# The minimal model of Hahn for the Calvin cycle

## TL;DR

This paper analyzes a minimal mathematical model of the Calvin cycle, demonstrating how photorespiration can stabilize the cycle's operation and providing rigorous proofs of the model's steady states and their stability.

## Contribution

The paper provides the first rigorous analysis of Hahn's minimal Calvin cycle model, including stability of steady states with and without photorespiration.

## Key findings

- In the no-photorespiration model, exactly one unstable positive steady state exists.
- In the photorespiration model, two positive steady states exist, one stable and one unstable.
- Photorespiration can stabilize the Calvin cycle, preventing concentrations from diverging.

## Abstract

There are many models of the Calvin cycle of photosynthesis in the literature. When investigating the dynamics of these models one strategy is to look at the simplest possible models in order to get the most detailed insights. We investigate a minimal model of the Calvin cycle introduced by Hahn while he was pursuing this strategy. In a variant of the model not including photorespiration it is shown that there exists exactly one positive steady state and that this steady state is unstable. For generic initial data either all concentrations tend to infinity at lates times or all concentrations tend to zero at late times. In a variant including photorespiration it is shown that for suitable values of the parameters of the model there exist two positive steady states, one stable and one unstable. For generic initial data either the solution tends to the stable steady state at late times or all concentrations tend to zero at late times. Thus we obtain rigorous proofs of mathematical statements which together confirm the intuitive idea proposed by Hahn that photorespiration can stabilize the operation of the Calvin cycle. In the case that the concentrations tend to infinity we derive formulae for the leading order asymptotics using the Poincar\'e compactification.