论文标题
在大型差异方程式的大型Lotka-Volterra系统中的平衡和幸存的物种
Equilibrium and surviving species in a large Lotka-Volterra system of differential equations
论文作者
论文摘要
Lotka-Volterra（LV）方程在各种生态，生物学和化学系统的数学建模中起关键作用。当物种数量（或取决于观点，化学成分）变成大，基本但基本的问题，例如计算存活物种的数量仍然缺乏理论答案。在本文中，我们考虑了一个大的LV方程系统，其中各种物种之间的相互作用是随机矩阵的实现。我们提供具有独特平衡的条件，并提出启发式方法来计算幸存物种的数量。这种启发式方法结合了随机矩阵理论，数学优化（LCP）和标准极值理论的论点。数值模拟以及一项经验研究，其中相互作用的强度随时间发展，说明了结果的准确性和范围。
Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.
