Lorenzen Natural Mortality Model — M

Computes size-dependent natural mortality following Lorenzen (1996, 2022). Supports both weight-based and growth-based formulations.

Usage

M_lorenzen(
  age,
  Linf,
  L0,
  k,
  k_vb = k,
  lw_fun = NULL,
  weight_based = FALSE,
  growth_model = c("vb", "gompertz", "logistic"),
  sample_params = FALSE
)

Arguments

age: Numeric vector of ages at which to compute mortality.
Linf: Asymptotic length.
L0: Length at birth.
k: Growth coefficient native to the fitted model (used for $L(t)$ prediction via growth_model).
k_vb: VB-equivalent growth coefficient for the growth-based formulation's $\ln(k)$ term. Defaults to k, which is correct for VB fits. For non-VB fits, use compute_k_vb_equivalent or the derivative-matching approach. Ignored for weight-based formulation.
lw_fun: Function mapping length to weight in grams: lw_fun(L). Required if weight_based = TRUE.
weight_based: Logical. If TRUE, uses weight-based formulation. If FALSE (default), uses growth-based formulation.
growth_model: Character. Growth model for length prediction: "vb", "gompertz", or "logistic". Default "vb".
sample_params: Logical. If TRUE, samples allometric/regression parameters from their distributions for each call. If FALSE, uses mean values.

Value

Numeric vector of instantaneous mortality rates.

Details

Two formulations are available:

Weight-based (Lorenzen 1996): $$M(W) = \alpha \cdot W^{\beta}$$ where $\alpha \sim N(3.69, 0.502)$ and $\beta \sim N(-0.305, 0.029)$. Body weight $W(t)$ is computed from predicted length via a user-supplied length-weight function using the native growth model trajectory.

Growth-based (Lorenzen 2022): $$\ln M = 0.28 - 1.30 \ln(L(t)/L_\infty) + 1.08 \ln(k_{VB})$$ This formulation was calibrated using von Bertalanffy parameters. The $L(t)/L_\infty$ ratio is computed from the native growth model for accurate relative-size estimates, while $k_{VB}$ must be the VB-equivalent growth coefficient (or native VB $k$) because the regression coefficients were calibrated against VB parameters.

Two Roles of k (Growth-Based Formulation)

Like Chen-Watanabe, the growth-based Lorenzen involves $k$ in two distinct roles:

Calibration coefficient: $\ln(k_{VB})$ enters the regression equation as a predictor. Since Lorenzen (2022) calibrated this term against VB parameters, it must be the VB growth coefficient or VB-equivalent. Passed via k_vb.
Growth trajectory: $L(t)/L_\infty$ represents relative body size. The native growth model produces more accurate predictions than a VB approximation. Controlled by k and growth_model.

References

Lorenzen, K. (1996). The relationship between body weight and natural mortality in juvenile and adult fish. Journal of Fish Biology, 49(4), 627-642.

Lorenzen, K. (2022). Size- and age-dependent natural mortality in fish populations. Fisheries Research, 255, 106454.

Examples

if (FALSE) { # \dontrun{
ages <- seq(0.5, 30, by = 0.5)
lw_fun <- function(L) 0.0001 * L^3.1

# Weight-based with Gompertz trajectory
M_wt <- M_lorenzen(
  age = ages, Linf = 100, L0 = 25, k = 0.15,
  lw_fun = lw_fun, weight_based = TRUE,
  growth_model = "gompertz"
)

# Growth-based with Gompertz trajectory + VB-equivalent k
M_gr <- M_lorenzen(
  age = ages, Linf = 100, L0 = 25,
  k = 0.15,                         # Native Gompertz k for L(t)
  k_vb = 0.08,                      # VB-equivalent k for coefficient
  growth_model = "gompertz"
)
} # }