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Understanding how climate can interact with other factors in determining patterns of species abundance is a persistent challenge in ecology. Recent research has suggested that the dynamics exhibited by some populations may be a non-additive function of climate, with climate affecting population growth more strongly at high density than at low density. However, we lack methodologies to adequately explain patterns in population growth generated as a result of interactions between intrinsic factors and extrinsic climatic variation in non-linear systems. We present a novel method (the Functional Coefficient Threshold Auto-Regressive (FCTAR) method) that can identify interacting influences of climate and density on population dynamics from time-series data. We demonstrate its use on count data on the size of the Soay sheep population, which is known to exhibit dynamics generated by nonlinear and non-additive interactions between density and climate, living on Hirta in the St Kilda archipelago. The FCTAR method suggests that climate fluctuations can drive the Soay sheep population between different dynamical regimes--from stable population size through limit cycles and non-periodic fluctuations.