Extinctions as a vestige of instability: The geometry of stability and feasibility

IF 1.9 4区 数学 Q2 BIOLOGY
Stav Marcus, Ari M. Turner, Guy Bunin
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引用次数: 0

Abstract

Species coexistence is a complex, multifaceted problem. At an equilibrium, coexistence requires two conditions: stability under small perturbations; and feasibility, meaning all species abundances are positive. Which of these two conditions is more restrictive has been debated for many years, with many works focusing on statistical arguments for systems with many species. Within the framework of the Lotka-Volterra equations, we examine the geometry of the region of coexistence in the space of interaction strengths, for symmetric competitive interactions and any finite number of species. We consider what happens when starting at a point within the coexistence region, and changing the interaction strengths continuously until one of the two conditions breaks. We find that coexistence generically breaks through the loss of feasibility, as the abundance of one species reaches zero. An exception to this rule - where stability breaks before feasibility - happens only at isolated points, or more generally on a lower dimensional subset of the boundary.
The reason behind this is that as a stability boundary is approached, some of the abundances generally diverge towards minus infinity, and so go extinct at some earlier point, breaking the feasibility condition first. These results define a new sense in which feasibility is a more restrictive condition than stability, and show that these two requirements are closely interrelated. We then show how our results affect the changes in the set of coexisting species when interaction strengths are changed: a system of coexisting species loses a species by its abundance continuously going to zero, and this new fixed point is unique. As parameters are further changed, multiple alternative equilibria may be found. Finally, we discuss the extent to which our results apply to asymmetric interactions.
作为不稳定遗迹的灭绝:稳定性和可行性的几何学
物种共存是一个复杂的、多方面的问题。在平衡状态下,共存需要两个条件:小扰动下的稳定性;可行性,也就是所有物种的丰度都是正的。这两种情况中哪一种更具限制性已经争论了很多年,许多工作都集中在具有许多物种的系统的统计论据上。在Lotka-Volterra方程的框架内,我们研究了相互作用强度空间中共存区域的几何形状,用于对称竞争相互作用和任何有限数量的物种。我们考虑从共存区域内的某一点开始,并不断改变相互作用强度,直到两个条件中的一个中断时发生的情况。我们发现,当一个物种的丰度达到零时,共存通常突破了可行性的丧失。这条规则的一个例外——稳定性在可行性之前就打破了——只发生在孤立的点上,或者更普遍地说,发生在边界的一个较低维度子集上。这背后的原因是,随着稳定边界的接近,一些丰度通常向负无穷发散,因此在某个更早的点上灭绝,首先打破可行性条件。这些结果定义了一种新的意义,即可行性比稳定性更具有限制性,并表明这两个要求是密切相关的。然后,我们展示了当相互作用强度改变时,我们的结果如何影响共存物种集的变化:共存物种系统的丰度不断趋于零,失去一个物种,这个新的不动点是唯一的。随着参数的进一步改变,可能会发现多个备选平衡。最后,我们讨论了我们的结果在多大程度上适用于不对称相互作用。
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来源期刊
CiteScore
4.20
自引率
5.00%
发文量
218
审稿时长
51 days
期刊介绍: The Journal of Theoretical Biology is the leading forum for theoretical perspectives that give insight into biological processes. It covers a very wide range of topics and is of interest to biologists in many areas of research, including: • Brain and Neuroscience • Cancer Growth and Treatment • Cell Biology • Developmental Biology • Ecology • Evolution • Immunology, • Infectious and non-infectious Diseases, • Mathematical, Computational, Biophysical and Statistical Modeling • Microbiology, Molecular Biology, and Biochemistry • Networks and Complex Systems • Physiology • Pharmacodynamics • Animal Behavior and Game Theory Acceptable papers are those that bear significant importance on the biology per se being presented, and not on the mathematical analysis. Papers that include some data or experimental material bearing on theory will be considered, including those that contain comparative study, statistical data analysis, mathematical proof, computer simulations, experiments, field observations, or even philosophical arguments, which are all methods to support or reject theoretical ideas. However, there should be a concerted effort to make papers intelligible to biologists in the chosen field.
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