Time delay effects on spatiotemporal pattern dynamics in directed networks

Pattern theory concerning networks has attracted substantial attention in recent years, proving its utility across diverse sub-disciplines, notably within the realm of reaction–diffusion (RD) systems. With real-world systems often characterized by directed connections, the traditional exploration of RD systems restricted to undirected networks has seamlessly extended to encompass directed networks. Prior studies have unveiled that in directed scenarios, the inherent asymmetry of the Laplacian matrix gives rise to intricate dispersion relations, thereby instigating the pursuit of further exploration and analysis. In this paper, we delve into time delay within RD systems operating on directed networks, which is not clearly revealed in previous work. Our investigations mainly focus on the emergence of spatially organized patterns resulting from the spatio-temporal dynamical processes. In detail, we linearize the RD system to derive the region of instability wherein patterns exhibit. This instability is time delay induced. Subsequent to this, we conduct extensive numerical simulations to illustrate wave patterns across different directed networks. Significantly, when the network connectivity is sufficiently high and the nodes achieve synchronized oscillations, we find that network properties — including the average degree and degree distribution — exert minimal influence on pattern formation. In contrast, time delay significantly shapes pattern periodicity. Furthermore, we establish a compelling correlation between time delay and pattern periodicity, elegantly captured by an exponential-type function.