Analysis of a model of the Calvin cycle with diffusion of ATP

Burcu Gürbüz; Alan D. Rendall

doi:10.3934/dcdsb.2021268

Article Contents

2022, Volume 27, Issue 9: 5161-5177. Doi: 10.3934/dcdsb.2021268

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Analysis of a model of the Calvin cycle with diffusion of ATP

Burcu Gürbüz and
Alan D. Rendall^,

Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

^* Corresponding author: Alan D. Rendall
^* Corresponding author: Alan D. Rendall

Received: July 2021

Revised: September 2021

Early access: November 2021

Published: September 2022

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Abstract

The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time.

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Mathematics Subject Classification: Primary: 92C40; Secondary: 35K57.

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Figure 1. Non-monotone behaviour near $ P_1 $

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Figure 2. Long-time behaviour of the solution in Fig. 1

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Figure 3. Long-time behaviour of a solution approaching the point $ P_2 $

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Figure 4. Plot in the $ x_{\rm GAP} $-$ x_{\rm Ru5P} $ plane of a solution starting near $ P_2 $

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