[Paper Review] Replicator equations and space

Artem S. Novozhilov, Vladimir P. Posvyanskii+1 more|arXiv (Cornell University)|Aug 26, 2013

Mathematical and Theoretical Epidemiology and Ecology Models19 references3 citations

TL;DR

This paper introduces a novel reaction-diffusion replicator equation with global regulation of the second kind to model spatially structured macromolecular systems. By incorporating diffusion and a spatially non-uniform regulation mechanism, the model enables coexistence of all macromolecules even when the non-spatial version predicts extinction, demonstrating that low diffusion promotes permanence and spatial heterogeneity stabilizes coexistence.

ABSTRACT

A reaction--diffusion replicator equation is studied. A novel method to apply the principle of global regulation is used to write down the model with explicit spatial structure. Properties of stationary solutions together with their stability are analyzed analytically, and relationships between stability of the rest points of the non-distributed replicator equation and distributed system are shown. A numerical example is given to show that the spatial variable in this particular model promotes the system's permanence.

Motivation & Objective

• To develop a spatially explicit replicator equation that preserves the conservation of total concentration while allowing for spatial structure.
• To analyze the stability and persistence of stationary solutions in the reaction-diffusion system under global regulation of the second kind.
• To investigate how spatial diffusion influences the long-term survival (permanence) of all macromolecular types, especially when the non-spatial model predicts extinction.
• To bridge theoretical replicator dynamics with experimental observations of RNA self-replicating systems where all types persist.

Proposed method

• Formulate a reaction-diffusion system where the replicator dynamics are governed by a modified equation incorporating spatial diffusion and a global regulation mechanism of the second kind.
• Use the principle of global regulation to maintain constant total concentration across space, ensuring consistency with the non-spatial replicator model.
• Analyze stationary solutions analytically by solving the system of partial differential equations under Neumann boundary conditions.
• Apply stability theory to compare the stability of non-spatial and spatially distributed equilibria, identifying conditions under which stability is preserved.
• Derive a sufficient condition for permanence based on the structure of the interaction matrix and diffusion coefficients.
• Conduct numerical simulations using an in vitro RNA self-replication interaction matrix to demonstrate coexistence under low diffusion.

Experimental results

Research questions

• RQ1 How does the global regulation of the second kind affect the stability of stationary solutions in a spatially extended replicator system?
• RQ2 Under what conditions does the spatially distributed system preserve the dynamical behavior of the non-spatial replicator equation?
• RQ3 Can the inclusion of spatial diffusion and non-uniform regulation lead to the permanence of all macromolecular types, even when the non-spatial model predicts extinction?
• RQ4 To what extent do numerical simulations of a biologically inspired RNA system confirm the theoretical predictions of coexistence under low diffusion?

Key findings

• For sufficiently large diffusion coefficients, the spatially distributed system behaves similarly to the non-spatial replicator equation, with three macromolecules persisting and three going extinct.
• When diffusion coefficients are reduced (e.g., d = (0.04, 0.05, 0.04, 0.05, 0.04, 0.05)), the system exhibits persistent coexistence of all six macromolecules, indicating a qualitative shift in behavior.
• Numerical results show that solutions no longer converge to a spatially uniform state under low diffusion, instead displaying sustained oscillations and spatial heterogeneity.
• The system with low diffusion coefficients is permanent, as all average concentrations remain bounded away from zero over time, confirming the analytical condition for permanence.
• The model with global regulation of the second kind successfully reproduces experimental observations of RNA self-replicating systems where all six types survive.
• The results suggest that spatial structure and low diffusion can act as mechanisms promoting coexistence, challenging the mean-field assumption of well-stirred reactors.