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

Abstract

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 higherorder 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.

Bridging parametric and nonparametric measures of species interactions unveils new insights of non‐equilibrium dynamics

“…However, the computational methods emerging from the parametric approach are difficult to distinguish (e.g. functional responses and higher‐order interactions) (AlAdwani and Saavedra 2019). Yet, relying upon the computational feasibility of the nonparametric approach (Deyle et al 2016, Martin et al 2018, Cenci and Saavedra 2019), we may be able to distinguish the nature of species interactions acting on a system.…”

“…Importantly, the differences between approaches (measures) can offer an opportunity to gain further insights about non‐equilibrium ecological dynamics and higher‐order interactions without modeling them (AlAdwani and Saavedra 2019). For example, focusing on dynamics and building from the classic complexity–stability debate (May 1972), it is assumed that a community can be dynamically stable only if most of the constant, direct, intraspecific terms are negative ( a ii < 0), i.e.…”

“…Both measures have their own strengths and weaknesses: within parametric approaches, measures have a mechanistic or phenomenological interpretation of a direct effect between species, but the magnitude and dimension of such parametric measures are model‐dependent (Cenci and Saavedra 2018a, AlAdwani and Saavedra 2019, Letten and Stouffer 2019). Instead, within the nonparametric approach, measures can be estimated directly from data (such as time series of species abundance) with statistical methods, but they are often hard to be biologically interpreted (Sugihara and May 1990, Ushio et al 2018, Cenci et al 2019).…”

Towards a system-level causative knowledge of pollinator communities

“…In fact, it has already been proved that it is impossible to write analytically (a closed-form algebraic solution) a polynomial system with degree five or higher with arbitrary coefficients (unknown values) [44]. Note that a simple 3-species system (e.g., two pollinators and one plant) with Type II functional responses (i.e., a non-linear response such as those observed in density-dependent processes arising from competition for floral resources or pathogen spillover) can already form a polynomial of degree eight [45]. This intractability of complex models implies that if the majority of their parameter values are not known a priori (reducing the system to a polynomial of degree four or lower), these models can only be used numerically (simulations).…”

“…Then, the problem that arises is that it becomes computationally impossible to differentiate the role played by each parameter (e.g., interactions, environmental conditions) in the solutions of the system [40]. While studies have attempted to tackle this complexity by using statistical methods such as Akaike Information Criterion [46], the number of solutions of a polynomial system does not necessarily depend on the number of parameters but on the polynomial degree [45]. Hence, it is not just the lack of data that limits the use of complex models, as it can be perceived [47], it is their intractability, especially in high-dimensional systems [40].…”