  Matrix population models have proved popular and useful for studying
stage-structured populations in quantitative ecology. The goal of this paper is
to develop and analyze a general matrix population model that is applicable to
the population dynamics of the invasive round goby fish that incorporates both
stage structure---larvae, juveniles, and adults---and dispersion. Specifically,
we address the issue of whether or not dispersion can amplify the net
reproductive number. To this end, we will first review the mathematics of
matrix population models with dispersion, particularly those with an Usher
demography matrix. Techniques for computing the net reproductive number, like
the graph reduction method of de-Camino-Beck and Lewis, will be discussed. A
common theme will be the usefulness of submatrices of relevant matrices
obtained by the expunging of rows and columns corresponding to non-newborns.
Finally, examples will be provided, including the calculation of the net
reproductive number for multiple regions using the graph reduction method of
de-Camino-Beck and Lewis, examples where dispersion results in a total
population flourishing when the populations would otherwise go extinct, and an
application to the invasive round goby fish.
