Chen-Watanabe Natural Mortality ($L_0$ Parameterization)

Computes age-specific natural mortality using the Chen & Watanabe (1989) model with an $L_0$ parameterization that eliminates dependence on the theoretical parameter $t_0$.

Usage

M_chen_watanabe_L0(
  age,
  Linf,
  L0,
  k,
  k_native = k,
  growth_model = c("vb", "gompertz", "logistic"),
  tmax = NULL,
  Linf_factor = 0.99,
  two_phase = TRUE,
  tmat = NULL,
  late_model = c("gompertz", "logistic"),
  tm_factor = 2/3,
  M_mult = 2,
  smooth_factor = 1/3
)

Arguments

age: Numeric vector of ages at which to compute mortality.
Linf: Asymptotic length.
L0: Length at birth.
k: VB-equivalent growth coefficient (used as $M_\infty$ in the CW formula). For VB fits, this is the native $k$. For Gompertz or Logistic fits, use compute_k_vb_equivalent or .derive_k_vb_derivative.
k_native: Growth coefficient native to the fitted model (used for $L(t)$ prediction). Defaults to k, which is correct for VB fits. For non-VB fits, pass the model-specific growth coefficient here.
growth_model: Character. Growth model for $L(t)$ prediction: "vb" (default), "gompertz", or "logistic".
tmax: Maximum age. If NULL, estimated from growth parameters as age when $L(t) = $ Linf_factor $\times L_\infty$ (using the VB equation with k for consistency with the CW theoretical framework).
Linf_factor: Numeric in (0, 1). Fraction of $L_\infty$ used to estimate $t_{max}$. Default 0.99.
two_phase: Logical. If TRUE, applies two-phase model with late-life senescence. Default TRUE.
tmat: Age at maturity. Required if two_phase = TRUE.
late_model: Character. Senescence model: "gompertz" (default) or "logistic".
tm_factor: Numeric. Fraction of $t_{mat}$ at which transition to senescence begins. Default 2/3.
M_mult: Numeric. Multiplier for senescence mortality plateau relative to mortality at $t_m$. Default 2.
smooth_factor: Numeric. Controls smoothness of transition between phases. Default 1/3.

Value

Numeric vector of instantaneous mortality rates (same length as age).

Details

The standard Chen-Watanabe formulation expresses mortality as: $$M(t) = \frac{k}{1 - e^{-k(t - t_0)}}$$

where $t_0$ is the theoretical age at length zero — a parameter with no direct biological interpretation that can take implausible values, particularly when growth data are sparse.

We reparameterize using the relationship between $t_0$ and $L_0$ (birth length) under von Bertalanffy dynamics: $$L_0 = L_\infty(1 - e^{kt_0})$$

After algebraic manipulation (see vignette), the $L_0$-parameterized form becomes: $$M(t) = \frac{k_{VB} \cdot L_\infty}{L(t)}$$

where $k_{VB}$ is the von Bertalanffy growth coefficient (or VB-equivalent for non-VB fits) and $L(t)$ is the predicted length at age $t$ from the native growth model.

Two Roles of k

The Chen-Watanabe equation involves $k$ in two conceptually distinct roles:

Asymptotic mortality rate: The VB-specific constant $M_\infty = k_{VB}$ that governs the theoretical minimum mortality as $t \to \infty$. This enters the CW formula as the numerator. Because the derivation assumes VB dynamics, this must always be the VB growth coefficient (or VB-equivalent for non-VB fits). Passed via the k argument.
Growth trajectory: The length-at-age prediction $L(t)$ that determines how far an individual is from asymptotic size. For non-VB growth models, the native trajectory with native $k$ produces more accurate body-size estimates than the VB approximation. Controlled by k_native and growth_model.

For VB fits, both roles use the same $k$, and the default k_native = k preserves full backward compatibility.

Two-Phase Extension

The original CW model produces unrealistic mortality trajectories at old ages (approaching zero asymptotically). When two_phase = TRUE, the model adds a senescence component where mortality increases after maturity, more realistically capturing late-life dynamics. Mortality follows the standard CW model until age $t_m$ (a fraction of $t_{mat}$), then transitions to a senescence model (Gompertz or logistic) that increases mortality toward $t_{max}$.

References

Chen, S., & Watanabe, S. (1989). Age dependence of natural mortality coefficient in fish population dynamics. Nippon Suisan Gakkaishi, 55(2), 205-208.

Examples

if (FALSE) { # \dontrun{
ages <- seq(0.1, 30, by = 0.5)

# VB fit (k serves both roles)
M_vb <- M_chen_watanabe_L0(
  age = ages, Linf = 100, L0 = 25, k = 0.1,
  two_phase = TRUE, tmat = 10
)

# Gompertz fit (separate k for each role)
M_gomp <- M_chen_watanabe_L0(
  age = ages, Linf = 100, L0 = 25,
  k = 0.08,                       # VB-equivalent k for M_inf
  k_native = 0.12,                # Native Gompertz k for L(t)
  growth_model = "gompertz",
  two_phase = TRUE, tmat = 10
)
} # }

Chen-Watanabe Natural Mortality (\(L_0\) Parameterization)