Pattern formation arising from diffusive instability under random versus directed spatial motion

Classical Turing patterns emerge from reaction–diffusion systems when a spatially homogeneous steady state undergoes a spatial instability due to diffusion. However, it is also possible to study pattern formation under reaction–diffusion–advection systems which allow for some combination of random diffusion and directed motion. We focus on the case of two reaction–diffusion–advection equations where motion is endogenous (such as motion along fecundity or fitness gradients) rather than exogenous (motion due to a flow or prevailing weather conditions), obtaining conditions under which two-species models of this form undergo Turing or wave instabilities leading to pattern formation. The wave instability typically only occurs for three-or-more-species models, so the emergence of this instability and related spatio-temporal patterning in a two-species model is a novel finding. We use this instability theory in conjunction with numerical simulations to explore spatial and spatio-temporal pattern formation in two case studies, a predator–prey model and a chemotaxis model, to demonstrate the effectiveness of the theory. Perhaps the most interesting finding is that moving patterns comprising predator aggregations which pursue prey aggregations are possible in the presence of the wave instability.