The clumping transition in niche competition: a robust critical phenomenon

Abstract

We show analytically and numerically that the appearance of lumps and gaps in the distribution ofn competing species along a niche axis is a robust phenomenon whenever the finiteness of the nichespace is taken into account. In this case, depending on whether the niche width of the speciesσ is above or below athreshold σc, whichfor large n coincideswith 2/n, there are twodifferent regimes. For σ > σc the lumpy pattern emerges directly from the dominant eigenvector of thecompetition matrix because its corresponding eigenvalue becomes negative. Forσ ≤ σc thelumpy pattern disappears. Furthermore, this clumping transition exhibits critical slowing down asσ is approached from above. We also find that the number of lumps of the species distribution versusσ displays a stair-step structure. The positions of these steps are distributed according to apower law. It is thus straightforward to predict the number of groups that can be packedalong a niche axis and this value is consistent with field measurements for a wide range ofthe model parameters.