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A Structure-Preserving Rational Integrator for the Replicator Dynamics on the Probability Simplex

Mario Pezzella

TL;DR

This work develops a high-order, structure-preserving integrator for the replicator dynamics on the probability simplex. By combining a two-stage rational discretization with a normalization step, the method unconditionally preserves positivity, mass, and simplex geometry while exactly reproducing corner and internal equilibria. An embedded auxiliary scheme enables reliable local error estimation and adaptive time-stepping via a PI controller, and a discrete quotient rule is shown to hold up to second order. Numerical experiments demonstrate quadratic convergence, equilibrium preservation, and superior performance in challenging, oscillatory regimes compared to standard solvers. The approach offers a robust tool for long-term simulations of frequency-dependent selection across applications in biology, ecology, and beyond.

Abstract

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.

A Structure-Preserving Rational Integrator for the Replicator Dynamics on the Probability Simplex

TL;DR

This work develops a high-order, structure-preserving integrator for the replicator dynamics on the probability simplex. By combining a two-stage rational discretization with a normalization step, the method unconditionally preserves positivity, mass, and simplex geometry while exactly reproducing corner and internal equilibria. An embedded auxiliary scheme enables reliable local error estimation and adaptive time-stepping via a PI controller, and a discrete quotient rule is shown to hold up to second order. Numerical experiments demonstrate quadratic convergence, equilibrium preservation, and superior performance in challenging, oscillatory regimes compared to standard solvers. The approach offers a robust tool for long-term simulations of frequency-dependent selection across applications in biology, ecology, and beyond.

Abstract

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.

Paper Structure

This paper contains 15 sections, 7 theorems, 51 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The continuous-time model
  3. The normalized rational integrator
  4. Structure-preserving properties
  5. Error Analysis and Convergence
  6. The discrete quotient rule
  7. Adaptive time-stepping via local error estimation
  8. Numerical experiments
  9. Test 1. Convergence order -
  10. Test 2. Equilibria preservation and dynamical consistency -
  11. Test 3. Discrete quotient rule -
  12. Test 4. Adaptive stepsize selection -
  13. Conclusions
  14. Data availability statement
  15. Acknowledgments

Key Result

Theorem 3.1

Let the initial value $\bm{x}^0$ belong to the probability simplex $\Delta^{N-1}$ defined in eq:simplex_def and let $\{\bm{x}^n\}_{n\in \mathbb{N}_0},$$\{\bm{x}^{n+\frac{1}{2}}\}_{n\in \mathbb{N}_0}$ be the sequences generated by the numerical method eq:Compact_Rational_Integrator. Then, for any cho Furthermore, if $\bm{x}^0$ lies on a face of the simplex $\Delta^{N-1}_\mathcal{I},$ as defined in

Figures (9)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 4 more figures

Theorems & Definitions (15)

  • Theorem 3.1: Simplex-confined discrete dynamics
  • proof
  • Theorem 3.2: Corner equilibria preservation
  • proof
  • Theorem 3.3: Internal equilibria preservation
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 5 more