Quantifying the impact of environmental changes on migratory species: a model perturbation framework

Migratory species use different habitats and pathways across their migratory route. Pathway contribution metrics are transient metrics of population growth, derived from population models, and quantify the predicted contribution of an individual, travelling along a specified migratory route, to the total population over a specified length of time. Environmental disturbances or management actions may occur temporally or spatially throughout the process of migration. The impact that a given perturbation may have on pathway contribution metrics is not always obvious owing to the propagation of the perturbation through the migratory cycle. Here, we develop a general modelling framework that incorporates perturbations into a class of matrix migratory population models, and which quantifies the effect that perturbations to the model, in terms of the transition rates of habitats and pathways, have on pathway contribution metrics. We also detail how to calculate the sensitivity of pathway contribution metrics to the perturbations considered.Our framework may be used to provide insights into the impact that environmental disturbances or management actions have on migratory populations. These insights may be used to inform management actions which either buffer against possible deleterious disturbances or increase the population size through targeted interventions. Our theoretical results are illustrated via hypothetical examples and a model inspired by the monarch butterfly; we uncover results that are not clear from the calculation of the pathway contribution metrics alone.