  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.
