Transitions in consumption behaviors in a peer-driven stochastic consumer network

Abstract We study transition phenomena between attractors occurring in a stochastic network of two consumers. The consumption of each individual is strongly influenced by the past consumption of the other individual, while own consumption experience only plays a marginal role. From a formal point of view we are dealing with a special case of a nonlinear stochastic consumption model taking the form of a 2-dimensional non-invertible map augmented by additive and/or parametric noise. In our investigation of the stochastic transitions we rely on a mixture of analytical and numerical techniques with a central role given to the concept of the stochastic sensitivity function and the related technique of confidence domains. We find that in the case of parametric noise the stochastic sensitivity of fixed points and cycles considered is considerably higher than in the case of additive noise. Three types of noise induced transitions between attractors are identified: (i) Escape from a stochastic fixed point with converge to a stochastic k-cycle, (ii) escape from the stochastic k-cycle to a stochastic fixed point, and (iii) cases in which the consumption process moves between the respective stochastic attractors for ever. The noise intensities at which such transitions are likely to occur tend to be smaller in the case of parametric noise than with additive noise.