The half-century old mathematics that ecologists use to understand how predator and prey populations rise and fall has received a revamp. Two scientists from Georgia Tech did this by crediting evolution for what it is but not commonly thought to be: fast, not slow.
The scientists, Joshua Weitz and Michael Cortez, applied a branch of mathematics called fast-slow dynamical systems theory to model how two populations could vary over time if they are evolving together. Until now, this has been the exclusive demesne of the Lotka-Volterra equations, derived by Alfred Lotka and Vito Volterra in the early 20th century. On a graph, these equations are visually striking for how they show predator and prey numbers rising in falling in continuous cycles.
For example, cheetahs eat baboons. In an ecosystem good for baboonkind, baboons will thrive. Cheetahs will eat them and thrive. As the number of baboons increases, so will the number of cheetahs. With too many cheetahs, the number of baboons will decline. As a result, the number of cheetahs will also decline. But the ecosystem is good for baboons. So after the number of cheetahs has declined, more baboons will appear. As the number of baboons increases, so does the number of cheetahs. And so on.
However, the Lotka-Volterra equations make several assumptions to get this far, many of which oversimplify natural conditions to the point that they no longer seem natural. Chief among them concerns the ignorance of genetic variations. Animals do possess them whether in the field or in the laboratory but the Lotka-Volterra equations assume the differences arising from them don’t exist. As a result, while “predators and their prey differ in their abilities acquire food or avoid capture,” the equations just overlook such traits, said Michael Cortez, a postdoc at Georgia Tech and first author on the paper describing the revamped equations. It was published in the Proceedings of the National Academy of Sciences May 5.
Turned on its head
In fact, Cortez and his postdoctoral mentor Joshua Weitz were particularly motivated by three studies, two from 2001 and one from 2011, whose findings gave rise to absurd implications if the Lotka-Volterra reasoning was applied. The equations – like depicted in the chart – require the prey population to peak first, followed by the predator population. The studies from 2001 and 2011 investigated gyrfalcon-rock ptarmigan, mink-muskrat and phage-V. cholerae pairs, and found the opposite: they showed the predator population peaked first, before the prey population did.
So are the prey eating the predators? “This is not the case,” Cortez explained. According to him, the reversal in peaking is driven by fluctuations in the abundance of different types of prey. One type of prey could be more or less able to avoid capture, while one type of predator could be more or less able to capture prey. Thus, these two kinds of animals are developing distinct genetic traits at the same time, i.e. coevolving.
To understand how coevolution influenced the number of predators and prey, Cortez and Weitz applied fast-slow dynamical systems theory. The ‘fast’ applies to the change in the number of types of predator or prey. The ‘slow’, to how the population as a whole is changing. Between them, says Cortez, “I was able to break the reverse cycles into pieces and study each piece of the cycle individually, allowing me to understand how coevolution was causing the reverse cycles.”
“The most surprising and exciting prediction from our work is that co-evolution between predators and prey can reverse this ordering, yielding cycles where peaks in prey abundance follow peaks in predator abundance,” Weitz added.
A different fast-slow
While this is not the first study to investigate what effects evolution has on changing populations, it is the first to accommodate fast rates of evolution, i.e. evolutionary changes that are more rapid and occur within a few generations. As a result, their implications are far-ranging, too, for the Lotka-Volterra equations were not restricted to ecology even though they were inspired by it. One other area of science in which a system could go back and forth between two stable states is chemistry and all its chemical reactions.
However, just like in ecology, the precise mathematics that governs them is computationally intensive. On May 6, researchers from Oxford University published a paper in The Journal of Chemical Physics explaining how the mathematics could be further simplified, making it easier to model them on computers. While this team also considers fast-slow systems, the designation is different. The Cortez-Weitz model compared how rapid evolutionary changes (fast) affected population (slow). The ‘Oxford model’, on the other hand, compares how changes in the sources of food (fast) affect the time taken for the predators to become extinct (slow).
To demonstrate, Maria Bruna, the first author on the paper, explained that in their system, she and her team consider whale and plankton populations. Plankton is an important food source for whales. While whales live and function over many years, plankton blooms can be fickle and change their yield of food on a daily basis. However, some environmental conditions can push the plankton blooms to take many years to shift their yield. “In such cases, the whales will ‘care’ about these metastable transitions in plankton, since they notice the changes in plankton abundance on a timescale which is relevant to them,” she said.
Weitz expressed interest in this work: “It would be very interesting to see what happens when their method is applied to more complex contexts, including in which populations are comprised of two or more variants.”
Cortez MH, & Weitz JS (2014). Coevolution can reverse predator-prey cycles. Proceedings of the National Academy of Sciences of the United States of America PMID: 24799689
Bruna M, Chapman SJ, & Smith MJ (2014). Model reduction for slow-fast stochastic systems with metastable behaviour. The Journal of chemical physics, 140 (17) PMID: 24811625