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

Amplification of the Net Reproductive Number by Dispersion for a Matrix Population Model Applicable to the Invasive Round Goby Fish