Elasticity analysis is a key tool in the analysis of matrix population models, which describe the dynamics of stage-structured populations in ecology and evolution. Elasticities of the dominant eigenvalue of a matrix model to matrix entries obey certain symmetries. Yet not all consequences of these symmetries are fully appreciated, as they are sometimes hidden in mathematical detail. Here, we propose a method to reason about these symmetries directly by visual inspection of the life cycle graph that corresponds to the matrix model. We present two applications of this method, one in ecology and one in evolution. First, we prove several conjectures about elasticities that were obtained from purely numerical results and that can support population managers in decision-making under scarce demographic information. Second, we show how to identify candidates for invariant trade-offs in evolutionary optimal life cycles. The method extends to the elasticity analysis of non-dominant eigenvalues, of the stochastic growth rate and, in next-generation matrices, of the basic reproduction number.

弹性分析是分析矩阵种群模型的关键工具，该模型描述了阶段结构种群在生态和进化中的动态。矩阵模型的主要特征值对矩阵项的弹性服从某些对称性。但是，由于有时会隐藏在数学细节中，因此并非完全理解这些对称性的所有后果。在这里，我们提出了一种通过直观检查与矩阵模型相对应的生命周期图来直接推理出这些对称性的方法。我们介绍了此方法的两种应用，一种在生态学中，一种在进化中。首先，我们证明了一些纯粹从数值结果获得的关于弹性的猜想，这些猜想可以支持人口管理者在人口统计信息匮乏的情况下进行决策。第二，我们展示了如何在进化的最佳生命周期中确定不变权衡的候选者。该方法扩展到非主要特征值，随机增长率以及在下一代矩阵中的基本繁殖数的弹性分析。