  Climate change impacts population distributions, forcing some species to
migrate poleward if they are to survive and keep up with the suitable habitat
that is shifting with the temperature isoclines. Previous studies have analyzed
whether populations have the capacity to keep up with shifting temperature
isoclines, and have mathematically determined the combination of growth and
dispersal that is needed to achieve this. However, the rate of isocline
movement can be highly variable, with much uncertainty associated with yearly
shifts. The same is true for population growth rates. Growth rates can be
variable and uncertain, even within suitable habitats for growth. In this paper
we reanalyze the question of population persistence in the context of the
uncertainty and variability in isocline shifts and rates of growth.
Specifically, we employ a stochastic integrodifference equation model on a
patch of suitable habitat that shifts poleward at a random rate. We derive a
metric describing the asymptotic growth rate of the linearized operator of the
stochastic model and yielding a threshold criterion for population persistence.
We demonstrate that the variability in the yearly shift and in the growth rate
has a significant negative effect on the persistence in the sense that it
decreases the threshold criterion for population persistence. Mathematically,
we show how the persistence metric can be connected to the principal eigenvalue
problem for a related integral operator, at least for the case where isocline
shifting speed is deterministic. Analysis of dynamics for the case where the
dispersal kernel is Gaussian leads to the existence of a critical shifting
speed. Finally we illustrate our different results for butterfly population
using numerical simulations and demonstrate how increased variances in isocline
shifts and growth rates translate into decreased likelihoods of persistence.
