Use of Piecewise Nonlinear Models to Estimate Variable Size‐Related Mortality Rates

Abstract

AbstractI present a method and examples of a piecewise function in nonlinear regression to estimate size‐related differential mortality using age‐ and length‐based models for data collected from a Tennessee River (Alabama) population of blue catfish Ictalurus furcatus. The model is similar to catch curve regression but computes two slope coefficients that can be used to estimate mortality; these slopes meet at a joint (or knot) that estimates the age or length at which differences in mortality occur. The model is described as loge(Ni) = b0 − (b1 + b2)ti, where Ni is the number of fish collected for the ith age or length interval; ti is the age or relative age of the ith age or length interval; b0 is the intercept coefficient; b1 is the slope coefficient and is equal to b*1 when ti is less than or equal to the knot and 0 otherwise; b2 is a second slope coefficient and is equal to b*1 − b*2 when ti is greater than the knot and 0 otherwise; and b*1 and b*2 are estimated by the model. For these data, fish ranged in age from 3 to 25 years and ranged in total length (TL) from 301 to 1,291 mm. With this piecewise nonlinear approach, the fit between age or length and loge(Ni) improved over that determined with linear catch curves. For both age‐ and length‐based structure data, annual mortality for small blue catfish ranged from 0.41 to 0.49 for younger and smaller fish (age ≤ 10 years; TL ≤ 684 mm). For larger, older fish (age = 11–25 years; TL > 685 mm), annual mortality estimates ranged from 0.08 to 0.16. Observations of the fishery indicated that anglers harvested smaller blue catfish (<562 mm TL); harvest of larger fish was nil, which probably accounted for the size‐related differences in mortality. I provide a computer program to derive estimates of size‐related mortality using piecewise nonlinear regression.