Karl Sigmund and Robert D. Holt; PNAS March 2, 2021 118 (9) e2100200118

As biologist Andrew Hendry recently wrote, “research initiatives in ecology and evolution have periodically dated but never married” (1). This also holds for the theoretical underpinnings of the two fields. Roughly speaking, the first mathematical models of population ecology are a century old, and the first stirrings of evolutionary game theory date from half a century ago. Yet, the seamless fusion of these fields, long desired (2), is still work in progress (3). In PNAS, Grunert et al. (4) provide a valuable step along this path. It analyses conditions for evolutionary stability in an ecologically fluctuating environment, driven by species interactions, and points the way toward a more intensive investigation of ecoevolutionary dynamics.

Fluctuations in numbers of predators and prey sparked mathematical approaches to ecology. It seems almost obvious: the more prey, the better for the predators. They multiply. However, then prey suffer and dwindle. With fewer prey, predators decline. With fewer predators, prey numbers pick up. Hence, times improve for the predators again—and so on, endlessly. Such a feedback loop makes intuitive sense. Indeed, many records of predator–prey interactions, some dating back to the venerable Hudson’s Bay Company, and others impeccably up to date, display stable and regular fluctuations consistent with this scenario. There are alternative explanations for the fluctuations, ranging from maternal effects to the swings of fashion, but predation is what first comes to mind.

The earliest, stylized differential equations for interacting predator–prey populations, due to Lotka and Volterra, duly produced periodic oscillations in predator and prey numbers, but with a peculiar property: These now-classical equations are not structurally stable. This means that an arbitrarily small change can generate radically different outcomes, for instance no periodic orbit at all. For any self-respecting model this is a drawback. It was overcome in more realistic models. It suffices to take into account that a predator’s intake is not proportional to the number of prey but flattens out, because each meal demands its handling time, and because factors other than the reciprocal interaction influence dynamics (e.g., competition for space).

In particular, the Rosenzweig–MacArthur model used in ref. 4 displays either a stable equilibrium or else a limit cycle: Eventually, the oscillations will have a well-specified frequency and amplitude, independent of the initial condition.

No doubt many details of the real-life interaction between these species are missing in this model, but being structurally stable it can accommodate perturbations, provided these are small enough, and it captures essential features of many natural enemy–victim interactions.

Self-regulatory feedback loops also occur in the evolution of a species interacting with itself (as well as with other species). Maynard Smith was the first to apply this insight to the frequency-dependent selection of phenotypic traits, using game theory (5). An individual can be viewed as a player, a trait as a strategy, and the resulting “fitness” (or reproductive success) as payoff. This fitness depends on the environment. If the trait is heritable, selection will increase fitness and thereby adapt the trait to the environment.

See https://www.pnas.org/content/118/9/e2100200118