Reduction Properties in Competitive Logistic Networks with Adaptation
Department of Mathematics, Politecnico di Torino
Last modified: September 14, 2007
A general N-node network is considered for which, in absence of interactions, each node is governed by a logistic equation. Interactions among the nodes take place in the form of competition, which also includes adaptive abilities through a (short term) memory effect. As a consequence the dynamics of the network is governed by a system of NxN nonlinear ordinary differential equations. As a first step, equilibria and their stability are investigated analytically for the general network in dependence of the relevant parameters, namely the strength of competition, the adaptation rate and the network size. The existence of classes of invariant subspaces, related to symmetries, allows the introduction of reduced models, where N appears as a parameter, which give full account of existence and stability for the equilibria in the network, Reduced models are found effective also in describing time-dependent regimes, both in the form of periodic oscillations and chaotic behavior and with remarkable properties of synchronization.