In this thesis we investigate the formation of patterns in a simple activator-inhibitor model supplemented with an inhibitory nonlocal coupling term. This model exhibits a wave instability for slow inhibitor diffusion, while, for fast inhibitor diffusion, a Turing instability is found. For moderate values of the inhibitor diffusion these two instabilities occur simultaneously at a codimension-2 wave-Turing instability. We perform a weakly nonlinear analysis of the model in the neighbourhood of this codimension-2 instability. The resulting amplitude equations consist in a set of coupled Ginzburg-Landau equations. These equations predict that the model exhibits bistability between travelling waves and Turing patterns. We present a study of interfaces separating wave and Turing patterns arising from the codimension-2 instability. We study theoretically and numerically the dynamics of such interfaces in the framework of the amplitude equations and compare these results with numerical simulations of the model near and far away from the codimension-2 instability. Near the instability, the dynamics of interfaces separating small amplitude Turing patterns and travelling waves is well described by the amplitude equations, while, far from the codimension-2 instability, we observe a locking of the interface velocities. This locking mechanism is imposed by the absence of defects near the interfaces and is responsible for the formation of drifting pattern domains, i.e. moving localised patches of travelling waves embedded in a Turing pattern background and vice versa.

In this thesis we investigate the formation of patterns in a simple activator-inhibitor model supplemented with an inhibitory nonlocal coupling term. This model exhibits a wave instability for slow inhibitor diffusion, while, for fast inhibitor diffusion, a Turing instability is found. For moderate values of the inhibitor diffusion these two instabilities occur simultaneously at a codimension-2 wave-Turing instability. We perform a weakly nonlinear analysis of the model in the neighbourhood of this codimension-2 instability. The resulting amplitude equations consist in a set of coupled Ginzburg-Landau equations. These equations predict that the model exhibits bistability between travelling waves and Turing patterns. We present a study of interfaces separating wave and Turing patterns arising from the codimension-2 instability. We study theoretically and numerically the dynamics of such interfaces in the framework of the amplitude equations and compare these results with numerical simulations of the model near and far away from the codimension-2 instability. Near the instability, the dynamics of interfaces separating small amplitude Turing patterns and travelling waves is well described by the amplitude equations, while, far from the codimension-2 instability, we observe a locking of the interface velocities. This locking mechanism is imposed by the absence of defects near the interfaces and is responsible for the formation of drifting pattern domains, i.e. moving localised patches of travelling waves embedded in a Turing pattern background and vice versa.