A mathematical model to study mutualism between species

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In ecosystems there are various types of ecological interactions between species. One of the best known is predation (one species eats another, simplifying), but there are other varieties, such as competition for resources, mutualism, commensalism, parasitism, etc. However, it is known that these are not permanent states, because there may be transitions between these interactions as a result of evolution. For example, a predatory relationship between species may end up becoming a mutualistic or symbiotic relationship between them in the course of evolution.

A team led by researchers from the Polytechnic University of Madrid (UPM) and the Carlos III University of Madrid (UC3M) in Spain has developed a new mathematical model to study changes in ecological interactions between populations of microbes. Researchers from the Rey Juan Carlos University of Madrid in Spain and the Universidad del Pacífico in Lima, Peru have also collaborated on the study.

One of the conclusions these researchers have reached after working with this model is that there is a tendency towards mutualism, that is, towards a relationship that provides benefits from one species to another, and vice versa.

“Ecological interactions can start out whatever, but in most cases they end up in a mutualistic relationship,” says one of the researchers, José Antonio Cuesta Ruiz, a professor in the Department of Mathematics at UC3M.

In this study, the researchers have also shown the high frequency with which these transitions occur in nature, discovering evolutionary trajectories that go through various intermediate states, according to the researchers: “Despite being a simple model, its emergent behavior is complex : shows transitions between different ecological relationships, being able to go through different stages of mutualism, predation and competition before reaching its final state”, add Javier Galeano and Juan Manuel Pastor, professors at the UPM and co-authors of the work.

Simplified concept of microbial ecosystem in which the arrows represent the subsistence relationships between the 4 species (the 4 circles). (Image: Amazings/NCYT)

For their study, the researchers have used classical models of population dynamics to which they have applied a standard technique in evolutionary theory, called adaptive dynamics. This technique allows finding dynamic equations for the parameters of the population model, which determine the nature of ecological interactions. With this, it is possible to study how these interactions change over time. “This type of models, despite being very simple models, are capable of capturing essential elements to provide mechanisms of emergent phenomena. They are very useful when studying complex systems”, says Cuesta.

The study is titled “Prevalence of mutualism in a simple model of microbial coevolution”. And it has been published in the academic journal Physical Review E. (Source: UPM)

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