论文标题
多个人群的灭绝和一组死亡的Lyapunov功能
Extinction of multiple populations and a team of Die-out Lyapunov functions
论文作者
论文摘要
物种的灭绝是大量文献引起关注的主要问题。我们的调查可以深入了解何时必须发生物种灭绝，重点是确定哪种物种可能死亡以及它们死亡的速度。我们研究了人口相互作用的微分方程模型，其目的是确定几个物种何时（\ ie，有限解决方案的坐标）必须消失或``灭绝''，并且必须迅速做到这一点。通常每个坐标代表不同物种的种群密度。对于我们的主要工具，我们创建所谓的``Die-out'''Lyapunov功能。给定的系统可能具有多个或多个这样的功能，每个功能都是不同坐标集的函数。该灭绝功能意味着其子集中的一个物种必须迅速地消失 - 对于系统系数的几乎所有选择。我们创建了一个``团队''的``团队''，该功能共同努力，以表明$ k $物种必须死亡，其中$ k $是单独确定的。其次，我们为广义Lotka-Volterra系统提供了``营养''条件，该条件确保有一个全球吸引人的诱捕区域。这意味着所有解决方案都是有限的。
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.
