  In food webs, many interacting species coexist despite the restrictions
imposed by the competitive exclusion principle and apparent competition. For
the generalized Lotka-Volterra equations, sustainable coexistence necessitates
nonzero determinant of the interaction matrix. Here we show that this
requirement is equivalent to demanding that each species be part of a
non-overlapping pairing, which substantially constrains the food web structure.
We demonstrate that a stable food web can always be obtained if a
non-overlapping pairing exists. If it does not, the matrix rank can be used to
quantify the lack of niches, corresponding to unpaired species. For the species
richness at each trophic level, we derive the food web assembly rules, which
specify sustainable combinations. In neighboring levels, these rules allow the
higher level to avert competitive exclusion at the lower, thereby incorporating
apparent competition. In agreement with data, the assembly rules predict high
species numbers at intermediate levels and thinning at the top and bottom.
Using comprehensive food web data, we demonstrate how omnivores or parasites
with hosts at multiple trophic levels can loosen the constraints and help
obtain coexistence in food webs. Hence, omnivory may be the glue that keeps
communities intact even under extinction or ecological release of species.
