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Summary Predicting and managing species’ responses to climate change is one of the most significant challenges of our time. Tools are needed to address problems associated with novel climatic conditions, biotic interactions and greater climate velocities. We present a spatially explicit moving‐habitat model (MHM) and demonstrate its versatility in tackling critical questions in climate change research, including dispersal in multiple spatial dimensions, population stage structure, interspecific interactions, asymmetric range shifts, Allee effects and the presence of infectious diseases. The model utilizes integrodifference equations to track changes in population density over time in a habitat that is moving. The model is quite flexible and can accommodate variation in demography, dispersal patterns, biotic interaction and stochasticity in the velocity of climate change. The methods provide a general mechanistic understanding of the underlying ecological processes that drive a system. Field data can be readily incorporated into the model to give insight into specific populations of interest and inform management decisions. Synthesis. Moving‐habitat models unite ecological theory, data‐centred modelling and conservation decision support under a single framework. Their ability to generate testable hypotheses, incorporate data and inform best management practices proves that these models provide a valuable framework for climate change biologists.

Environmental changes are forcing many species to track suitable conditions or face extinction. In this study, we use a two-dimensional integrodifference equation to analyze whether a population can track a habitat that is moving due to climate change. We model habitat as a simple rectangle. Our model quickly leads to an eigenvalue problem that determines whether the population persists or declines. After surveying techniques to solve the eigenvalue problem, we highlight three findings that impact conservation efforts such as reserve design and species risk assessment. First, while other models focus on habitat length (parallel to the direction of habitat movement), we show that ignoring habitat width (perpendicular to habitat movement) can lead to overestimates of persistence. Dispersal barriers and hostile landscapes that constrain habitat width greatly decrease the population's ability to track its habitat. Second, for some long-distance dispersal kernels, increasing habitat length improves persistence without limit; for other kernels, increasing length is of limited help and has diminishing returns. Third, it is not always best to orient the long side of the habitat in the direction of climate change. Evidence suggests that the kurtosis of the dispersal kernel determines whether it is best to have a long, wide, or square habitat. In particular, populations with platykurtic dispersal benefit more from a wide habitat, while those with leptokurtic dispersal benefit more from a long habitat. We apply our model to the Rocky Mountain Apollo butterfly (Parnassius smintheus).

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