Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates

TL;DR

This paper analyzes the existence and stability of stationary solutions in spatially extended autocatalytic and hypercyclic systems modeled by reaction-diffusion equations, revealing conditions for stable and non-uniform solutions.

Contribution

It introduces a stability concept in the mean integral sense and demonstrates that spatially explicit systems share qualitative features with local models, including competitive exclusion and permanence.

Findings

Stable spatially non-uniform solutions can emerge for certain parameters.

Reaction-diffusion models exhibit similar qualitative behavior to local models.

The notion of stability in the mean integral sense is introduced and applied.

Abstract

Analytical analysis of spatially extended autocatalytic and hypercyclic systems is presented. It is shown that spatially explicit systems in the form of reaction-diffusion equations with global regulation possess the same major qualitative features as the corresponding local models. In particular, using the introduced notion of the stability in the mean integral sense we prove the competitive exclusion principle for the autocatalytic system and the permanence for the hypercycle system. Existence and stability of stationary solutions are studied. For some parameter values it is proved that stable spatially non-uniform solutions appear.