Mathematical Modeling of Human Behaviors During Catastrophic Events: Stability and Bifurcations

Abstract

The aim of this paper is to present some mathematical results concerning the PCR system (Panic-Control-Reflex), which is a model for human behaviors during catastrophic events. This model has been proposed to better understand and predict human reactions of individuals facing a brutal catastrophe, in a context of an established increase of natural and industrial disasters. After stating some basic properties, that is positiveness, boundedness, and stability of the solutions, we analyze the transitional dynamic. We then focus on the bifurcation that occurs in the system, when one behavioral evolution parameter passes through a critical value. We exhibit a degeneracy case of a saddle-node bifurcation, in a larger context of classical saddle-node bifurcations and saddle-node bifurcations at infinity, and we study the inhibition effect of higher order terms.