{"ID":5937777,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-08T18:34:06.66078812Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.04456","arxiv_id":"2607.04456","title":"Mathematical Model of Evolution of Non-Degenerate Replicator Systems","abstract":"We propose and analyse a mathematical model of evolutionary adaptation for non-degenerate (permanent) replicator systems, in which the fitness landscape matrix evolves on a slow timescale -- the evolutionary time -- while the species dynamics unfold on a fast timescale. Under a two-timescale separation justified by Tikhonov's theorem, the adaptation problem reduces to maximising the mean fitness at steady state over a convex admissible set of fitness landscape matrices. We derive a fitness variation formula and establish necessary and sufficient conditions for a fitness maximum, showing that the optimisation reduces at each step to a linear programming problem. The algorithm is applied to four canonical replicator systems: the hypercycle, the bi-hypercycle, the anthill system, and the RNA molecule network. In all cases the evolutionary process follows a universal three-phase pattern: an initial phase of fitness growth without equilibrium shift, during which purely altruistic replication gives way to mixed altruistic-selfish behaviour; a second phase of dominant species emergence; and a stabilisation phase analogous to the error catastrophe threshold in quasispecies models. A key consequence is that all evolved systems acquire resistance to parasitic species. We further prove that without non-degeneracy constraints the process leads to sequential species annihilation, with a provable spectral lower bound on fitness increase by dimension reduction.","short_abstract":"We propose and analyse a mathematical model of evolutionary adaptation for non-degenerate (permanent) replicator systems, in which the fitness landscape matrix evolves on a slow timescale -- the evolutionary time -- while the species dynamics unfold on a fast timescale. Under a two-timescale separation justified by Tik...","url_abs":"https://arxiv.org/abs/2607.04456","url_pdf":"https://arxiv.org/pdf/2607.04456v1","authors":"[\"Alexander S. Bratus\",\"Sergey Drozhzhin\",\"Tatiana Yakushkina\"]","published":"2026-07-05T18:46:49Z","proceeding":"q-bio.PE","tasks":"[\"q-bio.PE\",\"math.DS\",\"math.OC\"]","methods":"[]","has_code":false}