Evolution is frequently rough and unforgiving; individuals within a species compete for food, reproductive partners, or other resources. Species fight each other for survival, especially when preying on one another.

Mathematical biologists often simplify these dynamics to predator versus prey. Real-world populations of predator and prey species within a given ecosystem cycle between booms and busts. In various cases, multiple species—including both predators and prey—coexist with similar diets. For example, a cubic meter of seawater can harbor several species of plankton, consisting of tiny plants and animals (see Figure 1).

Figure 1. Many species of plankton, comprised of minuscule plants and animals, successfully coexist in the ocean despite competing for the same food sources. Public domain image.

One would naively expect reproductive success (more offspring) or competitive performance (eating more than your neighbor) to lead to one species’ domination. But that does not occur. While many of these organisms consume the same food, one species does not out-eat the others; the plankton swarm’s overall diversity remains fairly constant. Biologists refer to this phenomenon as the “paradox of the plankton” or the “biodiversity paradox,” among similar terms.

A possible solution to the paradox is the “kill the winner” (KTW) hypothesis. “We have many species competing with each other, but for each one we also have a predator,” Chi Xue, a mathematical physicist at the University of Illinois at Urbana-Champaign, explained. “[If] one species grows to a large population and is poised to win out over the others, the predator will eventually come in and eat the winning species.”

In other words, evolution may punish runaway reproductive or competitive success in prey species. Higher population numbers expose organisms to the predators who consume them. KTW is a reasonable hypothesis, but testing it quantitatively to measure whether it matches real population biology is challenging.

Xue and her advisor, Nigel Goldenfeld, modeled multiple strains of bacteria and bacteriophages (or “phages”), which are viruses that prey on bacteria. They found that the standard mathematical version of KTW failed in realistically small populations. Some strains became extinct in the local environment, causing a cascade of other extinctions and the eventual demise of the entire ecosystem.

To fix the problem, Xue and Goldenfeld introduced a way for species to evolve mathematically. “We have an arms race,” Xue said. “The prey mutates to escape the predator, but the predator also mutates to catch the prey. New species are constantly introduced, thus avoiding the extinction problem.”

The KTW mathematical model is based on a set of equations first proposed for chemical reactions by Alfred J. Lotka in 1920, then adapted to fish populations by Vito Volterra in 1926 (other mathematically-minded biologists discovered similar equations around that time).

When applied to predator-prey systems, the so-called Lotka-Volterra model was simple, elegant, and completely incorrect. KTW adapts the Lotka-Volterra equations to include multiple predator and prey species. Each predator species targets a single prey species, while the prey species simultaneously compete with each other for food. The math produces a result that seems intuitively correct: if a prey species becomes too successful, the predator species targeting it has enough food to increase its own population, therefore restoring balance.

Many KTW formulations assume that all populations are quite large, making it possible to treat the number of organisms in the species as a continuous variable. Xue and Goldenfeld realized that this assumption does not hold in real-world situations. “We took into account the fact that population size is discrete,” Xue said. “Because the population size is integer, we found that demographic noise causes species’ coexistence to break down.”

Demographic Noise Plotting the number of adult women of

a given height on a graph yields very

different plots for small and large

sample sizes. For large numbers, the

distribution is Gaussian: symmetrical

around the population’s mean value.

Smaller populations, however, do not

exhibit a clear pattern. Because an

individual’s height does not correlate

with the height of anyone else,

fluctuations dominate the data when

the population is small; this is known

as “demographic noise.” Demographic

noise in species represents randomness

in reproduction, deaths from disease or

accident, and other factors arising from

classification of populations as discrete

rather than continuous.





“Demographic noise” is the ecologists’ version of “shot noise”: random fluctuations that arise simply because the number of objects in a model are integer (see accompanying inset). It is important even for abundant species like bacteria because a real-world ecosystem does not include every member of a bacterial species, and predator-prey interactions occur in relatively small local populations.

When Xue and Goldenfeld modified KTW to treat discrete populations, species became sequentially extinct and the model failed. Although this sort of thing does happen in nature sometimes, it is not a feature of normal ecosystems. Diversity exists among species, and our mathematical models should reflect that.

To solve the problem, they looked to evolution. “If we introduce coevolution into the system, we can avoid the extinction situation,” Xue said. “Coevolution here means that prey and predator are both mutating.”

In strict Lotka-Volterra or KTW models, all organisms within a specific group (prey species, for example) are identical. But according to Darwinian evolution, mutation ensures that descendants are not always identical to their parents, even in species like bacteria that reproduce asexually (bacteria can also transfer genes between individuals, which helps spread new traits). These small changes between generations are often enough to give the offspring an edge over other species.

Xue and Goldenfeld modeled descent with modification by allowing their various bacterial prey species to mutate into new strains at a certain rate. The predatory phage that targets the original strain cannot prey on the new strain, which then “wins.” However, phages also evolve at a similar rate, allowing them to consume the new strain (viruses are not “alive” in the normal biological sense, but they still evolve).

The simplified KTW model is a set of coupled differential equations

\[\dot{B_l} = B_i \Big(b_i - e\:\sum\limits^m_{j = 1} B_j - pV_i \Big) \\ \dot{V_l} = V_i (\beta p B_i - d),\]

where \(B_i\) and \(V_i\) are the populations of \(m\) species of bacteria and viruses respectively. Each virus strain preys on only one bacterial strain with rate \(p\), while the bacteria reproduce at rate \(b_i\) and compete with other strains at rate \(e\). The viruses can also “burst” with rate \(\beta\) and perish at rate \(d\). One can obtain the original Lotka-Volterra model by setting \(e\) to zero and restricting the equations to one predator and prey species.

The previous equations model populations as continuous. Xue and Goldenfeld modified them to handle discrete numbers for each population. They also included potential species in the overall species set and added a mutation rate, so that offspring from population \(i\) could be part of population \(i + 1\).

The researchers discovered several distinct outcomes upon adjusting mutation rates:

When the mutation rate was made too small, the demographic noise problem returned. The prey species did not evolve quickly enough to avoid extinction, and the whole ecosystem collapsed. Species maintained diversity at moderate mutation rates, but average populations remained low. Individual prey species sometimes spiked when mutation allowed them to reproduce without predatory interference, only to crash again when predators evolved to catch up.; A rapid mutation rate kept overall diversity high; no species had a chance to dominate another. Prey mutated quickly enough to avoid being eaten into extinction, while predators adapted to new strains before prey populations grew out of hand.

Xue and Goldenfeld focused on bacteria/phage competition, since they could study these diverse and rapidly-mutating species in the laboratory. But Xue argues that adjustment of mutation and prediction rates can extend the model to other systems, such as mammals. “If you look at an ordinary predator and prey—the fox and hare, for example—their coevolution rate is very slow compared to their predation rate,” she said.

Whatever those rates, coevolution is key to understanding how ecosystems maintain species diversity, even among competitors. “Evolution is the universal law behind everything in biology,” Xue said. “Diversity is universal for basically every ecosystem on Earth.”

Further Reading

[1] Murray, J.D. (1989). Mathematical Biology. Berlin, Germany: Springer-Verlag. *The Lotka-Volterra model, its problems, and its generalizations are available in chapter 3.

[2] Xue, C., & Goldenfeld, N. (2017). Coevolution Maintains Diversity in the Stochastic ‘Kill the Winner’ Model. Phys. Rev. Lett., 119, 268101.

Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.