关于复制子方程解的渐近性态
On the asymptotic behavior of the solutions to the replicator equation
摘要：
选择系统和相应的复制子方程对复制子的进化进行了高度的抽象。在本文中，我们应用新的选择系统的分析方法的复制子方程。为了适用于所建议的算法的相互作用矩阵的复制子方程应进行转换，特别是标准的奇异值分解，使我们能够改写的复制子方程在一个方便的形式。原来的$n$维问题减少到所谓的护送系统，在某些重要的情况下，可以显着小于原来的系统的尺寸的解决方案的渐近行为的分析。利用牛顿图方法研究了当相互作用矩阵秩为1或2时，护送系统解的渐近性态。分析了一般的相互作用矩阵为秩1的复制子方程，给出了渐近状态为多态平衡点的条件。作为具有秩2的相互作用矩阵的系统的示例，我们考虑来自[Adams，M.R.和Sornborger，A.T.，J Math Biol，54：357-384，2007]，为此，我们证明了，对于系统的任意维数和在一些适当的条件下，一般地，在单形的1-骨架上存在一个全局稳定的平衡点。
英文摘要：
Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm the interaction matrix of the replicator equation should be transformed; in particular the standard singular value decomposition allows us to rewrite the replicator equation in a convenient form. The original $n$-dimensional problem is reduced to the analysis of asymptotic behavior of the solutions to the so-called escort system, which in some important cases can be of significantly smaller dimension than the original system. The Newton diagram methods are applied to study the asymptotic behavior of the solutions to the escort system, when interaction matrix has rank 1 or 2. A general replicator equation with the interaction matrix of rank 1 is fully analyzed; the conditions are provided when the asymptotic state is a polymorphic equilibrium. As an example of the system with the interaction matrix of rank 2 we consider the problem from [Adams, M.R. and Sornborger, A.T., J Math Biol, 54:357-384, 2007], for which we show, for arbitrary dimension of the system and under some suitable conditions, that generically one globally stable equilibrium exits on the 1-skeleton of the simplex.
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