多个种群的灭绝与一组消亡李雅普诺夫函数
Extinction of multiple populations and a team of Die-out Lyapunov functions
AI总结：
本研究通过构造“消亡”李雅普诺夫函数组，判定微分方程模型中多个物种必然指数灭绝的条件，并给出广义Lotka-Volterra系统的营养级条件以确保解有界。
摘要：
物种灭绝是一个备受关注且文献众多的重要问题。我们的研究深入探讨了物种灭绝必然发生的条件，重点在于确定哪些物种可能灭绝以及它们灭绝的速度。我们研究了一个种群相互作用的微分方程模型，旨在确定多个物种（即有界解的坐标）何时必须灭绝或“走向灭绝”，并且必须以指数速度灭绝。通常，每个坐标代表一个不同物种的种群密度。作为我们的主要工具，我们构造了所谓的“消亡”李雅普诺夫函数。一个给定的系统可能拥有多个或许多这样的函数，每个函数都是不同坐标集合的函数。该消亡函数意味着其子集中的某个物种必须以指数速度灭绝——对于系统系数的几乎所有选择均如此。我们创建了一组协同工作的消亡函数，以证明k个物种必须灭绝，其中k是单独确定的。其次，我们为广义Lotka-Volterra系统提出了一个“营养级”条件，该条件保证存在一个全局吸引的捕获区域。这意味着所有解都是有界的。
英文摘要：
The extinction of species is a major problem of concern with a large literature. Our investigation gives insight into when species extinctions must occur, with an emphasis on determining which species might possibly die out and on how fast they die out. We investigate a differential equations model for population interactions with the goal of determining when several species (\ie, coordinates of a bounded solution) must die out or ``go extinct'' and must do so exponentially fast. Typically each coordinate represents the population density of a different species. For our main tool, we create what we call ``die-out'' Lyapunov functions. A given system may have several or many such functions, each of which is a function of a different set of coordinates. That die-out function implies that one of the species in its subset must die out exponentially fast -- for almost every choice of coefficients of the system. We create a ``team'' of die-out functions that work together to show that $k$ species must die, where $k$ is determined separately. Secondly, we present a ``trophic'' condition for generalized Lotka-Volterra systems that guarantees that there is a trapping region that is globally attracting. That implies that all solutions are bounded.
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