## Beyond 3D: The Dimensionality of Food Webs

Ecosystems are a chaotic battle royale, with predators and prey, plants and animals, competitors and allies all fighting it out to eat or be eaten. But the food webs scientists typically put together are deceptively tidy diagrams, with simple arrows connecting diners to their natural food options. Ecologists readily admit that a true representation of an ecosystem's network would be multi-dimensional, simultaneously taking into account multiple traits for each species involved. But just how many dimensions would such a model need to accurately depict the complexity of a large ecosystem? 10? 100? 1000?

In a new paper published this week in *Ecology Letters*, a team led by scientists at the Computation Institute and University of Chicago calculate that number – and find that it is surprisingly low. Using data collected by their co-authors on 200 different food webs, ranging from the Caribbean reef to New Zealand grasslands to an Arctic Ocean inlet, Anna Eklöf, Stefano Allesina and colleagues looked for the minimum number of dimensions and traits needed to accurately describe a food network. The findings may save ecologists time and effort in revealing the structure underlying an ecosystem, and also help scientists build computational models that can make predictions about an ecosystem's future.

"To collect this kind of data takes ages to do," said Eklöf. "If we can find some common rules about these networks, then we can apply them to larger networks. We can also learn about the function of networks, and what happens to networks when we disturb them in different ways."

Dimensionality gives a flavor of the complexity of a food network. Imagine a pond with 10 fish species of various sizes. If body size is the only trait that determines the food network and each fish just eats the fish that are smaller than itself, then the network is one-dimensional. If the fish are unevenly distributed and the big fish only eat the smaller fish in their part of the pond, then the network is two-dimensional. If blue fish only eat red fish, a third dimension must be added to fully describe the network, and so on.

In the real world, ecosystems are far more complex, with hundreds of species and tens of thousands of connections. Still, ecologists can sniff out these dimensions the old-fashioned way through exhaustive observation of the ecosystem. But Eklöf and colleagues approached the problem from a computational perspective, starting with a high number of theoretical dimensions and working downward. For each number of dimensions, their model was checked against the field data about the network structure. If it described the network with no errors (for example, suggesting a predator-prey link where none has been observed), one dimension was taken away, until they reached the lowest number of dimensions needed to describe the network without error.

Running all 200 food webs through this method, the researchers pegged the upper limit of dimensionality at 10 – even the most complicated network in their sample could be described with a mere 10 factors. But Eklöf and colleagues had reason to suspect that the real number was even lower.

"Because we have this very strict counting of error links that our model predicts are really not there, this means that if we get just one of these erroneous links, we say that is the upper bound for dimensionality," Eklöf said. "But in a network where you have maybe 16,000 links, one error isn't really that big of a deal. So we basically said that we would accept some amount of errors in the data, and when we do that we can squeeze down the number of dimensions."

Using this form of "model selection," the dimensionality needed to describe a given food network was reduced to four – a reassuringly low number for any ecologist seeking to grapple with the structure of a network.

"The take-home message is that for even very complex ecological networks, the dimensionality is fairly low," Eklöf said. "Which kind of gives us hope that we can eventually identify these dimensions."

The researchers then approached the problem from a different direction, one perhaps more directly relevant to how a food network is studied in the field. A researcher studying the imaginary pond described earlier would spend time measuring as many traits about the ten fish species as possible: size, color, habitat, anatomy, and on and on. In order to potentially save wasted effort trying to collect every last detail, Eklöf and colleagues used the datasets to ask just how few traits are necessary to sufficiently describe most of a food network.

Here, their results showed that a little bit of work -- combined with some cleverness -- can produce a lot of insight. In many cases, a single trait (such as body size) was sufficient to describe the majority of a network. But in even the most complex networks, a mere three traits could describe at least one-third of the network structure.

The key, the researchers found, was to not just choose any old trait. Matching traits reflecting the nature of partnerships within the network – such as bill size of fruit-eating birds and the size of the various fruits in the ecosystem – could predict more of a network than arbitrarily chosen characteristics. Though it sounds like common sense, this strategy of trait-picking goes against traditional ecological practice, Eklöf said.

"When you usually look at traits in food webs, you look at the predator perspective, how the predator chooses its prey," Eklöf said. "Here, we wouldn't assume anything like that. We looked in both directions, and what's striking – if not surprising – is that matching traits often come to explain most of the structure."

However, these magic numbers of 4 and 3 aren't to be taken as hard and fast upper limits for every ecosystem on Earth. Eklöf said that a back-of-the-envelope calculation using similar methods on a hypothetical food network containing every species on Earth (roughly 100 million species and 1 trillion connections) came up with a much higher dimensionality of 25. Still, that's lower than might be predicted for such a massive network, and shows how dimensionality increases slowly as more and more species are added.

That's helpful for computational ecologists who seek to model the structure of natural food networks in order to make predictions about what happens when they are disrupted. In one possible scenario, an ecologist could construct a food web model from field observations, then use it to simulate what would happen if a new species unexpectedly arrived in their environment.

"If you have an invasive species coming to a new place or new ecosystem, we could determine what kind of interactions it would form and how that would function or disturb the community," Eklöf said. "With the low dimensionality, it suggests that we can construct better models of ecological networks of all kinds."