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.

💡 Research Summary

The paper tackles the long‑standing difficulty of analyzing high‑dimensional replicator equations, which model the frequency dynamics of competing strategies or replicators in evolutionary game theory, ecology, and economics. The authors introduce a systematic reduction method based on the singular value decomposition (SVD) of the interaction matrix (A). By writing (A = U\Sigma V^{\top}), where (\Sigma) contains the non‑zero singular values (\sigma_{1},\dots,\sigma_{r}) (with (r=\operatorname{rank}(A))) and (U, V) are orthogonal, the original (n)-dimensional system can be transformed into a lower‑dimensional “escort system” governing the variables (\mathbf y = V^{\top}\mathbf x). The escort system has the compact form

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