| Nature is complex, unpredictable, sometimes even chaotic. Each ecosystem is its own world, a web of interactions within and among species and their environments. Coral reefs buzz with thousands of interdependent species — from fish and clams to algae and turtles — while temperate forests teem with woodpeckers, trees and mushrooms, with another world in the soil itself. Every life form or interaction within an ecosystem has rules and constraints. Species are woven into other species, and ecosystems are woven into other ecosystems, creating a highly dynamic, hyperconnected, global ecological network.
Often, a disturbance to one of these elements can have cascading effects. Seemingly minor changes — to nutrients, weather or disease — or the arrival of a new species can trickle and then rush through an ecosystem. Add in the significant alterations driven by humans, including deforestation, pollution, wildfire and drought, and you get an ever-evolving canvas. Nature can be so unpredictable, in fact, that it can be difficult to comprehend, never mind study.
In some cases, to understand these systems, we need to simplify them — and this is where math can help. Scientists use equations and models to distill nature's chaos and divine the fundamental rules that govern the interactions and species that compose it.
These models are, by their nature, simplifications of the real world; they don't capture all the nuances of an actual ecosystem. Still, they're important tools we can use both to improve our understanding of natural systems and to help keep them healthy.
What's New and Noteworthy A group of physicists and marine biologists broke down corals into geometric shapes to describe how the growth of individual polyps collectively forms complex structures. They thought of each individual coral as a conelike shape that extends out of a six-edged base, which they called a "hexacone." Then they identified simple mathematical rules to create a "universal model" that could explain why some corals branch, others expand into domes and still more grow into narrow columns. Such models could help keep corals alive as their environments rapidly shift due to climate change.
Some scientists predict that many ecosystems are close to a "tipping point" that, if reached as a result of disruptions such as deforestation, wildfires or climate change, could collapse them into new states from which they can't return. But modeling ecosystems and their tipping points is highly complicated, requiring countless calculations. In 2022, a group of ecologists replaced thousands of equations with a single one to explain how close ecosystems are to collapse. "With one equation, we know everything," said Jianxi Gao, a network scientist at Rensselaer Polytechnic Institute. "Before, you have a feeling. Now you have a number."
Despite making up only 3% of Earth's land area, peatlands, which form from dead vegetation that piles up for hundreds or thousands of years, store twice as much carbon as all of Earth's trees. However, predicting their underground shape and size — and therefore how much carbon they store — has proved difficult. So researchers developed a simple equation to calculate their three-dimensional shape. The new equation approximates a bog's depth with surprising accuracy using two satellite measurements: the shape of its boundary and its relative height. "It's absolutely nuts that you have this completely impenetrable, terrible, hard-to-get-through, complicated forest, and it's described by a simple equation," said Charles Harvey, an environmental engineer at the Massachusetts Institute of Technology who led the study. "I wouldn't have guessed it would work as well." | 
