Taylor's law and abrupt biotic change in a smoothly changing environment

Taylor's law (TL), widely verified in empirical ecology, states that the variance of population density approximates a power function of the mean population density, with exponent denoted b. A model of multiplicative increments in population density, where the increments are determined by a Markovian environment, predicts TL with an explicit formula for b. We give a simple theoretical example where, unexpectedly, smooth changes in environmental autocorrelation lead to an abrupt, infinite discontinuity in b. As the daily probability of change in environmental state increases from 0 to 1, b rises from 2 slowly at first, then explodes to +∞ when the population becomes critical, drops to -∞, and rises again to 2. In this model, an exponent b of large magnitude (positive or negative) signals the proximity of a population's criticality and of a singularity in b. A comparable real-world singularity in b could adversely affect fisheries, forestry, agriculture, conservation, and public health.