Rock-paper-scissors tournaments explain ecological diversity

The childhood game of rock-paper-scissors provides one solution to this puzzle, report researchers at the University of Chicago and the University of California, Santa Barbara in Proceedings of the National Academy of Sciences. A mathematical model designed around the game's dynamics produced the potential for limitless biodiversity, and suggested some surprising new ecological rules.

"If you have two competitors and one is better, eventually one of the two will be driven extinct," said co-author Stefano Allesina, PhD, assistant professor of ecology and evolution at the University of Chicago. "But if you have three or more competitors and you use this rock-paper-scissor model, you can prove that many of these species can co-exist forever."

The rock-paper-scissors rules are an example of an "intransitive" competition, where the participants cannot be simply ordered from best to worst. When placed in pairs, winners and losers emerge: rock beats scissors, paper beats rock, and scissors beat paper. But when all three strategies compete, an impasse is reached where no one element is the undisputed winner.

In nature, this kind of rock-paper-scissors relationship has been observed for three-species groups of bacteria and lizards. But scientists had not yet studied how more complex intransitive relationships with more than three players -- think rock-paper-scissors-dynamite, and beyond -- could model the more complex ecosystems.

"No one had pushed it to the limit and said, instead of three species, what happens if you have 4,000? Nobody knew how," Allesina said. "What we were able to do is build the mathematical framework in which you can find out what will happen with any number of species."

Allesina and co-author Jonathan Levine, PhD, professor of ecology, evolution & marine biology at UCSB, combined the advanced mathematics of game theory, graph theory, and dynamical systems to simulate the outcome when different numbers of species compete for various amounts of "limiting factors" with variable success. An example, Allesina said, is a group of tree species competing for multiple resources such as nitrogen, phosphorus, light, and water.

When more limiting factors are added to the model, the amount of biodiversity quickly increases as a "tournament" of rock-paper-scissors matches develops between species, eliminating some weak players but maintaining a stable balance between multiple survivors.

"What we put together shows that when you allow species to compete for multiple resources, and allow different resources to determine which species win, you end up with a complex tournament that allows numerous species to coexist because of the multiple rock-paper-scissors games embedded within," Levine said.

In some models, where each species' advantage in one limiting factor is coupled to a disadvantage on another, a mere two limiting factors is capable of producing maximal biodiversity -- which stabilizes at half the number of species originally put into the model, no matter how large.

"It basically says there's no saturation," Allesina said. "If you have this tradeoff and have two factors, you can have infinite species. With simple rules, you can create remarkable diversity."

The model also produced a strange result: when the limiting factors are uniformly distributed, the total number of species that survive is always an odd number. Adjusting the model's parameters to more closely model the uneven distribution of resources in nature removed this intriguing quirk.

Allesina and Levine tested the realism of their model by successfully reverse-engineering a network of species relationships from field data on populations of tropical forest trees and marine invertebrates. Next, they will test whether the model can successfully predict the population dynamics of an ecosystem. Recently, Allesina was awarded a $450,000 grant by the James S. McDonnell Foundation to conduct experiments on bacterial populations that test the rock-paper-scissors dynamics in real time.

In the meantime, the rock-paper-scissors model proposes new ideas about the stability of ecosystems -- or the dramatic consequences when only one species in the system is removed.

"The fact that many species co-exist could depend on the rare species, which are more likely to go extinct by themselves. If they are closing the loop, then they really have a key role, because they are the only ones keeping the system from collapsing," Allesina said.

"If you're playing rock-paper-scissors and you lose rock, you're going to end up with only scissors in the system," Levine said. "In a more complex system, there's an immediate cascade that extends to a very large number of species."

The paper, "Competitive network theory of species diversity," was published online by the Proceedings of the National Academy of Sciences on March 14, 2011. The research was supported by the James S. McDonnell Foundation and the National Science Foundation.