Dynamics of the reversible Gray-Scott model and convergence to its irreversible limit

TL;DR

This paper establishes well-posedness and analyzes the long-term behavior of a reversible Gray-Scott model, demonstrating convergence to its irreversible limit and identifying stable steady states using stability and center manifold theories.

Contribution

It introduces a reversible variant of the Gray-Scott model, proves its well-posedness, and shows convergence to the classical model under specific parameters, advancing understanding of its dynamics.

Findings

Trajectories converge to homogeneous steady states

Stable steady state is exponentially attractive

Convergence to classical Gray-Scott model is established

Abstract

Well-posedness of a reversible variant of the Gray-Scott model is shown, along with the convergence of each trajectory to one of the two spatially homogeneous steady states. The principle of linearized stability provides the local attractivity at an exponential rate of the stable steady state, while the long-term limit is identified with the help of center manifold theory. Finally, convergence to the classical Gray-Scott model is proved for an appropriate choice of parameters.