  In the present paper we analyze the linear stability of a hierarchical
size-structured population model where the vital rates (mortality, fertility
and growth rate) depend both on size and a general functional of the population
density ("environment"). We derive regularity properties of the governing
linear semigroup, implying that linear stability is governed by a dominant real
eigenvalue of the semigroup generator, which arises as a zero of an associated
characteristic function. In the special case where neither the growth rate nor
the mortality depend on the environment, we explicitly calculate the
characteristic function and use it to formulate simple conditions for the
linear stability of population equilibria. In the general case we derive a
dissipativity condition for the linear semigroup, thereby characterizing
exponential stability of the steady state.
