Tommaso Tonolo — GSSI (Gran Sasso Science Institute) #
Generalized Lotka Volterra model on the Bethe lattice #
Recent times have witnessed a burst of activity on the application of equilibrium  and non-equilibrium statistical mechanics to study the behaviour of large ecosystems, in particular their stability and the nature of their equilibria.  In particular, many results on the coexistence of many species have been obtained using the Generalized Lotka-Volterra model. The latter, under appropriate hypothesis  on the shape of the interaction matrix between species and the stochasticity concourring to  the dynamics (demographic noise), allows to recast the dynamical stability problem  in terms of equilibrium statistical mechanics. We present here for the first time results on  the equilibrium statistical mechanics of the Generalized Lotka-Volterra model with  an interaction network between species which is sparse (Bethe lattice). Our analysis, at variance with the standard approach which makes use of a dense interaction network, reveals novel and highly non-trivial heterogeneity effects in the populations distributions, as for instance strong deviations from Gaussianity when increasing the heterogeneity of intra-species interactions. These results are in accordance with data from real ecosystems and also with other different models for ecological communities, as in [J. Grilli, Nature communications 11(1), 4743 (2020)].  In this talk I will review how the effective Hamiltonian for species interactions derived in [A. Altieri, F. Roy, C. Cammarota, G. Biroli, Phys. Rev. Lett. 126, 258301 (2021)] can be used to generate local marginals for populations distribution abundance when interactions are sparse, and I will present the main results obtained varying both the temperature (strength of demographic noise) and the heterogeneity of species interactions.