  In his seminal work in the 1970s Robert May suggested that there was an upper
limit to the number of species that could be sustained in stable equilibrium by
an ecosystem. This deduction was at odds with both intuition and the observed
complexity of many natural ecosystems. The so-called stability-diversity debate
ensued, and the discussion about the factors making an ecosystem stable or
unstable continues to this day. We show in this work that dispersal can be a
destabilising influence. To do this, we combine ideas from Alan Turing's work
on pattern formation with May's random-matrix approach. We demonstrate how a
stable equilibrium in a complex ecosystem with two trophic levels can become
unstable with the introduction of dispersal in space. Conversely, we show that
Turing instabilities can occur more easily in complex ecosystems with many
species than in the case of only a few species. Our work shows that adding more
details to the model of May gives rise to more ways in which an equilibrium can
become unstable. Making May's simple model more realistic is therefore unlikely
to remove the upper bound on complexity.
