Resilience of Cyclic Ecosystems in the Presence of R- and P-Tipping

制度变迁是指“正常”状态突然出乎意料地转变为非常不同的状态（例如，通常丰富的种群变得稀少，森林让位给草原），这是气候变化下保护生态系统的一个重要问题。从数学上讲，这些状态变化是由临界点引起的：参数的一个小变化会导致动力系统从一个固定的基态切换到另一个。传统上，这些引爆事件被归因于b引爆，即参数穿过一个分岔值，在这个分岔值处，第一基态变得不稳定或消失。然而，最近发现了一种基于速率的倾倒：如果参数变化足够快，并且在没有任何经典分岔的情况下，固定基态之间会发生r -倾倒。甚至在最近，我和我的同事发现了P-tipping，一种介于振荡和静止基态之间的新型tipping：这里是参数变化时刻振荡的相位。这些动态倾倒行为通常发生在预期的b倾倒之前，因此可能对预测生态系统对气候变化的响应至关重要。我的研究项目是对捕食者-猎物和食物网模型中的R-和p -倾斜进行全面调查。由于目前对R-和p -倾倒的研究还很少，这些动态倾倒过程在生态系统生存中的作用在很大程度上是未知的。因此，我和我的学生将对关键生态系统模型中的R-和p -倾斜度进行彻底的数学研究。首先，我们将扩展P-tipping的初始工作，该工作主要集中在两个经典的捕食者-猎物模型上，通过研究更复杂的模型，包括生态系统的其他关键特征，如捕食者行为的季节性变化，其他物种（例如，三物种食物链和食物网模型），以及栖息地斑块之间的运动。在应对气候变化时，决策者需要知道生态系统中何时会发生R-和p -临界点事件。然而，生态模型中结构敏感性的存在使得预测变得极其困难，因为我们不知道什么样的数学函数能准确地代表被建模的物种相互作用和行为。此外，气候变化预计会显著改变气候敏感模式参数的平均值和变率（自相关、幅度）。因此，我的研究计划包括对上述生态系统模型范围内气候敏感参数值的结构敏感性和时间变化进行详细的数学研究。这项工作将为我们预测生态系统何时容易受到R-和p -倾斜的影响，以及这些事件在气候变化下发生的可能性提供重要信息。我恭敬地承认，拟议的研究将对西勒克斯奥肯那根民族传统的、祖传的和未被割让的领土进行。

Regime shifts, where a "normal" state suddenly and unexpectedly shifts to a very different one (e.g., a population that is normally abundant becomes scarce, a forest gives way to grassland), are a significant concern in the conservation of ecosystems under climate change. Mathematically, these regime shifts are caused by tipping points: A small change in a parameter causes a dynamical system to switch from one stationary base state to another. Traditionally, these tipping events are ascribed to B-tipping, i.e., the parameter crosses through a bifurcation value at which the first base state becomes unstable or disappears. Recently, however, a tipping based on rate was discovered: R-tipping between stationary base states occurs if the parameter change is fast enough, and in the absence of any classical bifurcations. Even more recently, my colleagues and I discovered P-tipping, a new type of tipping between oscillatory and stationary base states: Here it is the phase of the oscillation at the moment of parameter change that matters. These dynamic tipping behaviours typically occur before B-tipping is expected, and may thus be critically important in predicting ecosystem response to climate change. My research program is a comprehensive investigation of R- and P-tipping in predator-prey and food web models. As R- and P-tipping are only lightly explored to date, the role of these dynamic tipping processes in the survival of ecosystems is largely unknown. My students and I will therefore carry out a thorough mathematical study of R- and P-tipping in key ecosystem models. We will begin by will extending the initial work on P-tipping, which focussed on two classic predator-prey models, by examining more complex models that include other key features of ecosystems such as seasonal changes in predator behaviours, additional species (e.g., three-species food chain and food web models), and movement between habitat patches. In tackling climate change, decision-makers need to know when R- and P-tipping events will occur in ecosystems. The presence of structural sensitivity in ecological models, however, makes prediction extremely difficult, as it is unknown what mathematical functions accurately represent the species interactions and behaviours being modeled. In addition, climate change is predicted to significantly alter the mean and variability (autocorrelation, amplitude) of climate-sensitive model parameters. My research program thus includes a detailed mathematical study of structural sensitivity and temporal variation of climate-sensitive parameter values across the range of ecosystem models studied above.  This work will provide important information regarding our ability to predict when ecosystems are vulnerable to R- and P-tipping, and how likely these events are under climate change. I respectfully acknowledge that the proposed research will be done on the traditional, ancestral and unceded territory of the Syilx Okanagan Nation.