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Understanding animal movement behaviour is key to furthering our knowledge on intra- and inter-specific competition, group cohesion, energy expenditure, habitat use, the spread of zoonotic diseases or species management. We used a radial basis function surface approximation subject to minimum description length constraint to uncover the state-space dynamical systems from time series data. This approximation allowed us to infer structure from a mathematical model of the movement behaviour of sheep and red deer, and the effect of density, thermal stress and vegetation type. Animal movement was recorded using GPS collars deployed in sheep and deer grazing a large experimental plot in winter and summer. Information on the thermal stress to which animals were exposed was estimated using the power consumption of mechanical heated models and meteorological records of a network of stations in the plot. Thermal stress was higher in deer than in sheep, with less differences between species in summer. Deer travelled more distance than sheep, and both species travelled more in summer than in winter; deer travel distance showed less seasonal differences than sheep. Animal movement was better predicted in deer than in sheep and in winter than in summer; both species showed a swarming behaviour in group cohesion, stronger in deer. At shorter separation distances swarming repulsion was stronger between species than within species. At longer separation distances inter-specific attraction was weaker than intra-specific; there was a positive density-dependent effect on swarming, and stronger in deer than in sheep. There was not clear evidence which species attracted or repelled the other; attraction between deer at long separation distances was stronger when the model accounted for thermal stress, but in general the dynamic movement behaviour was hardly affected by the thermal stress. Vegetation type affected intra-species interactions but had little effect on inter-species interactions. Our modelling approach is useful in interpreting animal interactions, in order to unravel complex cooperative or competitive behaviours, and to the best of our knowledge is the first modelling attempt to make predictions of multi-species animal movement under different habitat mosaics and abiotic environmental conditions. [ABSTRACT FROM AUTHOR]