We consider the number of trophic levels in a food chain given by the
equilibrium state for a simple mathematical model with ordinary differential
equations which govern the temporal variation of the energy reserve in each
trophic level. When a new trophic level invades over the top of the chain,
the chain could lengthen by one trophic level.
We can derive the condition that such lengthening could
occur, and prove that the possibly longest chain is globally stable.
In some specific cases, we find that the possibly longest chain is such that
the lower trophic level has a greater energy reserve than the higher has,
so that the distribution of energy reserves can be regarded to have a
pyramid shape, whereas, if any of its trophic levels is removed, the pyramid
shape cannot be maintained.
Further, we find the condition that arbitrary long chain can be established.
In such unbounded case, we prove that any chain could not have
the pyramid shape of energy reserve distribution.