The Relevance of Cross-diffusion in the Formation of Turing Patterns

Enrique Peacock-Lopez, Williams College, Williamstown, MA, USA

**Abstract: **Over the years, the Rosenzweig-MacArthur (RM) model has been used to study simple prey-predator systems. It has been observed, however, that the RM model cannot sustain Turing patterns when using a diagonal diffusion tensor. As a result, researchers have introduced changes to the RM model that induce stable Turing patterns. In most cases, the changes have been made to the so-called response function, changing the interspecies interaction, or by adding an intraspecies interaction to the model. In this communication, we study the original RM model but we include cross-diffusion, which considers off diagonal elements in the diffusion tensor. Although cross-diffusion is well characterized in multicomponent solutions, including electrolytes, it has an apparent counterintuitive meaning in predator-prey systems. We observe, however, that in plant and fish systems, the lack of predator mobility is compensated by their ability to camouflage and attract their prey, which yields a negative cross-diffusion coefficient. We show that negative cross-diffusion is enough to trigger stable Turing patterns in the RM model.

*Keywords: *ecology, Turing pattern, prey-predator, spatial diffusion