Professor Laurent Pujo-Menjouet of the University of Lyon I and other researchers around the world have been developing mathematical models that treat romantic relationships as dynamical systems. In other words, they study how feelings evolve, respond to outside pressures, and either recover or decline over time.

In his book Love! Valour! Equations!, which brings together mathematics, psychology, literature, cinema and everyday life, Pujo-Menjouet writes: "There is no mysterious enchantment or magic potion. By now, that would be known. Is all hope lost then? Certainly not. That is precisely where everything becomes interesting."

From Population Models to Romantic Relationships

The mathematical path to understanding relationships starts in an unexpected place: with rabbits.

Some of the ideas now being applied to couples were originally developed to study changes in animal populations. Malthusian models and predator-prey equations, for example, have been used to examine the spread of Australian rabbits and the shifting populations of Canadian lynxes and snowshoe hares.

Another important idea is the Allee effect, which describes what happens when a population drops below a critical level. Below that threshold, survival becomes much more difficult. Above it, the population has a better chance of growing and remaining stable.

Researchers have found that a similar threshold concept can be useful when thinking about romantic relationships.

"Just as we can predict population growth or endangered species, we can model the evolution of feelings within a couple," explains Pujo-Menjouet. "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."

The framework draws on a long history of mathematical work. That includes Thomas Malthus's population models from the 18th century, the Lotka-Volterra predator-prey equations developed in the 1920s, and later efforts by psychologist John Gottman, economist José Manuel Rey, physicist Sergio Rinaldi, and mathematician Steven Strogatz, who was among the first to apply dynamical systems theory directly to romantic relationships.

A Mathematical Threshold for Relationship Stability

Psychologist Dr. John Gottman, widely known as 'Doctor Love', helped show that relationship behavior could be analyzed mathematically. By coding interactions between couples and translating those exchanges into graphs, his work demonstrated that mathematical models could help predict relationship outcomes.

More recent approaches have added additional layers of complexity. One example is delayed reactions (introduced by Polish researchers Bielezyk, Forys, and Marek Bodnar), which account for the fact that people do not always respond immediately during disagreements. The amount of time someone takes to react or reflect can either calm a conflict or make it worse.

Despite the growing sophistication of the models, Pujo-Menjouet says one central idea remains relatively simple. Each relationship may have a critical threshold representing the minimum level of relationship strength needed for the bond to remain stable.

"If your feelings fall below this threshold and stay there, the mathematics predicts an inevitable decline toward separation," says Pujo-Menjouet. "However, understanding this threshold gives couples a powerful tool: they can recognize danger zones and take corrective action."

His team has also created models designed to account for everyday disruptions, including stress, arguments, and pressures coming from outside the relationship. The key issue is not whether those disruptions happen, but how effectively a couple rebounds from them.

"The important question isn't whether couples argue," says Pujo-Menjouet. "It's whether they have enough resilience to return to equilibrium afterwards."

Three Forces That Shape Long-Term Love

The model highlights three major factors that help determine whether a relationship remains stable over time.

The first is the mutual attraction between partners. The second is the gradual weakening of feelings that can naturally occur over time. The third is how each person responds emotionally to the other.

Together, these forces influence how resilient the relationship becomes when it encounters stress or conflict.

Pujo-Menjouet compares that resilience to the story of The Three Little Pigs: "Life will always send its wolves: stress, illness, work, children, financial difficulties. Mathematics helps us understand how to build the brick house rather than the straw one."

Could AI Become a Relationship Early Warning System?

Researchers are also considering how these mathematical models might be developed further. One possibility is combining traditional mathematical modeling with artificial intelligence to create an early warning system for couples.

Such a system could potentially detect when a relationship is moving toward an unstable state and provide enough warning for the couple to change course.

Pujo-Menjouet says this could give partners "time to course-correct."

"Think of it as a relationship GPS," says Pujo-Menjouet. "It shows you where you are, where you're heading, and suggests routes to reach your desired destination. The models don't tell you who to love, it helps explain how love evolves and what couples can do to protect it."

Why Love Cannot Be Reduced to an Equation

The researchers also emphasize that these models have important limitations.

Most have been developed largely from patterns observed in Western relationships. The reasons relationships succeed or fail can vary greatly across cultures, religions, socioeconomic circumstances, and generations. Building a model that works universally would therefore require far more complexity.

Human behavior also does not always follow predictable rules. People can act irrationally, change their habits, seek therapy, strengthen their commitment, or grow in ways that a mathematical model cannot fully anticipate.

For that reason, the equations are designed to identify tendencies rather than predict an unavoidable future. Couples can still behave in ways that defy what the models suggest.

"Mathematics can illuminate patterns and provide guidance," Pujo-Menjouet explains, "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."