How do fluctuating ecological dynamics impact the evolution of hosts and parasites?

Theoretical models of the evolution of parasites and their hosts have shaped our understanding of infectious disease dynamics for over 40 years. Many theoretical models assume that the underlying ecological dynamics are at equilibrium or constant, yet we know that in a great many systems there are fluctuations in the ecological dynamics owing to a variety of intrinsic or extrinsic factors. Here, we discuss the challenges presented when modelling evolution in systems with fluctuating ecological dynamics and summarize the main approaches that have been developed to study host-parasite evolution in such systems. We provide an in-depth guide to one of the methods by applying it to two worked examples of host evolution that have not previously been studied in the literature: when cycles occur owing to seasonal forcing in competition, and when the presence of a free-living parasite causes cycles, with accompanying interactive Python code provided. We review the findings of studies that have explored host-parasite evolution when ecological dynamics fluctuate, and point to areas of future research. Throughout we stress the importance of feedbacks between the ecological and evolutionary dynamics in driving the outcomes of infectious disease systems. This article is part of the theme issue 'Infectious disease ecology and evolution in a changing world'.