Patterns are ubiquitous, ranging from nano-structures, tissue organization, and population dynamics to geophysical phenomena. Scientists often model these by reaction-diffusion equations based on ideal diffusion, e.g., the seminal Turing model. In contrast, natural systems also exhibit physical interactions between their constituents, which might affect patterns.
Spatiotemporal patterns with physical interactions
We study how physical interactions affect cyclic dominant reactions, like the seminal rock-paper-scissors game, which exhibits spiral waves for ideal diffusion. We find that weak physical interactions change the length- and time-scales of spiral waves. In contrast, strong repulsive interactions can generate oscillating lattices (see movie), and strong attraction leads to an interplay of phase separation and chemical oscillations, like droplets co-locating with cores of spiral waves. Our work suggests that physical interactions are relevant for forming spatiotemporal patterns in nature, and it might shed light on how biodiversity is maintained in ecological settings.
Turing patterns with physical interactions
Regular spatial patterns are ubiquitous, yet the underlying principles are often mysterious. Many theoretical descriptions invoke Turing patterns, which combine ideal diffusion with non-linear reactions. However, the cooperativity required for complex reactions should also affect spatial fluxes, which is inconsistent with ideal diffusion. To alleviate this, we investigate how physical interactions affect patterns. We find that even weak interactions change patterns and whether they form. Moreover, strong interactions can lead to a wider range of patterns, since such interactions allow spontaneous accumulation of material by phase separation. Our theory thus suggests physical interactions are a crucial aspect of pattern formation.