Abstract: In this work we study numerical homotopy continuation, a method used for solving nonlinear equations and nonlinear systems of equations. We focus on detecting and resolving error in a fixed step predictor-correct method for numerically solving univariate polynomials. We verify the homotopy paths with a Lipschitz condition to detect path jumping that leads to false multiplicities of roots and missed roots; we show that mutual path jumping cannot occur. At path ends we may obtain spurious roots, which are easy to detect for univariate polynomials when we check for non-zero residuals at the final path points. We show that pseudozeros---roots of nearby polynomials---provide another means of understanding error at path end points and give insight on how to optimize a homotopy continuation algorithm for a specific problem. We demonstrate that a change of basis can improve homotopy continuation results when finding ill-conditioned roots.

Abstract: Wild turkeys (Meleagris gallopavo) were over-hunted and faced near extinction in the United States in the 1930s. Hunting practices have since adapted, but the US still experiences unexplained turkey population declines. Nest success is a primary driver of annual population change. Female turkeys (hens) are the sole caretakers of the nest, so factors affecting their behaviour influence population dynamics. Little is known about the effect of hunting on hens, as most hunting-related studies of turkeys focus on males. To investigate the effect of hunting on hens during their incubation period, we analyze GPS movement data from over 600 hens in sites across the southeastern US. Hunting occurs in most sites and is prohibited in one. Hen behaviour can be inferred from location data. We are interested in the ways in which hunting affects both the behaviour and physiological state of the hens. We develop a hidden Markov model, treating physiological state as the hidden variable and behaviour as the observed variable. The model assumes that behaviour varies with physiological state and hunting presence. We found that hunting has an effect on hen behaviour, and this effect changes depending on their physiological state. Further, the model predicts that, on average, hens in hunted sites leave their nest for longer consecutive periods than hens in non-hunted sites. Results suggest that hens in hunted sites place their nest at greater risk and prioritize their own survival. These results should be considered when developing hunting policies.