Pattern formation in a predator–prey system with a finite interaction range in a channel-like region using the Fick–Jacobs diffusion approach

In this work, we present a diffusive predator–prey model with a finite interaction scale between species and an external flow. The system is confined to a two-dimensional domain with one coordinate larger than another, which allows us to use the one-dimensional projection of the diffusion operator, known as the Fick–Jacobs projection, here with an external force. Within this approach, we obtain analytical results for an exponential-shaped channel showing that patterns can emerge through the diffusion-driven instability mechanism. We show that the range of unstable modes where patterns can appear is modified by the species interaction’s spatial scale and an effective advection term that includes external velocity and the shape parameter that characterizes the channel-like region.