A general framework for the generation of long wavelength patterns in multi-cellular (dis- crete) systems is proposed, which extends beyond conventional reaction-diffusion (continu- um) paradigms [2]. The standard partial differential equations of reaction-diffusion frame- work can be considered as a mean-field like ansatz which corresponds, in the biological set- ting, to sending to zero the size (or volume) of each individual cell. By relaxing this approxi- mation and, provided a directionality in the flux is allowed for, we demonstrate here that in- stability leading to spatial pattern formation can always develop if the (discrete) system is large enough, namely, composed of sufficiently many cells, the units of spatial patchiness. The macroscopic patterns that follow the onset of the instability are robust and show oscilla- tory or steady state behavior.
[1] M. Asllani, T. Carletti, D. Fanelli, P. K. Maini, Eur. Phys. J. B 93 135 (2020).
[2] A. M. Turing, Phil. Trans. R. Soc. B 237 37 (1952).