Narrow Results

We extend the two-dimensional model of drug use introduced in [Behrens et al., 1999, 2000, 2002] by considering two additional states that represent in more detail newly initiated ("light") users' response to the drug experience. Those who dislike the drug quickly "quit" and briefly suppress initiation by others. Those who like the drug progress to ongoing ("moderate") use, from which they may or may not escalate to "heavy" or dependent use. Initiation is spread contagiously by light and moderate users, but is moderated by the drug's reputation, which is a function of the number of unhappy users (recent quitters + heavy users). The model reproduces recent prevalence data reasonably well from the U.S. cocaine epidemic, with one pronounced peak followed by decay toward a steady state. However, minor variation in parameter values yields both long-run periodicity with a period akin to the gap between the first U.S. cocaine epidemic (peak ~1910) and the current one (peak ~1980), as well as short-run periodicity akin to that observed in data on youthful use for a variety of substances. The combination of short- and long-run periodicity is reminiscent of the elliptical burstors described by Rubin and Terman [2002]. The existence of such complex behavior including cycles, quasi periodic solutions, and chaos is proven by means of bifurcation analysis.

Minimal models composed of two ordinary differential equations are considered in this paper to mimic the dynamics of the feelings between two persons. In accordance with attachment theory, individuals are divided into secure and non-secure individuals, and synergic and non-synergic individuals, for a total of four different classes. Then, it is shown that couples composed of secure individuals, as well as those composed of non-synergic individuals can only have stationary modes of behavior. By contrast, couples composed of a secure and synergic individual and a non-secure and non-synergic individual can experience cyclic dynamics. In other words, the coexistence of insecurity and synergism in the couple is the minimum ingredient for cyclic love dynamics. The result is obtained through a detailed local and global bifurcation analysis of the model. Supercritical Hopf, fold and homoclinic bifurcation curves are numerically detected around a Bogdanov–Takens codimension-2 bifurcation point. The existence of a codimension-2 homoclinic bifurcation is also ascertained. The bifurcation structure allows one to identify the role played by individual synergism and reactiveness to partners love and appeal. It also explains why ageing has a stabilizing effect on the dynamics of the feelings. All results are in agreement with common wisdom on the argument. Possible extensions are briefly discussed at the end of the paper.

In this paper, we analyse a differential game describing the interactions between a potential offender and the law enforcement agency. We assume that both players want to maximise their welfare expressed in monetary units, and compare the results obtained by applying the Nash equilibrium concept under symmetric with that under asymmetric information. The comparison reveals that under asymmetric information the offence rate is lower, due to the deterrence caused by the activities of the law enforcement agency. Both players' controls start at a steady state value and stick to it until close to the end of the planning horizon, when they leave the steady state to take into account the scrap value; this can be interpreted as a turnpike property of Nash equilibria. Furthermore, a sensitivity analysis is carried out. Among others, it turns out that a myopic offender tends to a higher offence level.