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

TL;DR

This paper analyzes the evolution and disappearance of multi-pulse patterns in a reaction-diffusion system with time- and space-varying parameters, combining asymptotic analysis and numerical simulations to predict pattern dynamics and pulse disappearance.

Contribution

It introduces a reduced N-dimensional dynamical system for pulse patterns and develops a hybrid asymptotic-numerical method to predict pulse disappearance and pattern transitions.

Findings

Patterns can be reduced to an N-dimensional manifold for analysis.

Disappearance of pulses can be predicted by boundary crossing in the manifold.

Irregular patterns lose pulses gradually, while regular patterns undergo abrupt transitions.

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 $\mathcal{M}_N$. Next, we determine the local stability of $\mathcal{M}_N$ via the quasi-steady spectrum associated to evolving $N$-pulse patterns, which provides provides explicit information on the boundary $\partial\mathcal{M}_N$. Following the dynamics on $\mathcal{M}_N$, a $N$-pulse pattern may move through $\partial\mathcal{M}_N$ and `fall off' $\mathcal{M}_N$. A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern `jumps' to another invariant manifold $\mathcal{M}_M$, 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 loose 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 $\partial\mathcal{M}_N$ typically evolve towards regularity.