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We consider the evolution of multi-pulse patterns in an extended Klausmeierequation with parameters that change in time and/or space. We formally showthat the full PDE dynamics of a $N$-pulse configuration can be reduced to a$N$-dimensional dynamical system describing the dynamics on a $N$-dimensionalmanifold $\mathcal{M}_N$. Next, we determine the local stability of$\mathcal{M}_N$ via the quasi-steady spectrum associated to evolving $N$-pulsepatterns, which provides provides explicit information on the boundary$\partial\mathcal{M}_N$. Following the dynamics on $\mathcal{M}_N$, a $N$-pulsepattern may move through $\partial\mathcal{M}_N$ and `fall off'$\mathcal{M}_N$. A direct nonlinear extrapolation of our linear analysispredicts the subsequent fast PDE dynamics as the pattern `jumps' to anotherinvariant manifold $\mathcal{M}_M$, and specifically predicts the number $N-M$of pulses that disappear. Combining the asymptotic analysis with numericalsimulations of the dynamics on the various invariant manifolds yields a hybridasymptotic-numerical method describing the full process that starts with a$N$-pulse pattern and typically ends in the trivial homogeneous state withoutpulses. We extensively test this method against PDE simulations and deducegeneral conjectures on the nature of pulse interactions with disappearingpulses. We especially consider the differences between the evolution ofirregular and regular patterns. In the former case, the disappearing process isgradual: irregular patterns loose their pulses one by one. In contrast,regular, spatially periodic, patterns undergo catastrophic transitions in whicheither half or all pulses disappear. However, making a precise distinctionbetween these two drastically different processes is quite subtle, sinceirregular $N$-pulse patterns that do not cross $\partial\mathcal{M}_N$typically evolve towards regularity.

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