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Résumé |
We introduce a spatially-extended ecosystem of Generalized Lotka-Volterra type,
where species can diffuse and interactions are nonlocal. We compute the criterion
for the loss of stability of the spatially homogeneous ecosystem, and we show that
the stability of the uniform state crucially depends on the most abundant species,
and on the interplay between space exploration during the time scale of
reproduction and the interaction range. Focusing on the spectrum of the
interaction matrix weighted by the species abundances, we identify a Baik-Ben
Arous-Péché transition that translates into a transition in the final patterns of
the species repartition. Finally assuming that the disorder is small, we exhibit
an explicit solution of the dynamical mean-field equation for the species density,
obtained as the fixed point of nonlocal Fisher-Kolmogorov-Petrovski-Piskounov
equations. Our work paves the way of future combined approaches at the frontier of
active matter and disordered systems, with the hope of better understanding
complex ecosystems like bacterial communities. |
