The clumping transition in niche competition: a robust critical phenomenon

We show analytically and numerically that the appearance of lumps and gaps in the distribution of n competing species along a niche axis is a robust phenomenon whenever the finiteness of the niche space is taken into account. In this case, depending on whether the niche width of the species σ is above or below a threshold σ c , which for large n coincides with 2/ n , there are two different regimes. For σ > σ c the lumpy pattern emerges directly from the dominant eigenvector of the competition matrix because its corresponding eigenvalue becomes negative. For σ ≤ σ c the lumpy pattern disappears. Furthermore, this clumping transition exhibits critical slowing down as σ is approached from above. We also find that the number of lumps of the species distribution versus σ displays a stair-step structure. The positions of these steps are distributed according to a power law. It is thus straightforward to predict the number of groups that can be packed along a niche axis and this value is consistent with field measurements for a wide range of the model parameters.