Spontaneous periodic travelling waves in oscillatory systems with cross-diffusion

TL;DR

This paper discovers a new pattern formation mechanism in active systems with predator-prey interactions, where spontaneous traveling waves emerge from uniform oscillations due to cross-diffusion effects.

Contribution

It introduces a novel type of pattern formation driven by cross-diffusion in oscillatory predator-prey systems, leading to spontaneous traveling waves in large domains.

Findings

Traveling waves form spontaneously without boundary influence.

Stable wavelength range is bounded and distinct from uniform oscillation instability bands.

Pattern formation occurs due to instability of uniform oscillations in the presence of cross-diffusion.

Abstract

We identify a new type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently large domain, spatially uniform oscillations in such systems are unstable with respect to small perturbations. This instability, through a transient regime appearing as spontanous focal sources, leads to establishment of periodic traveling waves. The traveling waves regime is established even if boundary conditions do not favor such solutions. The stable wavelength are within a range bounded both from above and from below, and this range does not coincide with instability bands of the spatially uniform oscillations.