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 sigma is above or below a threshold sigma(c), which for large n coincides with 2/n, there are two different regimes. For sigma > sigma(c) the lumpy pattern emerges directly from the dominant eigenvector of the competition matrix because its corresponding eigenvalue becomes negative. For sigma <= sigma(c) the lumpy pattern disappears. Furthermore, this clumping transition exhibits critical slowing down as sigma is approached from above. We also find that the number of lumps of the species distribution versus sigma 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.