Is the addition of higher-order interactions in ecological models increasing the understanding of ecological dynamics?

Recent work has shown that higher-order interactions can increase the stability, promote the diversity, and better explain the dynamics of ecological communities. Yet, it remains unclear whether the perceived benefits of adding higher-order terms into population dynamics models come from fundamental principles or a simple mathematical advantage given by the nature of multivariate polynomials. Here, we develop a general method to quantify the mathematical advantage of adding higher-order interactions in ecological models based on the number of free-equilibrium points that can be engineered in a system (i.e., equilibria that can be feasible or unfeasible by tunning model parameters). We apply this method to calculate the number of free-equilibrium points in Lotka-Volterra dynamics. While it is known that Lotka-Volterra models without higher-order interactions only have one free-equilibrium point regardless of the number of parameters, we find that by adding higher-order terms this number increases exponentially with the dimension of the system. Our results suggest that while adding higher-order interactions in ecological models may be good for prediction purposes, they cannot provide additional explanatory power of ecological dynamics if model parameters are not ecologically restricted.