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Temperature and nutrients are fundamental, highly nonlinear drivers of biological processes, but we know little about how they interact to influence growth. This has hampered attempts to model population growth and competition in dynamic environments, which is critical in forecasting species distributions, as well as the diversity and productivity of communities. To address this, we propose a model of population growth that includes a new formulation of the temperature–nutrient interaction and test a novel prediction: that a species' optimum temperature for growth, Topt, is a saturating function of nutrient concentration. We find strong support for this prediction in experiments with a marine diatom, Thalassiosira pseudonana: Topt decreases by 3–6 °C at low nitrogen and phosphorus concentrations. This interaction implies that species are more vulnerable to hot, low-nutrient conditions than previous models accounted for. Consequently the interaction dramatically alters species' range limits in the ocean, projected based on current temperature and nitrate levels as well as those forecast for the future. Ranges are smaller not only than projections based on the individual variables, but also than those using a simpler model of temperature–nutrient interactions. Nutrient deprivation is therefore likely to exacerbate environmental warming's effects on communities.Phytoplankton form the base of most aquatic ecosystems. We show that when nutrients are low, phytoplankton are less able to tolerate extreme temperatures. The temperature at which they grow most quickly (their optimum temperature) also decreases. This means that hot, low-nutrient conditions – commonly found in the tropical oceans – are a major challenge to species' survival. Many species' ranges may therefore be smaller than previously recognized. In the future, warming will increase the area of the ocean in which these challenging conditions are found, reducing these ranges even further.