Minkowski valuations under volume constraints

Abardia-Evéquoz, Judit; Colesanti, Andrea; Gómez, Eugenia Saorín

doi:10.1016/j.aim.2018.05.033

“…The copyright holder for this preprint (which was not this version posted April 1, 2019. ; https://doi.org/10.1101/595140 doi: bioRxiv preprint of the Minkowski summation and those of forming convex hulls are commuting (Abardia-Evquoz et al, 2018), that is…”

Section: S1mentioning

confidence: 99%

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

AlAdwani

¹

,

Saavedra

²

2019

Mathematical Biosciences

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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. 2.

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“…Furthermore, let us define e i to be the point in space with its i th component to be 1 and the rest are all zeros. It is also important to note that the operations S1 of the Minkowski summation and those of forming convex hulls are commuting (Abardia-Evquoz et al, 2018), that is…”

Section: Supplementary Informationmentioning

confidence: 99%

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

AlAdwani

¹

,

Saavedra

²

2019

Preprint

View full text Add to dashboard Cite

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.

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“…In the 1930s, Blaschke started a systematic investigation, and then Hadwiger obtained the famous Hadwiger's characterization theorem. The Hadwiger's characterization theorem provides the connection between rigid motion invariant set functions and symmetric polynomials (see [4]) for further results and generalizations, see [1], [2], [3], [5], [20], [29]. The following Minkowski endomorphism was introduced by Schneider in [21].…”

Section: Introductionmentioning

confidence: 99%