取更新函数的混合方法，该方程机械地捕获宿主和寄生蜂之间的生态相互作用。这种混合方法用于研究寄生蜂对寄主密度的抑制。我们的分析表明，当寄生蜂以恒定速率攻击宿主时，宿主密度不能被抑制超过某一点而不会使种群动态不稳定。相比之下，当寄生蜂的攻击率随着宿主密度的增加而增加时（III 型功能反应），则可以在保持系统稳定性的同时将宿主种群密度抑制到任意低的水平。这些结果对于引入寄生物以消灭作为寄生物宿主物种的害虫的生物控制具有重要意义。当寄生蜂的攻击率随着寄主密度的增加而增加时（III 型功能反应），则可以在保持系统稳定性的同时将寄主种群密度抑制到任意低的水平。这些结果对于引入寄生物以消灭作为寄生物宿主物种的害虫的生物控制具有重要意义。当寄生蜂的攻击率随着寄主密度的增加而增加时（III 型功能反应），则可以在保持系统稳定性的同时将寄主种群密度抑制到任意低的水平。这些结果对于引入寄生物以消灭作为寄生物宿主物种的害虫的生物控制具有重要意义。