Adaptive competition in social influence: a dynamical systems approach

<jats:p>Social influence plays a central role in shaping human behavior, governing how individuals adjust attitudes and actions in response to group norms. Extensive evidence shows that the type of reinforcement provided by peers (whether prosocial or antisocial) critically affects behavioral adoption. To formalize these mechanisms, we develop a ratio-dependent compartmental model describing the competition between prosocial influence and negative peer pressure within a population of fixed size. Individuals are classified as Undecided, Prosocials, or Pressurers, and transitions occur through social interaction. The model incorporates nonlinear, proportion-dependent transmission terms that reflect the comparative effectiveness of each influence rather than absolute group sizes. Analytical techniques are applied to characterize equilibrium points, determine their stability, and identify critical thresholds associated with the persistence or extinction of negative peer pressure. Parametric exploration reveals conditions under which prosocial influence becomes dominant and promotes cooperative behavioral outcomes. The proposed framework provides a mathematically tractable representation of competing social forces and offers new insights into how relative prevalence and influence strength shape population-level behavioral dynamics.</jats:p>