A navigational guide to variable fitness: common methods of analysis, where they break down, and what you can do instead

Our methods for analyzing stochastic fitness are mostly approximations, and the assumptions behind these approximations are not always well understood. Furthermore, many of these approximations break down when fitness variance is high. This review covers geometric mean growth, diffusion approximations, and Markov processes. It discusses where each is appropriate, the conditions under which they break down, and their advantages and disadvantages, with special attention to the case of high fitness variance. A model of sessile and site-attached coastal species is used as a running example, and fully worked calculations and code are provided. Summary: The logarithm of geometric mean growth is usually only appropriate when (a) an invader growth rate is needed and (b) fitness variability is driven by environmental fluctuations. The usual approximation breaks down when fitness variance is high. Diffusion approximations can provide a reasonable guide to the expected change in frequency over a time step if expected fitnesses and fitness variances are appropriately scaled by the average expected fitness. Diffusion approximations can perform less well for fixation probabilities, especially since further approximations may be required. Fixation probabilities can be calculated exactly using a Markov process, regardless of how large fitness variance is, although an analytic expression is frequently not possible. If an analytic expression is desired, it may be worth using a diffusion approximation and checking it with a Markov process calculation.