  The geometric approach to nonequilibrium thermodynamics is promising for
understanding the role of dissipation and thermodynamic trade-off relations.
This paper proposes a geometric framework for studying the nonequilibrium
thermodynamics of reaction-diffusion systems. Based on this framework, we
obtain several decompositions of the entropy production rate with respect to
conservativeness, spatial structure, and wavenumber. We also generalize optimal
transport theory to reaction-diffusion systems and derive several trade-off
relations, including thermodynamic speed limits and thermodynamic uncertainty
relations. The thermodynamic trade-off relations obtained in this paper shed
light on thermodynamic dissipation in pattern formation. We numerically
demonstrate our results using the
Fisher$\unicode{x2013}$Kolmogorov$\unicode{x2013}$Petrovsky$\unicode{x2013}$Piskunov
equation and the Brusselator model and discuss how the spatial pattern affects
unavoidable dissipation.
