Resilience for stochastic systems interacting via a quasi-degenerate network

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process, which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods. The standard approach to population dynamics relies on deterministic modeling. Beyond the traditional deterministic view, stochastic effects can prove fundamental in shaping the ensuing system dynamics. Furthermore, in many cases of interest, dynamical models are hosted on a network which accounts for the intricate pattern of interlinked interactions. In this paper, we explore the nontrivial interplay between stochastic forcing and network arrangement, in a model of reactive populations. In particular, we show that a resonant amplification of the noisy component can be seeded, when replicating a non-normal reactive scheme on the nodes of a directed, quasidegenerate, network. This observation, that we here substantiate analytically, calls for a revised concept of resilience.