摘要
When biological systems undergo hysteretic behavior of stress and recovery due to global warming or other changes in environment and resources, modeling hysteresis becomes increasingly important. A system displays hysteresis if the output depends on past history of changes in the input. Moreover, the trajectory of rate-dependent hysteresis may be viewed as a loop when the period is known. Consider the thermoregulatory response of an animal, hysteresis appears as the delay in body temperature when an animal experiences heat stress produced by elevated air temperature. If air temperature is controlled as a sinusoidal, the hysteresis loop shows an elliptical pattern. The parameters of such loops are useful for describing rhythmic biological processes. Three analytic methods, linear, nonlinear, and two-step harmonic, are developed to fit an elliptical hysteretic process. Formulas for parameters that characterize the dynamics are obtained. Statistical efficiency of parameter estimation is evaluated by simulations using both the delta method and bootstrap. Overall, two-step simple harmonic regression with bootstrap produces the most efficient estimates for parameters of the elliptical hysteresis loop. Application of this procedure to control-chamber data reveals differences between animals in response to heat stress. Keywords Circadian physiology Constrained least squares Ellipse-specific Nonlinear regression Generalized eigen-system Heat transfer Thermo-regulation