Observed species interactions, such as in pollination, food webs, and parasitism, arise by a combination of individual species densities, trait-matching, and neutral effects. In our work, we have investigated the use of optimal transportation theory to model species interaction networks. In optimal transportation, one tries to find a coupling between a set of sources and a set of sinks by minimizing a linear cost function. This theory has been around since the Second World War, where the mathematician Kantorovich developed it for military logistics. Recent variations also maximize the entropy of the coupling, in addition to minimizing a cost. These formulations are both conceptually more sound and admit to highly efficient algorithms for computing the solution, leading to a recent surge of optimal transportation in computer vision machine learning.
We translated ecological interaction modeling as finding a coupling between two trophic levels maximizing a combination of utility and entropy, with the marginal species abundances as constraints. This framework relates to serval established ecological theories, such as neutrality, optimal foraging, and MaxEnt. Furthermore, it allows one to anticipate how a species interaction networks will rewire themselves, given that the species distributions change.
Results/Conclusions
We formulated species interaction modeling as an optimization problem where one can interpolate between utility-driven versus entropy-driven. We distinguished four settings, depending on whether one fixes two, one, or none of the species abundances. Besides, we provided an efficient way of estimating the utility matrix from one or several observed couplings.
Experimental results show that optimal transportation theory can predict species couplings more accurately, as measured by Kullback-Leibler divergence, compared to a neutral model. We showed how the effect of honeybee spillover could be modeled based on observed networks at a different time. An experiment on 52 host-parasites networks over Eurasia showed that the method could generalize over space as well.
Optimal transportation is an elegant mathematical framework that can yield new insights on ecosystems. Besides, it leads to a new class of predictive models that separate abundance effects from trait-matching.