Differential Equation Modeling as a Source of Theoretical Insight: Four Disparate Examples

We explore differential equations involving alcoholism, socialmobility, excess female mortality, and international arms competition.In each of these instances we show that the initial equation, orsystem of equations, has a sociological plausibility comparable tothat of the associated solutions; the solutions do indeed describetime-series trajectories that seem to represent important and uniquesocial processes. We argue that the central challenge of differentialequation modeling is to use experimentation to clarifyrelationships between, on the one hand, the equations andtheir coefficients and, on the other, the solutions and thetime-series orbits created by them. Such feedback interaction ofdifferential equations and their solutions appears to be the basis forfurther theoretical insight, and rapid assessments of theseinteractions are now possible largely because modern softwareencourages experimentation with many combinations of inputcoefficients.This paper expands on an argument made by Nielsen and Rosenfeld(1981, p. 161), who recommend that differential equations be interpreted in a way that emphasizes their solutions, i.e., the time-seriestrajectories of Yvalues, the orbits of Y, taken torepresent behavior of dependent variables through time. We concludethat the most edifying interpretations of differential equations focuson the equations themselves, the resulting trajectories, therelationships between equations and trajectories, and the theoreticalsignificance of all three.