This paper proposes a systematic approach to analyze the stability of a food chain system with a free delay parameter $\tau$ and a free system parameter $\zeta$. Our objective is to study the stability in the whole $\zeta$-$\tau$ plane. First, we consider the case when $\zeta$ is fixed and analyze the stability along the whole $\tau$ axis. Such a problem, called the complete stability problem w.r.t. delay, can be solved through addressing the corresponding auxiliary characteristic roots. Next, we proceed to consider the case when $\zeta$ is also a free parameter. We will obtain the complete classification for the auxiliary characteristic roots. As a result, we can characterize the stability property (more precisely, we can determine the possible number of stability $\tau$-intervals and analytically calculate them) over the whole $\zeta$-$\tau$ plane. The methodology presented in this paper offers a new way to design $\zeta ,$ with the associated stability $\tau$-intervals explicitly available. Finally, a numerical example is given to illustrate our methodology. It is interesting to see that, for some values of $\zeta$ the system may have many stability $\tau$-intervals, and all such values can be identified by our methodology.

本文提出了一种系统方法来分析具有自由延迟参数的食物链系统的稳定性 $\tau$ 和一个免费的系统参数 $\zeta$. 我们的目标是研究整体的稳定性$\zeta$——$\tau$飞机。首先，我们考虑以下情况$\zeta$ 固定并分析整体稳定性 $\tau$轴。这样的问题称为延迟完全稳定问题，可以通过解决相应的辅助特征根来解决。接下来，我们继续考虑以下情况$\zeta$也是一个自由参数。我们将获得辅助特征根的完整分类。因此，我们可以表征稳定性属性（更准确地说，我们可以确定可能的稳定性数$\tau$-intervals 并分析计算它们）在整个 $\zeta$——$\tau$飞机。本文介绍的方法提供了一种新的设计方法$\zeta ,$ 与相关的稳定性 $\tau$-intervals 明确可用。最后，给出了一个数值例子来说明我们的方法。有趣的是，对于某些值$\zeta$ 系统可能有很多稳定性 $\tau$-intervals，所有这些值都可以通过我们的方法来识别。