Opinion dynamics with Lotka-Volterra type interactions

Abstract

We investigate a class of models for opinion dynamics in a population with two interacting families of individuals. Each family has an intrinsic mean field "Voter-like" dynamics which is influenced by interaction with the other family. The interaction terms describe a cooperative/conformist or competitive/nonconformist attitude of one family with respect to the other. We prove chaos propagation, i.e., we show that on any time interval [0, T ], as the size of the system goes to infinity, each individual behaves independently of the others with transition rates driven by a macroscopic equation. We focus in particular on models with Lotka-Volterra type interactions, i.e., models with cooperative vs. competitive families. For these models, although the microscopic system is driven a.s. to consensus within each family, a periodic behaviour arises in the macroscopic scale. In order to describe fluctuations between the limiting periodic orbits, we identify a slow variable in the microscopic system and, through an averaging principle, we find a diffusion which describes the macroscopic dynamics of such variable on a larger time scale.

Delay-Induced Periodic Behaviour in Competitive Populations

“…For the second case we show that the microscopic process reaches a small interval containing 1/2 in a finite time and remains close to it for a time exponentially large in N . Theorem 2.1 For every > 0 and N sufficiently large, (i) if φ is strictly increasing, then ∀ m < 1 2 there exist C > 0 and T > 0 such that…”

“…Then, observing that, for any m(0) < 1 2 we have lim t→+∞ m(t) = 0 and choosing T such that m(T ) < 4 , we can use the same argument as above and, for any m = m(0) < 1 2 , we get…”

“…. , 1 2 , 1 2 ) in R 2n+3 . The points x 0 and x 1 are traps for the microscopic process, while the equilibrium point x * appears only in the macroscopic model.…”

“…At the same time, they may adopt irrational behaviours due to social influence or to an imitation mechanism, adapting their choices to those of their neighbours [4,25]. A similar situation may occur in the context of opinion dynamics or behavioral economics (e.g., [1,13,14,16,24]). We shall compare this model to the one where agents are cooperative and a fast convergence to consensus occurs.…”

Rhythmic behavior of an Ising Model with dissipation at low temperature

“…cells or individual animals) but it is the result of the interactions within the network. In recent years many stylized models have been proposed to identify possible origines of time-periodicity (Giacomin and Poquet, 2015;Lindner et al, 2004;Scheutzow, 1985a,b;Shinomoto and Kuramoto, 1986;Aleandri and Minelli, 2019). Existing examples are mostly restricted to mean-field interaction, i.e., the interaction network is the complete graph (Andreis and Tovazzi, 2018;Dai Pra et al, 2013;Collet et al, 2015Collet et al, , 2016Jahnel and Külske, 2014;Luçon and Poquet, 2020+;Luçon and Poquet, 2020).…”