Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model

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

“…Therefore, the synchronization condition ( 26) is more likely to be fulfilled in small domains, connected with strong boundary couplings. It is worth noting that the synchronization of complex networks with point-wise couplings of the form ( 7)-( 9) is not influenced by the size of the domains, as proved in [8] for instance. Roughly speaking, boundary couplings of the form (3) are able to synchronize the local dynamics in a neighborhood of the boundaries; if the domains are small, this boundary synchronization can extend to the whole domain.…”

“…Indeed, various forms of synchronization, such as identical synchronization, have been studied in [3], [27], [35] or [37] for complex networks determined by reaction-diffusion systems. 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.…”

“…Notably, the modeling of the couplings is often reductive. For instance, displacements of individuals are roughly modeled by point-wise couplings in [8] and [9], whose adequateness with biological observations can legitimately be criticized (see Figure 1(a)). Furthermore, the local dynamics of the complex networks studied in these papers are often described by identical systems defined in identical domains; the equations of these complex networks can be written in the following form:…”

“…Compared with the model determined by ( 1)-( 2)-( 3)-( 4), we observe that the couplings are here integrated into the reaction-diffusion equation (7), whereas they are defined by the Robin boundary condition (3) in our new model. Furthermore, the boundary condition (8) of Neumann type is seen as incompatible with natural displacements of biological individuals through the boundary of their habitat. Thus it appears highly relevant to study the dynamics of complex networks for which a refined modeling of the couplings is expected.…”

“…. , Ω n underlying the complex network model ( 1)-( 2)-( 3)-( 4) are non-identical, we are led to transport the solutions in a common arbitrary domain, hence the following definition differs from the identical synchronization studied for instance in [8].…”