  In this article, we study the growth and spread of a new species in a river
network with two or three branches via the Fisher-KPP advection-diffusion
equation over some simple graphs with every edge a half infinite line. We
obtain a rather complete description of the long-time dynamical behavior for
every case under consideration, which can be loosely described by a trichotomy
(see Remark 1.7), including two different kinds of persistence states as
parameters vary. The phenomenon of "persistence below carrying capacity"
revealed here appears new, which does not occur in related models of the
existing literature where the river network is represented by graphs with
finite-lengthed edges, or the river network is simplified to a single infinite
line.
