Nicol-Harper Alex, Dooley Claire, Packman David, Mueller Markus, Bijak Jakub, Hodgson David, Townley Stuart, Ezard Thomas
1Ocean and Earth Science, National Oceanography Centre, University of Southampton Waterfront Campus, Southampton, UK.
2Environment and Sustainability Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter in Cornwall, Penryn Campus, Cornwall, UK.
Popul Ecol. 2018;60(1):185-196. doi: 10.1007/s10144-018-0620-y. Epub 2018 Jun 5.
In our increasingly unstable and unpredictable world, population dynamics rarely settle uniformly to long-term behaviour. However, projecting period-by-period through the preceding fluctuations is more data-intensive and analytically involved than evaluating at equilibrium. To efficiently model populations and best inform policy, we require pragmatic suggestions as to when it is necessary to incorporate short-term transient dynamics and their effect on eventual projected population size. To estimate this need for matrix population modelling, we adopt a linear algebraic quantity known as non-normality. Matrix non-normality is distinct from normality in the Gaussian sense, and indicates the amplificatory potential of the population projection matrix given a particular population vector. In this paper, we compare and contrast three well-regarded metrics of non-normality, which were calculated for over 1000 age-structured human population projection matrices from 42 European countries in the period 1960 to 2014. Non-normality increased over time, mirroring the indices of transient dynamics that peaked around the millennium. By standardising the matrices to focus on transient dynamics and not changes in the asymptotic growth rate, we show that the damping ratio is an uninformative predictor of whether a population is prone to transient booms or busts in its size. These analyses suggest that population ecology approaches to inferring transient dynamics have too often relied on suboptimal analytical tools focussed on an initial population vector rather than the capacity of the life cycle to amplify or dampen transient fluctuations. Finally, we introduce the engineering technique of pseudospectra analysis to population ecology, which, like matrix non-normality, provides a more complete description of the transient fluctuations than the damping ratio. Pseudospectra analysis could further support non-normality assessment to enable a greater understanding of when we might expect transient phases to impact eventual population dynamics.
在我们这个日益不稳定和不可预测的世界中，人口动态很少能统一稳定为长期行为。然而，通过之前的波动逐期预测比在平衡状态下评估需要更多的数据且分析过程更复杂。为了有效地对人口进行建模并为政策提供最佳依据，我们需要关于何时有必要纳入短期瞬态动态及其对最终预测人口规模影响的务实建议。为了估计矩阵人口建模的这种需求，我们采用一种称为非正态性的线性代数量。矩阵非正态性与高斯意义上的正态性不同，它表示给定特定人口向量时人口投影矩阵的放大潜力。在本文中，我们比较并对比了三种备受认可的非正态性度量指标，这些指标是针对1960年至2014年期间来自42个欧洲国家的1000多个年龄结构人口投影矩阵计算得出的。非正态性随时间增加，这与在千年前后达到峰值的瞬态动态指标相呼应。通过对矩阵进行标准化以关注瞬态动态而非渐近增长率的变化，我们表明阻尼比对于判断一个种群在规模上是否容易出现瞬态繁荣或衰退并无信息价值。这些分析表明，用于推断瞬态动态的种群生态学方法常常依赖于次优的分析工具，这些工具侧重于初始人口向量而非生命周期放大或抑制瞬态波动的能力。最后，我们将伪谱分析这一工程技术引入种群生态学，它与矩阵非正态性一样，比阻尼比能更完整地描述瞬态波动。伪谱分析可以进一步支持非正态性评估，以便更深入地理解何时可能预期瞬态阶段会影响最终的种群动态。
