Berezovskaya Faina S, Novozhilov Artem S, Karev Georgy P
Howard University, 6-th Str., Washington, DC 20059, USA.
Math Biosci. 2007 Jul;208(1):270-99. doi: 10.1016/j.mbs.2006.10.006. Epub 2006 Nov 10.
A class of models of biological population and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-called elliptic sector. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. An algorithmic approach to analyze system behavior with parameter changes is presented. The developed methods and algorithm are applied to existing mathematical models of biological systems. In particular, we analyze a model of anticancer treatment with oncolytic viruses, a parasite-host interaction model, and a model of Chagas' disease.
分析了一类在原点具有奇异平衡点的生物种群和群落模型；结果表明，这些模型可能具有确定性灭绝的动态机制，从生物学角度来看这至关重要。这种机制对应于一族到原点的同宿轨道的存在，即所谓的椭圆扇形。给出了原点邻域内可能的拓扑结构以及趋于该点的轨道的渐近性的完整分析。提出了一种分析参数变化时系统行为的算法方法。所开发的方法和算法应用于现有的生物系统数学模型。特别是，我们分析了溶瘤病毒抗癌治疗模型、寄生虫 - 宿主相互作用模型以及恰加斯病模型。
