On the existence of stable pairing distributions

We formulate and analyze pair-formation models for multiple groups with general pairing rates and arbitrary mixing probabilities. Under the assumption of constant recruitment rates and equal average duration of all types of partnerships, we have shown that the dynamics are relatively simple because of the monotonicity properties of the dynamical system associated with the pairing/mixing of heterogeneous populations of male and female individuals. In fact, we have shown that the corresponding asymptotic stable paired distribution is given precisely by the asymptotic values of the matrices that prescribe the mixing/contact structure. In other words, if the sizes of the mixing subpopulations of males and females are asymptotically constant and if the average durations of partnerships are about the same regardless of type, then the matrices that describe the mixing between subpopulations also characterize the distribution of paired types. Alternatively, if the distribution of the average duration of relationships between individuals has a large variance then it may be impossible to detect any relationship between the mixing/contact structure and the observed distribution of paired types. The study of models with constant per-capita recruitment rates give rise to homogeneous systems of degree one. The analysis of the dynamics of pairs for models with exponentially growing populations of singles is complicated. So far, we are only able to classify the stability of all non-strictly positive boundary exponential solutions. From our incomplete analysis, it is not possible to detect necessary and sufficient conditions for the existence and stability of strictly interior exponential solutions. We cannot rule out the possibility of oscillations. The mathematical problems associated with the stability of exponential solutions of dynamical systems of degree one are of relevance in demography, epidemiology, and population dynamics.On leave from University of Alabama in Huntsville