Dynamical Models of Dyadic Interactions with Delay
When interpersonal interactions between individuals are described by the (discrete or continuous) dynamical systems, the interactions are usually assumed to be instantaneous: the rates of change of the actual states of the actors at given instant of time are assumed to depend on their states at the same time. In reality the natural time delay should be included in the corresponding models. We investigate a general class of linear models of dyadic interactions with a constant discrete time delay. We prove that in such models the changes of stability of the stationary points from instability to stability or vice versa occur for various intervals of the parameters which determine the intensity of interactions. The conditions guaranteeing arbitrary number (zero, one ore more) of switches are formulated and the relevant theorems are proved. A systematic analysis of all generic cases is carried out. It is obvious that the dynamics of interactions depend both on the strength of reactions of partners on their own states as well as on the partner's state. Results presented in this paper suggest that the joint strength of the reactions of partners to the partner's state, reflected by the product of the strength of reactions of both partners, has greater impact on the dynamics of relationships than the joint strength of reactions to their own states. The dynamics is typically much simpler when the joint strength of reactions to the partner's state is stronger than for their own states. Moreover, we have found that multiple stability switches are possible only in the case of such relationships in which one of the partners reacts with delay on their own state. Some generalizations to triadic interactions are also presented.