Mathematical models of consumer-resource systems are used to explore the evolution of traits related to resource acquisition in a generalist consumer species that is capable of exploiting two resources. The analysis focuses on whether evolution of traits determining the capture rates of two resources by a consumer species produce one generalist, two specialists, or all three types, when all types are characterized by a common fitness function. In systems with a stable equilibrium, evolution produces one generalist or two specialists, depending on the second derivative of the trade-off relationship. When there are sustained population fluctuations, the nature of the trade-off between the consumer's capture rates of the two resources still plays a key role in determining the evolutionary outcome. If the trade-off is described by a choice variable between zero and one that is raised to a power n, polymorphic states are possible when n > 1, which implies a positive second derivative of the curve. These states are either dimorphism, with two relatively specialized consumer types, or trimorphism, with a single generalist type and two specialists. Both endogenously driven consumer-resource cycles, and fluctuations driven by an environmental variable affecting resource growth are considered. Trimorphic evolutionary outcomes are relatively common in the case of endogenous cycles. In contrast to a previous study, these trimorphisms can often evolve even when new lineages are constrained to have phenotypes very similar to existing lineages. Exogenous cycles driven by environmental variation in resource growth rates appear to be much less likely to produce a mixture of generalists and specialists than are endogenous consumer-resource cycles.