Impact of spatial dimension on spatio-temporal pattern formations

The study of prey–predator interactions has been a cornerstone in understanding ecological dynamics. Traditional models have primarily focused on one or two-dimensional spatial representations. However, many organisms move in three dimensions, significantly influencing species interaction dynamics and pattern formation. In this research, we explore the formation of spatial patterns in a prey–predator system within one, two and three spatial dimensions. Utilizing a system of reaction–diffusion equations, we analyse the conditions under which various spatial patterns emerge, including Turing patterns. Through numerical simulations, we demonstrate how three-dimensional spatial dynamics can lead to complex and rich patterns not observable in one or two-dimensional models. In addition, we explore the influence of initial conditions, parameter variations, and mode sections on pattern diversity. Furthermore, we have obtained gyroid-type patterns in a three-dimensional spatial domain that create a highly interconnected network of spaces analogous to the interconnected habitats in an ecosystem, which does not occur in a lower-dimensional domain. These findings provide insights into the underlying mechanisms driving spatial organization in ecological systems and contribute to the broader understanding of pattern formation in three-dimensional environmental and biological contexts.