On the asymptotic behavior of the solutions to the replicator equation

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.Comment: 23 pages, 1 figure, several small changes are added, together with the new titl

The Processing of Ambiguous Degree Constructions in German

Based on a thorough semantic analysis of German degree quantifiers within the degree approach (cf. e.g. von Stechow, 1984; Heim, 2001), we derived hypotheses on the processing of sentences of the form “A kennt(VERB.3SG) einen besseren(ADJ.COMP) B als C” (‘A knows a better B than C’). These sentences are ambiguous between the DP-internal reading, INT (A knows a B who is a better B than C) and the DP-external reading, EXT (A knows a B who is a better B than a B that C knows). We show that INT is less complex than EXT in several respects. From this we hypothesize that the difference in complexity affects processing such that (i) INT is preferred over EXT in reading the ambiguous sentence (Study 1), that the sentence, if disambiguated by context towards INT rather than EXT, (ii) is judged more acceptable (Study 2) and (iii) that INT takes less long to read than EXT (Study 3). The first hypothesis was confirmed, but not the second and the third one