J. Math. Biol. (2015) 70:549–590 DOI 10.1007/s00285-014-0774-y Mathematical Biology

© The Author(s) 2014. This article is published with open access at Springerlink.com Abstract To fully understand population persistence in river ecosystems, it is neces-sary to consider the effect of the water flow, which varies tremendously with seasonal fluctuations of water runoff and snow melt. In this paper, we study integrodifference models for growth and dispersal in the presence of advective flow with both peri-odic (alternating) and random kernel parameters. For the alternating kernel model, we obtain the principal eigenvalue of the linearization operator to determine popula-tion persistence and derive a boundary value problem to calculate it. For the random model, we establish two persistence metrics: a generalized spectral radius and the asymptotic growth rate, which are mathematically equivalent but can be understood differently, to determine population persistence or extinction. The theoretical frame-work and methods for calculations are provided, and the framework is applied to calculating persistence in highly variable river environments