[Original paper: “Ian A. Hatton et al. (2015) “The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes” Science vol. 349 iss. 6252 p. 1070”]
Sometimes finding patterns in data helps answer questions; sometimes it asks questions. The study we’ll discuss today does the latter, and the question it asks is a big one: what is the relationship between an ecological community and the individuals that make up that community?
One of the most beloved and hotly debated theories in biology is Kleiber’s law, which poses a relationship between an individual’s mass (m) and its metabolism (b), i.e. how much energy it uses up. The mathematical formula is b = a*m^k, where a is a constant corresponding to the metabolism of an organism that weighs one gram (this in general varies from species to species), and k is an exponent that’s 3/4 if you ask a biologist, 2/3 if you ask a physicist, but definitely less than one. An animal’s mass, then, (e.g. how much biologically active tissue they have) has a huge influence on its overall metabolism.
Fig. 1: A gorilla and a mouse, animals with two very different metabolic rates.
The important thing about k < 1 is that per unit mass, metabolism decreases as organism size increases. A gorilla typically weighs about 8000 times more than a mouse; we can probably guess that a single gorilla uses up more total energy and eats more than a single mouse (Fig. 1). However, with Kleiber’s law we can predict that if we had 8000 mice instead of the one gorilla, they’d collectively have a metabolism about 10 times that of the gorilla, and eat about 10 times more. Now imagine replacing the gorilla with the equivalent amount of bacteria – that’s a hungry bacterio-ape!
Fig. 2: Data for Kleiber’s law. Despite a huge range of individual sizes [.00000000001 to 100000000 grams], over 1500 organisms fall very closely along the same relationship of how fast they can grow (“Maximum production”) and how big they are. Theory geeks like myself go crazy for straight lines on logarithmic plots [by ‘logarithmic’ I mean how things change along the axes], especially with so many points!
Kleiber’s law, however, is a theory about individuals, and the relationship between body mass and metabolism has been known and tested experimentally for decades. What the authors of this study uncovered was a really interesting link between this individual-level phenomenon and similar relationships at the ecosystem level. By compiling data from 2260 ecosystem studies across the globe (Fig. 3), they uncover similar k-exponent relationships for ecosystem ‘production’ (one can most simply think of this as growth) and predator-to-prey ratios.
Fig. 3: Each point corresponds to a place data was taken from for this study, all over the world; each color represents lakes/coasts/rivers (blue), open ocean (yellow), land plants (green), and land animals (red). It’s… a lot of data.
Whether it’s a lake, a river, an ocean, a forest, a savannah – same stuff. As we look at communities with more mass in them, i.e. more or bigger organisms, we see the same sort of decline in growth as with Kleiber’s law, and the same decline in how many predators a community of prey can support. Because the pattern is repeated across all kinds of ecosystems and scales, it suggests that the phenomenon is very general and doesn’t depend on the details of any particular ecosystem, but is somehow universal to the way ecosystems are structured. As you throw more algae into a pond, more gazelles into a savannah, or more fish into a bay, fewer predators per capita of prey can survive there according to the same k-formula, and each individual grows more slowly in the same way.
Fig. 4: All of the data from Fig. 3 are plotted here and show the key findings of the study. The slope k varies somewhat, but the relationship stays the same. With more prey/plants/algae comes slower individual growth and fewer predators per prey that can be sustained.
The exact value for k varies slightly, but this sort of data is hard to get good measurements for, and extracting a k-value from data is not as simple a mathematical process as it’s generally thought to be (a rant for another day). The general pattern is a lot more important than the specific number. Why this metabolism-mass relationship for individuals should translate to a growth-mass relationship or a predator-prey relationship for communities is not obvious!
We can bicker about what the particular slope k should be and other details, but this impressive data set convincingly shows that a general pattern emerges for (almost, if not) all of life, at both individual and community levels. Just as metabolism decreases (per unit mass) with size in a consistent way for individual organisms, growth decreases with size in a similar manner across a range of ecosystems, as does the ratio between predators and prey.
Such robust laws are rare in ecology. The ones that emerge from this data set are quite compelling, and beg for a convincing theory as to why they happen. Such a theory would elucidate fundamental links between physiology and ecology, and would be of immense applicability to a vast range of ecosystems.
At the same time, this study points out something really fascinating about the oceans. Ocean ecology lies at the small end of all of these mass scales, being filled mostly with microorganisms – after all, last I checked a tree is generally a bit bigger than a plankton cell. Based on this study, this means that ocean life generally supports one of the highest predator-to-prey ratios around, and that per unit mass, ocean life is hundreds if not thousands of times more productive than terrestrial life.
More than that, it suggests these differences are the result of something fundamental about how ecosystems structure themselves. Besides making ocean ecosystems markedly different than terrestrial ones, if we also consider that the ocean takes up a lot more of the earth’s surface than the continents, this also points to how ocean life has so much greater an impact on the earth system and climate than terrestrial life does.
Fig. 5: What’s so special about ocean life? Well for one, because ocean life is in general composed of very small life forms, the predator-prey ratio is very high, and the amount of mass actually increases up the food chain, unlike how we think of most ecosystems. However, that little bit sitting at the bottom of the food chain is growing like wild compared to its terrestrial counterparts!
Cael was once told by a professor that applied mathematicians are ‘intellectual dilettantes,’ which has been a proud self-identification for Cael since that moment. Cael is a graduate student at MIT & Woods Hole, & studies the ocean from a mathematical perspective; right now Cael is trying to figure out how detailed our measurements of phytoplankton communities can be if we detect them from space. Otherwise, Cael plays accordion, gardens, & reads instead of sleeping like it’s still fifth grade.