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Mathematician Says Relationships Have a Critical Threshold—and Crossing It Can Predict When Love Falls Apart
Equations once used to predict whether populations of rabbits would boom, stabilize, or collapse are now being applied to something potentially even more complicated: romantic relationships.
According to mathematician Laurent Pujo-Menjouet of Claude Bernard University Lyon 1, the forces that determine whether a couple survives life’s inevitable stresses can be studied much like other complex systems. Attraction can strengthen or fade, outside pressures can push a relationship away from stability, and perhaps most importantly, there may be a critical mathematical threshold beyond which recovery becomes increasingly difficult.
In his new book, Love! Valour! Equations! The Mathematics of the Couple, published by CRC Press, Pujo-Menjouet draws on mathematical ecology, psychology, literature, and even film to explore why some couples weather years of conflict, stress, children, illness, and financial hardship while others gradually drift apart.
The goal isn’t to produce an equation that predicts who will fall in love with whom. Instead, Pujo-Menjouet treats relationships as dynamical systems—systems that change over time, respond to disturbances, and can move toward either stability or decline.
“There is no mysterious enchantment or magic potion. By now, that would be known,” Pujo-Menjouet writes in the book, which draws on mathematics, psychology, literature, and film. “Is all hope lost then? Certainly not. That is precisely where everything becomes interesting.”
From Rabbits to Relationships
The mathematical models Pujo-Menjouet uses originated in ecology. His work uses Malthusian growth models and the Lotka-Volterra equations, originally developed to describe population dynamics in animals such as rabbits. He also studies the Allee effect, a phenomenon that explains how populations struggle to recover if they drop below a particular size.
Pujo-Menjouet adapts these concepts to relationships, using the idea of a critical threshold to examine when a couple may become increasingly vulnerable to separation. He also draws on the research of psychologist John Gottman, who showed that relationship behaviors can be quantified as data.
“Just as we can predict population growth or endangered species, we can model the evolution of feelings within a couple,” Pujo-Menjouet explains. “We’re not trying to predict who you’ll fall in love with, we’re trying to understand why some couples remain stable despite life’s inevitable shocks while others slowly drift apart.”
A Threshold for Falling Apart
The basic idea is simple: a relationship may have a critical threshold, or minimum level of stability needed to keep the bond intact. Pujo-Menjouet’s model considers three main forces: mutual attraction, the gradual weakening of feelings over time, and each partner’s emotional response to the other.
“If your feelings fall below this threshold and stay there, the mathematics predicts an inevitable decline toward separation,” Pujo-Menjouet says. “However, understanding this threshold gives couples a powerful tool: they can recognize danger zones and take corrective action.”
These models emphasize how couples recover from drifting apart. The key question is not whether couples experience conflict, but whether they have the resilience to restore balance afterward.
“Life will always send its wolves: stress, illness, work, children, financial difficulties,” Pujo-Menjouet says, “mathematics helps us understand how to build the brick house rather than the straw one.”
Could AI Become Love’s Early-Warning System?
Researchers such as Pujo-Menjouet are considering whether they could combine these mathematical models with artificial intelligence to create an early-warning system for relationships. Such a system could detect the signs of instability before they become apparent to the people involved.
Pujo-Menjouet compares this method to a GPS for relationships, showing where a couple currently is, where its dynamics may be heading, and the various ways they could arrive at a healthier place. Although the models don’t decide who somebody will love, they can help explain how relationships evolve and what actions might help couples maintain their connection. Early detection of problems would allow partners to make changes before reaching a stage where recovery becomes unlikely.
Why Love Resists a Formula
Pujo-Menjouet admits that present-day mathematical models have limitations. Most of these models are based on data from Western relationships. In contrast, the factors that impact whether a relationship succeeds or fails can differ from one culture, religion, or generation to another. He points out that the model must account for additional factors to be applicable everywhere.
Human behavior also introduces unpredictability that mathematical models cannot fully account for. People can change their habits, strengthen their commitment, seek therapy, or otherwise alter the dynamics of a relationship in ways the equations cannot anticipate. The equations identify tendencies rather than determine an unavoidable outcome.
“Mathematics can illuminate patterns and provide guidance,” Pujo-Menjouet says, “but ultimately, the success of a relationship depends on two people choosing, every day, to nurture their connection. The equations can show you the path, but you still have to walk it together.”
Austin Burgess is a writer and researcher with a background in sales, marketing, and data analytics. He holds an MBA, a Bachelor of Science in Business Administration, and a data analytics certification. His work focuses on breaking scientific developments, with an emphasis on emerging biology, cognitive neuroscience, and archaeological discoveries.
