Resonant and Attenuant 2-Cycles In The Smith-Slatkin Population Model

We study the responses of discretely reproducing populations to periodic fluctuations in three parameters, the carrying capacity and two demographic characteristics of the species (three parameters). We prove that small 2-periodic fluctuations of the three parameters generate 2-cyclic oscillations of the population. We develop a signature function for predicting the responses of populations to 2-periodic fluctuations. Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the three parameters. Periodic environments are deleterious for populations when the signature function is negative, while positive signature functions signal favorable environments. We compute the signature function for the Smith-Slatkin model, and use it to determine regions in parameter space that are either favorable or detrimental to the species.