Bounds for the critical speed of climate-driven moving-habitat models

Integrodifference equations have recently been used as models for populations undergoing climate-driven habitat movement. In these models, the persistence of a population is governed by the maximal or dominant eigenvalue of a Fredholm integral equation with an asymmetric kernel; this eigenvalue determines the critical translational speed for extinction of the population. Since direct methods for finding eigenvalues are often analytically or computationally expensive, we explored the extensive literature on alternative methods for localizing maximal eigenvalues. We found that a sequence of iterated row sums provide upper and lower bounds for the maximal eigenvalue. Alternatively, arithmetic and geometric symmetrization yield upper and lower bounds. Geometric symmetrization is especially valuable and leads to a simple Rayleigh quotient that can be used to analytically approximate the critical-speed curve. Our research sheds new light on the interpretation and limitations of the average-dispersal-success approximation; it also provides a generalization of this useful tool for asymmetric kernels.