In this note, we present a condition which guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction-diffusion PDE with Neumann boundary conditions in one dimension, using the Jacobian matrix of the reaction term and th
e first Dirichlet eigenvalue of the Laplacian operator on the given spatial domain. We also derive an analog of this PDE result for the synchronization of a network of identical ODE models coupled by diffusion terms.
This note works out an advection-diffusion approximation to the density of a population of E. coli bacteria undergoing chemotaxis in a one-dimensional space. Simulations show the high quality of predictions under a shallow-gradient regime.
