# Transient Instability and Patterns of Reactivity in Diffusive-Chemotaxis Soil Carbon Dynamics

## TL;DR

This paper explores how patterns in soil carbon dynamics can form due to transient instability, even when the system is not in a Turing unstable regime.

## Contribution

The study reveals that transient growth in spatially uniform states can lead to pattern formation in a subcritical Turing bifurcation.

## Key findings

- Pattern formation can occur outside of Turing unstable regimes due to transient dynamics.
- Bistable regions for pattern formation exist in two dimensions but not in one.
- Reactivity analysis shows how geometry and transient growth interact to create multistable patterns.

## Abstract

We study pattern formation in a chemotaxis model of bacteria and soil carbon dynamics as an example system where transient dynamics can give rise to pattern formation outside of Turing unstable regimes. We use a detailed analysis of the reactivity of the non-spatial and spatial dynamics, stability analyses, and numerical continuation to uncover detailed aspects of this system’s pattern-forming potential. In addition to patterning in Turing unstable parameter regimes, reactivity of the spatial system can itself lead to a range of parameters where a spatially uniform state is asymptotically stable, but exhibits transient growth that can induce pattern formation. We show that this occurs in the bistable region of a subcritical Turing bifurcation. Intriguingly, such bistable regions appear in two spatial dimensions, but not in a one-dimensional domain, suggesting important interplays between geometry, transient growth, and the emergence of multistable patterns. We discuss the implications of our analysis for the bacterial soil organic carbon system, as well as for reaction-transport modeling more generally.