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Who Will Win?

Two theoretical ecologists show that predicting survivors sometimes depends on a fractal¿a curve so detailed that no measurement guarantees an accurate forecast
aging



J. W. Stewart
diatom
Image: Courtesy of JEF HUISMAN

DIATOMS, such as Asterionella [above], are often pitted against green algae in the lab under different conditions to study how species compete for limited resources. When silicon is scarce, the algae proliferate and drive the diatoms the way of the dodo. If phosphorus is limiting, the diatoms outcompete the algae, which are phosphorus fumblers. When silicon and phosphorus are both scarce, diatoms and algae both prosper.

Jef Huisman of the University of Amsterdam and Franjo Weissing of the University of Groningen in the Netherlands remember the phone calls and e-mails that arrived from fellow theorists when their findings appeared in the journal Nature in 1999. "Oh, how stupid we were!" their colleagues exclaimed. "We never thought of this possibility!" Using a simple trick, the pair had exposed a wild and unforeseen side of a set of equations that govern a famous model in ecology¿one that represents how competing needs for shared resources sort species (not unlike contestants on Survivor) into winners and losers.

Researchers had assumed that the equations always enacted the same scenario, which on a computer unfolds as asymptotic curves crossing the screen. In other words, adept species multiply while unfit species dwindle, until the number of species matches the diversity of scarce resources or sinks below it. The model put a theoretical limit on biodiversity¿which Huisman and Weissing's results shattered. Indeed, they showed how the very same competitive circumstances could send opponents into cyclical and chaotic swaps of ascendancy.

More recently the duo has discovered that when some adversaries square off, the instabilities eventually end and a few specialized winners emerge. But there's a catch: who survives depends on a "fractal"¿a curve so detailed that no measurement guarantees an accurate forecast of the winners. "The fittest," they write in the May issue of American Naturalist, "can be as unpredictable as a roll of the dice."

These results have ecologists rethinking the very nature of competition among species and the origins of biodiversity. "We thought we had some understanding of what the limits on species diversity are," says James Grover of the University of Texas. "They [Huisman and Weissing] show things might be quite a bit more complicated than we thought."

Not-So-Model Behavior

It's easy to see why the model has proven so attractive for so long. Consider phytoplankton¿the typically microscopic organisms of oceans and lakes that, by harvesting sunlight, support almost all life in the water via the food chain. Every species needs sunlight, everyone needs CO2, all need the same nutrients, and where one cell floats looks as good a place as any other. Wouldn't just one of these species do? Why are there so many? Is no one here the fittest?

A contest in a beaker among two kinds of phytoplankton, a green alga and a diatom illustrates how the model equates limiting resources with a habitat's biodiversity. As conventionally staged in the lab, hoses circulate nutrients and CO2 at concentrations that a researcher specifies. When silicon is scarce, the algae proliferate and drive diatoms the way of the dodo. If phosphorus is limiting, the diatoms outcompete the algae, which are phosphorus fumblers. When silicon and phosphorus are both scarce, diatoms and algae march toward modest prosperity together.

fyto
Image: Courtesy of JEF HUISMAN

PHYTOPLANKTON from Lake Kinselmeer in the Netherlands is dominated by filamentous cyanobacteria but contains many other varieties as well. In fact, the average tablespoon of lake or ocean water contains around 50 times more species of phytoplankton than traditional ecological models would predict.

According to the model, a species secures its existence through a few key skills that keep it a step ahead of its competitors. It's hard to imagine Darwin calling such a scheme newfangled: species exist through specialization. Yet ecologists have had to bend or circumvent the model more often than they have been able to apply it. The diversity of phytoplankton in the field typically exceeds the number of limiting nutrients around them. In an average tablespoonful of lake or ocean, for example, there are about 50 times more species than the model would predict.

In doctoral research at Amsterdam, Huisman studied how algae compete for sunlight. In addition to experimental observations, he described the competition theoretically under Weissing's supervision at Groningen. He used the standard model for competition, obtained more or less standard results and had already filed his dissertation when something very nonstandard occurred.

Huisman had accepted a postdoctoral fellowship at Stanford University, but with a few weeks left in Amsterdam, he sat down again to play with the model. He and Weissing had once mused about abstract parallels between Huisman's model and models of natural selection that Weissing used. Weissing knew that in his models some trios of species coexisted perpetually in a rock-scissors-paper cycle of ascendancy: the first drove down the second, which held back the third, which on resurgence suppressed the first, so that the second recovered and so on.

The arrangement was immune to equilibrium. On a whim, Huisman tried assembling a trio of complementary species with the competition model and three limiting resources. He succeeded easily. Observing cycles with one set of parameters after another, it occurred to Huisman that he had tried something no one had thought to try before (or that if they had, they couldn't have tried very hard). The secret was three resources. "In a sense, it was really surprising that nobody had discovered it before, about say 10 or 20 years ago," Huisman says.

Another Personality Emerges

fyto fyto
Image: Courtesy of JEF HUISMAN

BI-CYCLE RACES among five competing species grouped in two overlapping trios yielded fractal patterns when graphed. For each of the first 160,000 races, the researchers varied the abundances of the same one or two species. They made one species' starting population the x-axis and the other the y-axis, and they colored each point either yellow for a Trio 1 victory or blue for a Trio 2 victory (top). When they varied the same species' abundances over a narrower range in another 160,000 races (bottom), they got the same pattern magnified.

At Stanford, Huisman continued collaborating via e-mail with Weissing on this non-equilibrium angle. Based on several influential theories and experiments, they knew that a buffeting by external factors¿predators, parasites or spurts of nutrients¿kept competitors from equilibrating and enabled more species than resources to coexist. Yet in their rock-scissors-paper oscillations, the competitors buffeted themselves.

At their respective keyboards, across 10 time zones, Huisman and Weissing cracked their knuckles and started tossing more species into the standard model with three resources to see what would happen. One worked while the other slept. Using various combinations of parameters, they defined different species, assembled them into groups and watched the results on their PCs. Plenty went extinct, Huisman says. "Most parameter sets lead to a worthless species that is unable to survive at all," he notes. "Some parameter combinations may lead to a superspecies that wipes out all resident species."

But soon Huisman and Weissing had nine virtual species coexisting on three resources in the model that said it couldn't happen. Meanwhile they had also decided to pursue chaos¿the seemingly random vicissitudes that arise in physical and other rule-governed processes such as weather. Weissing, who had come to ecology from mathematics, says he knew that chaos had to emerge sooner or later. And they finally found it by combining certain species on five limiting resources.

In this arrangement, the numbers of the various competitors undulated erratically without pattern. A reenactment of any match always proceeded similarly at first, but following even a slight adjustment to the size of one subpopulation, it charted a rapidly diverging path. The researchers knew well that such sensitivity to initial conditions was the hallmark of chaos. Finally they showed that chaos too supported a diversity of species beyond resources.

Fleeting niches in time, Huisman says, create the increased opportunity for biodiversity. At equilibrium, the abundance of resources and the size of opponents' forces are constant. If you are alive, it's because your skills are suited to the prosperity you have obtained. Yet chaos and oscillations create changing conditions in which many species encounter circumstances that give them the edge only once in a while. Not every adversary will be thrown off its game. "If one species is the best in all resources, it just wins," Huisman says. "If all species are more or less equal competitors, then we see this chaotic phenomenon."

Does this unruly chaotic behavior undo the standard model? Huisman, Weissing and other ecologists don't think so. On publication of the Nature report, ecologist Mark Ritchie of Utah State University said that Huisman and Weissing had given ecologists a valuable tool: "They're identifying a new way in which multiple species can coexist that we've never thought of before."

One More Surprise

Huisman and Weissing's May report describes further encounters with chaos. In a turn of events that surprised no mathematician, the two turned up a fractal. Huisman thought up a strategy for producing chaos using not five, but three resources. He again used the rock-scissors-paper pattern but this time in the form of a "bi-cycle." He formed two trios from five species, making one a common member of both groups. While one trio cycles, the other two species vie aimlessly. Yet in doing so, they jostle the trio in play, sometimes ending up ideally positioned to engage their missing partner. In that case they jump in, and the other two "extras" vacillate. These exchanges occur chaotically, Huisman and Weissing found.

chart
Image: Courtesy of JEF HUISMAN

TRANSIENT CHAOS emerges when five species, grouped in two trios, compete for three resources. Depending on the initial conditions, the system winds up such that species 1, 2 and 3 (black, red and blue, respectively) are winners of the competition (top), or species 1, 4 and 5 (black, green and yellow, respectively) are winners (bottom). This kind of sensitivity to initial conditions is a hallmark of chaos.

All the previous species assemblages oscillated or chaotically wandered in seeming perpetuity¿even over simulations lasting 10,000 plankton generations. But when Huisman and Weissing tinkered with the bi-cycles, they saw chaos that would unexpectedly and permanently cease. The reason was simple: each dancing trio shaped the environment differently. When Huisman and Weissing linked very dissimilar trios and watched them alternate, one eventually pushed two members of the other to extinction. A single two thousandths of a percent change in one species' abundance could give the other trio the advantage.

Huisman and Weissing made two graphs to illustrate the situation (see image). They enacted 160,000 matches among the same five competitors, each time altering only slightly the abundances of the same one or two species. They made one species' starting population the x-axis and the other the y-axis. And they made the color of each point they graphed such that it showed the outcome of the contest¿yellow for a Trio 1 victory, blue for Trio 2. The graph appeared to be a fractal.

For reassurance, the duo staged another 160,000 bi-cycle races, in which they varied the same two species' abundances over a narrower range. This run produced a second graph that was, in effect, a 25-fold magnification of a tile from the first¿a zoom-in view. As was true of the first diagram, there were many places in the second where blue and yellow points stood side by side. There was no sign that detail was diminished with magnification and no way to guess what colors would reside in any one spot on closer inspection. This figure had "fractal" written all over it.

The fractal nature of this pattern means that anticipating winners is like forecasting the weather, Weissing remarks. "The most you can hope for is a kind of a probabilistic answer," he says. Near the fractal curve boundaries, the proportion of blue to yellow dots helps to set the two outcomes' odds, but certainty is not an option. Says plankton ecologist Dan Roelke of Texas A & M University, "The work is very cool."

Beyond Cool

Huisman says that his lab has jobs open for the work still left to do. Verification of even just the basic prediction of oscillations and chaos¿whether pursued in the field or in the lab¿demands daunting amounts of groundwork and ultimately lots of day-by-day measurements. The model too calls for further elaboration, he says. The unpredictable finales that he and Weissing showed in their report unfolded only after about 1,000 days, or about three years. But the precipitous toll that winters regularly take on plankton might perpetually "reset the clock" before such events occur, Huisman fears.

In fact, just the jiggling influences of the outside world might upset the dynamics that he and Weissing engineered with careful parameter choices¿although he suspects that adding such details can only make the model's behavior even more complicated. "These are long-term questions that are going to take decades to resolve," says ecologist Steve Hubbell of the University of Georgia. Meanwhile Huisman will recruit applicants. "A theoretical physicist would be really helpful," he says.

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