Department of Mathematics, University of Vienna, Vienna, Austria.
Department of Evolution and Ecology, University of California, Davis, USA.
J Math Biol. 2022 Oct 18;85(5):54. doi: 10.1007/s00285-022-01815-2.
To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When do the signs of the invasion growth rates determine coexistence? (ii) When signs are sufficient, which invasion growth rates need to be positive? We focus on deterministic models and equate coexistence with permanence, i.e., a global attractor bounded away from extinction. For models satisfying certain technical assumptions, we introduce invasion graphs where vertices correspond to proper subsets of species (communities) supporting an ergodic measure and directed edges correspond to potential transitions between communities due to invasions by missing species. These directed edges are determined by the signs of invasion growth rates. When the invasion graph is acyclic (i.e. there is no sequence of invasions starting and ending at the same community), we show that permanence is determined by the signs of the invasion growth rates. In this case, permanence is characterized by the invasibility of all [Formula: see text] communities, i.e., communities without species i where all other missing species have negative invasion growth rates. To illustrate the applicability of the results, we show that dissipative Lotka-Volterra models generically satisfy our technical assumptions and computing their invasion graphs reduces to solving systems of linear equations. We also apply our results to models of competing species with pulsed resources or sharing a predator that exhibits switching behavior. Open problems for both deterministic and stochastic models are discussed. Our results highlight the importance of using concepts about community assembly to study coexistence.
为了理解物种共存的机制，生态学家经常研究理论和数据驱动模型的入侵增长率。这些增长率对应于一个遍历测度支持其他物种的情况下，一种物种的平均个体增长率。在生态学文献中，共存通常等同于入侵增长率为正。直观地说，正的入侵增长率确保了物种从稀有状态中恢复。为了为这种方法提供一个严格的数学框架，我们证明了两个问题的定理：（i）入侵增长率的符号何时决定共存？（ii）当符号充分时，哪些入侵增长率需要为正？我们专注于确定性模型，并将共存等同于持久性，即一个远离灭绝的有界全局吸引子。对于满足某些技术假设的模型，我们引入了入侵图，其中顶点对应于支持遍历测度的物种（群落）的适当子集，有向边对应于由于缺失物种的入侵而导致群落之间的潜在转变。这些有向边由入侵增长率的符号决定。当入侵图无环时（即没有从同一群落开始和结束的入侵序列），我们表明持久性由入侵增长率的符号决定。在这种情况下，持久性的特征是所有 [Formula: see text] 群落的可入侵性，即没有物种 i 的群落，其中所有其他缺失物种的入侵增长率都为负。为了说明结果的适用性，我们表明耗散的 Lotka-Volterra 模型通常满足我们的技术假设，并且计算它们的入侵图可以简化为求解线性方程组。我们还将我们的结果应用于具有脉冲资源或共享具有开关行为的捕食者的竞争物种模型。讨论了确定性和随机模型的未解决问题。我们的结果强调了使用群落组装的概念来研究共存的重要性。
