A turing model with correlated random walk

If in the classical Turing model the diffusion process (Brownian motion) is replaced by a more general correlated random walk, then the parameters describing spatial spread are the particle speeds and the rates of change in direction. As in the Turing model, a spatially constant equilibrium can become unstable if the different species have different turning rates and different speeds. Furthermore, a Hopf bifurcation can be found if the reproduction rate of the activator is greater than its rate of change of direction, and oscillating patterns are possible.