The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space

Highlights: • Movement and coarsening of N-pulse patterns using hybrid numerical–asymptotic method. • Desertification process viewed as fast transitions between slow invariant manifolds. • Study of the method's validity and limitations by comparison with PDE simulations. • Irregular patterns degrade gradually; regular patterns via catastrophic transitions. • Downhill moving vegetation patterns observed in reaction–diffusion ecosystem model. -- Abstract: We consider the evolution of multi-pulse patterns in an extended Klausmeier equation with parameters that change in time and/or space. We formally show that the full PDE dynamics of a $N$-pulse configuration can be reduced to a $N$-dimensional dynamical system describing the dynamics on a $N$-dimensional manifold $M_N$. Next, we determine the local stability of $M_N$ via the quasi-steady spectrum associated to evolving $N$-pulse patterns, which provides explicit information on the boundary $\partial M_N$. Following the dynamics on $M_N$, a $N$-pulse pattern may move through $\partial M_N$ and 'fall off' $M_N$. A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern 'jumps' to another invariant manifold $M_M$, and specifically predicts the number $N-M$ of pulses that disappear. Combining the asymptotic analysis with numerical simulations of the dynamics on the various invariant manifolds yields a hybrid asymptotic–numerical method describing the full process that starts with a $N$-pulse pattern and typically ends in the trivial homogeneous state without pulses. We extensively test this method against PDE simulations and deduce general conjectures on the nature of pulse interactions with disappearing pulses. We especially consider the differences between the evolution of irregular and regular patterns. In the former case, the disappearing process is gradual: irregular patterns lose their pulses one by one. In contrast, regular, spatially periodic, patterns undergo catastrophic transitions in which either half or all pulses disappear. However, making a precise distinction between these two drastically different processes is quite subtle, since irregular $N$-pulse patterns that do not cross $\partial M_N$ typically evolve towards regularity. hybrid asymptotic-numerical method