Extinction of Populations and the Schr\"{o}dinger Equation: Analytic Calculations for Abrupt Transitions

We study bifurcations in a spatially extended nonlinear system representing population dynamics with the help of analytic calculations based on the time-independent Schr\"{o}dinger equation for a quantum particle subjected to a uniform gravitational field. Despite the linear character of the Schr\"{o}dinger equation, the result we obtain helps in the understanding of the onset of abrupt transitions leading to extinction of biological populations. The result is expressed in terms of Airy functions and sheds light on the behavior of bacteria in a Petri dish as well as of large animals such as rodents moving over a landscape.