Stochastic nonlinear dynamics of interpersonal and romantic relationships

Current theories from biosocial (e.g., the role of neurotransmitters in behavioral features), ecological (e.g., cultural, political, and institutional conditions), and interpersonal (e.g., attachment) perspectives have grounded interpersonal and romantic relationships in nor-mative social experiences. However, these theories have not been developed to the point of providing a solid theoretical understanding of the dynamics present in interpersonal and romantic relationships, and integrative theories are still lacking. In this paper, mathematical models are used to investigate the dynamics of interpersonal and romantic relationships , via ordinary and stochastic differential equations, in order to provide insight into the behaviors of love. The analysis starts with a deterministic model and progresses to non-linear stochastic models capturing the stochastic rates and factors (e.g., ecological factors, such as historical, cultural and community conditions) that affect proximal experiences and shape the patterns of relationship. Numerical examples are given to illustrate various dynamics of interpersonal and romantic behaviors with particular emphases placed on sustained oscillations and transitions between locally stable equilibria that are observable in stochastic models (closely related to real interpersonal dynamics), but absent in determin-istic models. Interpersonal relationships appear in many contexts, such as in family, kinship, acquaintance, work, and clubs, to name a few. The manifestation of interpersonal relationships in society comes in many forms ranging from romantic, parent–child, friendships, comradeship, casual, friend-with-benefits, soul-mates, dating to more recently Internet relationships. The most intriguing of all these interpersonal relationships, which is also a dominant phenomenon and fundamental in human social life and interaction, is romantic relationship [1,2]. Romantic relationships refer to the mutually ongoing interactions among two or more individuals. Recent works show that romantic relationships are more common among adolescents than has previously been presumed, with more than half of adolescents in the United States being involved in some form of romantic relationships [3,4]. More than 70% of high school and college students report having had a special romantic relationship in the previous years, and also report more frequent interactions with romantic partners than with parents, siblings, and/or friends [4,5]. For adult, the study of romantic behaviors may provide invaluable insight of why majority of romantic relationships fail or do not make it to engagement and/or marriage [6]. Surra and Hughes [7] found that more than half (54%) of couples in their studies exhibit unpredictable and non-linear relational trajectories involving large number of turning and tipping points. Partners identified events such as new rivals, unresolved …