A Structure-preserving rational integrator for the replicator dynamics on the probability simplex

• A quadratically convergent rational integrator is developed for replicator dynamics on the probability simplex. • The method unconditionally preserves positivity, mass conservation, invariant faces, and both corner and internal equilibria. • A discrete analogue of the quotient rule governing component ratios evolution is established. • An embedded first-order auxiliary scheme enables adaptive time-stepping via efficient local error estimation. • Extensive numerical experiments confirm second-order accuracy and show superior performance compared to standard solvers in preserving qualitative dynamics. In this work, we introduce a quadratically convergent and dynamically consistent integrator specifically designed for the replicator dynamics. The proposed scheme combines a two-stage rational approximation with a normalization step to ensure confinement to the probability simplex and unconditional preservation of non-negativity, invariant sets and equilibria. A rigorous convergence analysis is provided to establish the scheme’s second-order accuracy, and an embedded auxiliary method is devised for adaptive time-stepping based on local error estimation. Furthermore, a discrete analogue of the quotient rule, which governs the evolution of component ratios, is shown to hold. Numerical experiments validate the theoretical results, illustrating the method’s ability to reproduce complex dynamics and to outperform well-established solvers in particularly challenging scenarios.