When animals can choose from a range of feeding options, often those options with a higher energetic gain carry a higher risk of predation. This paper analyses the optimal trade-off between food and predation. We are primarily interested in how an animal’s decisions and its state change over time. Our models are very general. They can be applied to growth decisions, such as choice of habitat, in which case we might consider how the state variable size changes over an animal’s lifetime. Equally our models are applicable to short-term foraging decisions, such as vigilance level, in which case we might consider how energy reserves vary over a day. We concentrate on two cases: (i) the animal must reach a fixed state, its fitness depending on when this is attained; (ii) the animal must survive to a fixed time, its fitness depending on its final state. In case (i) minimization of m ortality per unit increase of state is optimal under certain baseline conditions. In case (ii) behaviour is constant over time under baseline conditions (the 'Risk-spreading Theorem’). We analyse how these patterns are modified by complicating factors, e.g. time penalties, premature termination of the food supply, stochasticity in food supply or in metabolic expenditure, and state-dependence in the ability to obtain food, in metabolic expenditure and in predation risk. From this analysis we obtain a variety of possible explanations for why an animal should reduce its intake rate over time (i.e. show satiation). We show how earlier work can be viewed as special cases of our results.