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

Abstract

I 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 log_e(N_i) = b₀ − (b₁ + b₂)t_i, where N_i is the number of fish collected for the ith age or length interval; t_i 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 t_i 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 t_i 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 log_e(N_i) 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.