Sensitivity of the stable distribution vector for a size-classified population model

We consider the class of projection matrices arising from the size-classified matrix model for population growth. Suppose that such a matrix M is irreducible, and that the corresponding stable distribution vector is x. We give formulae for the partial derivatives of the entries in x with respect to the demographic parameters in M. Those formulae only require knowledge of the entries in M and the Perron value of M. For the special case that M is a Leslie matrix, we discuss various concavity and order properties of the partial derivatives, and for a large subclass of Leslie matrices, provide bounds on the sizes of the relevant partial derivatives.