Consensus Building by Committed Agents

One of the most striking features of our time is the polarization, nationally and globally, in politics and religion. How can a society achieve anything, let alone justice, when there are fundamental disagreements about what problems a society needs to address, about priorities among those problems, and no consensus on what constitutes justice itself? This paper explores a model for building social consensus in an ideologically divided community. Our model has three states: two of these represent ideological extremes while the third state designates a moderate position that blends aspects of the two extremes. Each individual in the community is in one of these three states. A constant fraction of individuals are committed agents dedicated to the third, moderate state, while all other moderates and those from either extreme are uncommitted. The states of the uncommitted may change as they interact, according to prescribed rules, at each time step with their neighbors; the committed agents, however, cannot be moved from their moderate position, although they can influence neighbors. Our main objective is to investigate how the proportion of committed agents affects the large-scale dynamics of the population: in other words, we examine the special role played by those committed to embracing both sides of an ideological divide. A secondary but equally important goal is to gently introduce important dynamical systems concepts in a natural setting. Finally, we briefly outline a model with different interaction rules, a fourth state representing those who loathe the other three states, and agents who may be committed to any one of the four states

Pattern formation in reaction-diffusion models with nonuniform growth

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth