A nonlinear threshold model for community response to environmental hazards

Abstract

In Predescu et al. 2002, the authors present a model for the interaction between the number of mosquito breeding sites in a community and that community's “level of consciousness” of the environmental problem. We build upon this concept here by generalizing the model and incorporating a threshold relationship between the environmental hazard, , and community response, ; the hazard must exceed a critical level ω before a community intervenes.

The model exhibits three different regimes of behavior as a function of the parameters: a zero-consciousness equilibrium, a positive consciousness (and low hazard) regime, and an oscillatory intermediate regime. This paper establishes global and local stability results, characterizations of trajectories and invariant regions of phase space for each of the regimes, and presents a complete characterization of the behavior of the special case in which community awareness increases at a constant rate once the threshold is reached.

Keywords:

• Difference equations
• Stability
• Oscillations
• Threshold
• Nonlinear
• Step function

Keywords:

• 39A11
• 40A05

Acknowledgements

The model presented in this paper builds on the conceptual foundation established in Predescu et al. [6], and subsequent discussions with R. Levins and T. Awerbuch-Friedlander have been both insightful and encouraging. The commentary of G. Verghese significantly improved the manuscript. This work was supported by the Martin Family Fellowship for Environmental Sustainability at MIT.