Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
Institute of Biomathematics and Biostatistics, Helmholtz Center Munich, Ingolstädter, Landstrasse 1, 85764, Neuherberg, Germany.
Bull Math Biol. 2019 Oct;81(10):3889-3917. doi: 10.1007/s11538-019-00659-0. Epub 2019 Aug 23.
Most studies of ecological interactions study asymptotic behavior, such as steady states and limit cycles. The transient behavior, i.e., qualitative aspects of solutions as and before they approach their asymptotic state, may differ significantly from asymptotic behavior. Understanding transient dynamics is crucial to predicting ecosystem responses to perturbations on short timescales. Several quantities have been proposed to measure transient dynamics in systems of ordinary differential equations. Here, we generalize these measures to reaction-diffusion systems in a rigorous way and prove various relations between the non-spatial and spatial effects, as well as an upper bound for transients. This extension of existing theory is crucial for studying how spatially heterogeneous perturbations and the movement of biological species involved affect transient behaviors. We illustrate several such effects with numerical simulations.
大多数关于生态相互作用的研究都研究渐近行为，如稳定状态和极限环。瞬态行为，即解在趋近渐近状态之前和期间的定性方面，可能与渐近行为有很大的不同。了解瞬态动力学对于预测生态系统对短期时间尺度上的扰动的响应至关重要。已经提出了几种衡量常微分方程系统中瞬态动力学的方法。在这里，我们以严格的方式将这些方法推广到反应扩散系统，并证明了非空间和空间效应之间的各种关系，以及瞬态的上限。现有理论的这种扩展对于研究空间异质扰动和所涉及的生物物种的运动如何影响瞬态行为至关重要。我们通过数值模拟说明了几种这样的效应。
