  Living species, ranging from bacteria to animals, exist in environmental
conditions that exhibit spatial and temporal heterogeneity which requires them
to adapt. Risk-spreading through spontaneous phenotypic variations is a known
concept in ecology, which is used to explain how species may survive when faced
with the evolutionary risks associated with temporally varying environments. In
order to support a deeper understanding of the adaptive role of spontaneous
phenotypic variations in fluctuating environments, we consider a system of
non-local partial differential equations modelling the evolutionary dynamics of
two competing phenotype-structured populations in the presence of periodically
oscillating nutrient levels. The two populations undergo spontaneous phenotypic
variations at different rates. The phenotypic state of each individual is
represented by a continuous variable, and the phenotypic landscape of the
populations evolves in time due to variations in the nutrient level. Exploiting
the analytical tractability of our model, we study the long-time behaviour of
the solutions to obtain a detailed mathematical depiction of evolutionary
dynamics. The results suggest that when nutrient levels undergo small and slow
oscillations, it is evolutionarily more convenient to rarely undergo
spontaneous phenotypic variations. Conversely, under relatively large and fast
periodic oscillations in the nutrient levels, which bring about alternating
cycles of starvation and nutrient abundance, higher rates of spontaneous
phenotypic variations confer a competitive advantage. We discuss the
implications of our results in the context of cancer metabolism.
