  An integro-differential equation on a tree graph is used to model the
evolution and spatial distribution of a population of organisms in a river
network. Individual organisms become mobile at a constant rate, and disperse
according to an advection-diffusion process with coefficients that are constant
on the edges of the graph. Appropriate boundary conditions are imposed at the
outlet and upstream nodes of the river network. The local rates of population
growth/decay and that by which the organisms become mobile, are assumed
constant in time and space. Imminent extinction of the population is understood
as the situation whereby the zero solution to the integro-differential equation
is stable. Lower and upper bounds for the eigenvalues of the dispersion
operator, and related Sturm-Liouville problems are found, and therefore
sufficient conditions for imminent extinction are given in terms of the
physical variables of the problem.
