  We study a mathematical model inspired by a common scenario in northern
Patagonia consisting of humans and their livestock living together with native
wildlife. The main production system in this area is sheep farming, which due
to wrong historical management has led to desertification of the habitat, with
impact on both native wildlife and livestock. Particularly the largest native
herbivore, the guanaco, has reflected in their numbers this alteration induced
in the environment. We analyze a mathematical model that captures the main
characteristics of the interaction between sheep and guanaco: the hierarchical
competition and the advantage granted by humans to the herds, and also
incorporates a dynamic for the habitat. Using the metapopulation formalism, the
trophic web of two herbivores is extended over a patchy landscape that
considers two characteristic times for the dynamics of the resource. Our study
stresses the dependence of the metapopulations dynamics on the recovery time of
the resource. These results are backed up by a deterministic mean field model
which shares some similarities with the stochastic and spatially extended one.
We find different regimes depending on the parameters considered: coexistence
of both species, survival of a single species and extinction of the other, and
extinction of both. Remarkably, in some regions of parameters space we detect
the presence of periodical spatio-temporal patterns, with persistent
oscillations of constant amplitude. Based on these results, we perform a
characterization of the observed scenarios in order to gain insight about the
system.
