Architecture of attractor determines dynamics on mutualistic complex networks

A mathematical system of differential equations for the modelization of mutualistic networks in Ecology has been proposed in Bastolla et al. (2007). Basically, it is studied how the complex structure of cooperation interactions between groups of plants and pollinators or seed dispersals affects to the whole network. In this paper we prove existence and characterization of the global attractor associated to the model. The description of the geometrical internal structure of the attractor becomes the proper complex network describing all the possible future scenarios of the phenomena. The arguments show a Morse Decomposition of the attractors, leading to the existence of a global Lyapunov function for the associated gradient semigroup. In particular, we are able to prove topological structural stability of the system, i.e., the associated attracting complex networks are robust under (autonomous and non-autonomous) perturbation of parameters.