On a generalized diffusion problem: A complex network approach

Abstract

In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution via operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.

Synchronization of Turing patterns in complex networks of reaction–diffusion systems set in distinct domains

“…The dynamics of such competing species models have also been studied in non-convex domains admitting a 'dumbbell' shape [12,24], which resembles a simple two-nodes network. However, to the best of our knowledge, the dynamics of the nonlinear predator-prey model (11) have never been studied in a complex network with boundary couplings of the form (3). In the numerical part of our paper, we focus on the synchronization of heterogeneous Turing patterns, which have been proved to appear in the Leslie-Gower model, in a diffusion-driven instability process.…”

“…The stability of persistence or extinction equilibria in meta-population models have been studied in [9] for a panic model, in [8] for a competing species system or in [34] for an epidemiological model. In [11], it has been proved that the spatial diffusion of individuals in such meta-population models acts as a combination of short and long range diffusion. In parallel, the dynamics of chemical reactions networks have been studied in [14], [15] via the entropy framework; synchronization of unstable patterns in other chemical reactions networks has been investigated in [25].…”

“…The dynamics of such competing species models have also been studied in non-convex domains admitting a "dumbbell" shape [24,12], which resembles a simple twonodes network. However, to the best of our knowledge, the dynamics of the nonlinear predator-prey model (11) have never been studied in a complex network with boundary couplings of the form (3).…”