The Chen-Watanabe Reparameterization

Overview

This vignette documents the mathematical framework underlying vitalBayes’s implementation of the Chen & Watanabe (1989) natural mortality model. Two methodological design choices distinguish this implementation from standard practice:

\(L_0\) parameterization: Expressing the CW model directly in terms of length at birth \(L_0\) and asymptotic length \(L_\infty\), eliminating dependence on the theoretical parameter \(t_0\).
Growth-model-agnostic estimation: Defining mortality through a normalized growth coefficient \(G(t) = L(t) / L_\infty\) and an asymptotic mortality rate \(M_\infty\) that is equivalent to the von Bertalanffy \(k\), regardless of which growth model is used to describe the data.

Together, these choices mean that users can fit whichever growth model best describes their data — von Bertalanffy, Gompertz, or Logistic — and still use the Chen-Watanabe mortality framework without theoretical compromise. For the mathematical details of the \(L_0\)-based growth model equations themselves, see vignette("fit_bayesian_growth").

The Chen-Watanabe Model

\(L_0\)-Parameterized Form

The Chen & Watanabe (1989) model relates instantaneous natural mortality \(M(t)\) to the von Bertalanffy growth coefficient \(k\):

\[M(t) = \frac{k \cdot L_\infty}{L(t)}\]

where \(L(t) = L_\infty - (L_\infty - L_0)e^{-kt}\) is the von Bertalanffy predicted length at age \(t\), \(L_0\) is the length at birth, \(L_\infty\) is asymptotic length, and \(k\) is the VB growth coefficient.

This compact form reveals the core biological mechanism: mortality is inversely proportional to body size. Small, young individuals are more vulnerable to predation, starvation, and environmental stochasticity; as an organism grows toward its asymptotic size, its instantaneous mortality declines toward a lower bound.

The Normalized Growth Coefficient \(G(t)\)

To generalize the CW model beyond von Bertalanffy growth, it is useful to define the normalized growth coefficient:

\[G(t) = \frac{L(t)}{L_\infty}\]

This is the fraction of asymptotic size achieved at age \(t\). At birth (\(t = 0\)), \(G(0) = L_0 / L_\infty\). As \(t \to \infty\), \(G(t) \to 1\) for all three growth models — the organism approaches its asymptotic size.

Using \(G(t)\), the CW model becomes:

\[M(t) = \frac{M_\infty}{G(t)}\]

where \(M_\infty = k\) is the asymptotic mortality rate — the mortality rate an individual would experience at asymptotic length. The entirety of the age-specific mortality pattern is captured by \(G(t)\): how quickly an organism reaches its full size determines how quickly its mortality declines.

Why \(M_\infty\) Equals the Von Bertalanffy \(k\)

The identification \(M_\infty \equiv k_{VB}\) emerges directly from the Chen-Watanabe derivation, which was built on von Bertalanffy growth dynamics. As age increases, the VB growth equation approaches \(L_\infty\) and the CW mortality equation approaches \(k\):

\[\lim_{t \to \infty} M(t) = \lim_{t \to \infty} \frac{k \cdot L_\infty}{L_\infty - (L_\infty - L_0)e^{-kt}} = k\]

The von Bertalanffy \(k\) governs the rate at which an organism approaches its asymptotic size; organisms with higher \(k\) grow faster but have shorter lifespans and higher baseline mortality (Beverton & Holt, 1959; Charnov, 1993). The CW model formalizes this life history trade-off: \(k\) simultaneously determines the growth rate (how fast \(G(t)\) approaches 1) and the floor of the mortality curve.

Critically, this relationship holds regardless of which growth model was used to fit the data. The Gompertz and Logistic models have their own growth coefficients \(k_g\) and \(k_l\), but these are not \(M_\infty\). When we use Gompertz or Logistic fits, we do not substitute their native \(k\) into the CW equation — instead, we compute the VB-equivalent \(k\) that encodes the same biological growth information (described below).

Growth-Specific \(G(t)\) Functions

Although \(M_\infty = k_{VB}\) is common to all growth models, the shape of the mortality curve is governed by \(G(t)\), which differs across models. Each growth model implies a different trajectory toward asymptotic size, and therefore a different pattern of declining mortality. Below are the normalized growth coefficients for each \(L_0\)-based growth model implemented in vitalBayes.

Von Bertalanffy

\[L_{VB}(t) = L_\infty - (L_\infty - L_0) \, e^{-k_{VB} \, t}\]

\[G_{VB}(t) = 1 - \left(1 - \frac{L_0}{L_\infty}\right) e^{-k_{VB} \, t}\]

The VB model describes growth as a constant exponential approach to \(L_\infty\). The absolute growth rate \(dL/dt\) is highest at birth and declines monotonically. This produces a mortality curve that is steepest in early life and decelerates smoothly — a pattern consistent with strong predation pressure on neonates that diminishes as individuals grow through vulnerable size classes.

Gompertz

\[L_{Gomp}(t) = L_\infty \exp\!\left[-\ln\!\left(\frac{L_\infty}{L_0}\right) e^{-k_g \, t}\right]\]

\[G_{Gomp}(t) = \exp\!\left[-\ln\!\left(\frac{L_\infty}{L_0}\right) e^{-k_g \, t}\right]\]

The Gompertz model describes growth where the rate of deceleration itself decelerates — an exponentially declining growth rate rather than a linearly declining one. Growth is initially rapid but slows earlier and more abruptly than under VB dynamics. In terms of mortality, the Gompertz \(G(t)\) rises faster in early life than \(G_{VB}(t)\), producing a steeper initial decline in mortality but a slower final approach to \(M_\infty\).

Logistic

\[L_{Log}(t) = \frac{L_\infty}{1 + \left(\frac{L_\infty}{L_0} - 1\right) e^{-k_l \, t}}\]

\[G_{Log}(t) = \frac{1}{1 + \left(\frac{L_\infty}{L_0} - 1\right) e^{-k_l \, t}}\]

The Logistic model is symmetric around its inflection point at \(L_\infty / 2\), with growth accelerating before the midpoint and decelerating after. This produces the most distinct mortality pattern: mortality declines slowly in very early life (when growth is still accelerating), then drops steeply as the organism passes through the inflection point, before leveling off toward \(M_\infty\). This pattern may suit species where size-dependent mortality relief is concentrated in a relatively narrow size window.

Comparing \(G(t)\) Across Models

The three \(G(t)\) functions all share the same boundary conditions (\(G(0) = L_0/L_\infty\) and \(G(\infty) = 1\)), but they differ in how they behave between these bounds. Because \(M(t) = M_\infty / G(t)\), the shape of \(G(t)\) directly determines the mortality schedule:

VB: Steady exponential approach. Mortality declines fastest at young ages and progressively slows.
Gompertz: Rapid early approach, slower later convergence. Mortality drops quickly in early life but the final decline toward \(M_\infty\) is drawn out.
Logistic: Sigmoidal approach. Mortality is relatively flat in very early life, then drops sharply around the growth inflection, and converges toward \(M_\infty\).

In practice, for many elasmobranch species and typical ratios of \(L_0/L_\infty\) (roughly 0.15–0.40), the differences in \(G(t)\) between models are modest provided the models fit the data comparably well. The choice of growth model has a larger effect on the early-age mortality curve (where \(G(t)\) values are small and thus \(M(t)\) values are large) than on mid- to late-life mortality.

library(ggplot2)
library(data.table)

# Example parameters
Linf <- 100; L0 <- 25
k_vb <- 0.08; k_g <- 0.12; k_l <- 0.15
ages <- seq(0, 40, by = 0.5)

# Compute G(t) for each model
G_vb   <- 1 - (1 - L0/Linf) * exp(-k_vb * ages)
G_gomp <- exp(-log(Linf/L0) * exp(-k_g * ages))
G_log  <- 1 / (1 + (Linf/L0 - 1) * exp(-k_l * ages))

dt <- data.table(
  age  = rep(ages, 3),
  G    = c(G_vb, G_gomp, G_log),
  model = rep(c("von Bertalanffy", "Gompertz", "Logistic"), each = length(ages))
)

ggplot(dt, aes(x = age, y = G, color = model)) +
  geom_line(linewidth = 1) +
  labs(x = "Age (years)", y = expression(G(t) == L(t)/L[infinity]),
       title = "Normalized Growth Coefficient by Model") +
  theme_bw() +
  theme(legend.position = "top")

Computing VB-Equivalent \(k\) from Any Growth Model

The Biological Milestones Approach

Since \(M_\infty = k_{VB}\), users who fit a Gompertz or Logistic model need a way to obtain the VB-equivalent \(k\) without fitting a separate VB model. The solution is found in the fact that all three growth models estimate the same biological quantities — \(L_\infty\), \(L_0\), \(L_{mat}\), and \(t_{mat}\) — regardless of their functional form. A shark has a true birth size, a true size and age at maturity, and a true asymptotic size; these do not change based on which equation we use to model growth.

Given these shared biological milestones, the VB-equivalent \(k\) is the growth coefficient that would produce a von Bertalanffy curve passing through \((0, L_0)\) and \((t_{mat}, L_{mat})\) with asymptote \(L_\infty\). Substituting \(L(t_{mat}) = L_{mat}\) into the VB equation and solving:

\[k_{VB}^{equiv} = \frac{1}{t_{mat}} \ln\!\left(\frac{L_\infty - L_0}{L_\infty - L_{mat}}\right)\]

In vitalBayes, compute_k_vb_equivalent() performs this computation and get_stochastic_mortality() calls it automatically when processing posteriors from non-VB growth fits.

Verification: Consistency When Using VB

When the underlying growth model is von Bertalanffy with maturity-based parameterization, \(k_{VB}^{equiv}\) is algebraically identical to the estimated \(k\). This serves as an internal consistency check:

library(vitalBayes)

# Extract VB posterior draws
params <- extract_growth_parameters(vb_fit, sex = 1, n_draws = 2000)

# Compare native k to VB-equivalent k
all.equal(params$k, params$k_vb_equiv, tolerance = 1e-10)
#> [1] TRUE

For Gompertz and Logistic fits, the VB-equivalent \(k\) and native \(k\) will be correlated but not identical:

params_gomp <- extract_growth_parameters(gomp_fit, sex = 1, n_draws = 2000)

# Native Gompertz k vs VB-equivalent k
cor(params_gomp$k, params_gomp$k_vb_equiv, use = "complete.obs")
#> [1] 0.89  # Correlated but distinct

Late-Life Senescence Modifications

The Problem with Asymptotic Mortality

The single-phase CW model predicts that mortality monotonically decreases toward \(M_\infty\) as \(G(t) \to 1\). This is biologically unrealistic for long-lived species: senescence, accumulated physiological damage, and declining immune function all contribute to increasing mortality at advanced ages. The monotonically declining CW curve captures juvenile and sub-adult mortality patterns well, but fails to represent late-life dynamics.

Two-Phase Extension

vitalBayes implements a two-phase CW model that blends the standard declining-mortality trajectory with a senescence component. The transition occurs at a user-defined age \(t_m\), typically set as a fraction of the age at maturity (\(t_m = \frac{2}{3} t_{mat}\) by default). Before \(t_m\), mortality follows the standard CW model. After \(t_m\), a senescence function gradually increases mortality toward \(t_{max}\).

Two senescence models are available:

Gompertz senescence (late_model = "gompertz"):

\[M_{late}(t) = M_s \exp\!\left[r \cdot (t - t_{max})\right]\]

where \(M_s = c \cdot M(t_m)\) is the mortality plateau at \(t_{max}\) (with \(c\) = M_mult, default 2), and \(r\) is solved from continuity at \(t_m\):

\[r = \frac{\ln\left(M(t_m) / M_s\right)}{t_m - t_{max}}\]

This produces an accelerating increase in mortality — modest at first, then more pronounced as the organism approaches maximum age. This pattern aligns with actuarial models of biological senescence in which the hazard function increases exponentially (Gompertz, 1825).

Logistic senescence (late_model = "logistic"):

\[M_{late}(t) = \frac{K}{1 + \exp\!\left[-r \cdot (t - t_{max})\right]}\]

where \(K = 2 \, c \cdot M(t_m)\) and \(r\) is again solved from continuity at \(t_m\).

Logistic senescence produces a bounded mortality increase that plateaus at \(K\) rather than growing without limit. This may be more appropriate when there is reason to believe that late-life mortality stabilizes rather than continuing to accelerate — for example, in populations subject to a physiological upper bound on hazard rates.

Smooth Blending

The transition between the early-phase CW model and the late-phase senescence model is smoothed using a logistic weighting function:

\[w(t) = \frac{1}{1 + \exp\!\left[-\frac{2(t - t_m)}{s \cdot |t_{max} - t_m|}\right]}\]

where \(s\) is the smooth_factor (default \(\frac{1}{3}\)). The blended mortality at any age is:

\[M(t) = (1 - w) \cdot M_{CW}(t) + w \cdot M_{late}(t)\]

This ensures a smooth, differentiable transition rather than a sharp kink at \(t_m\).

ages <- seq(0.1, 35, by = 0.25)

# Single-phase: monotonic decline
M_single <- M_chen_watanabe_L0(
  age = ages, Linf = 100, L0 = 25, k = 0.08,
  two_phase = FALSE
)

# Two-phase: Gompertz senescence
M_gomp <- M_chen_watanabe_L0(
  age = ages, Linf = 100, L0 = 25, k = 0.08,
  two_phase = TRUE, tmat = 12, late_model = "gompertz"
)

# Two-phase: Logistic senescence
M_logis <- M_chen_watanabe_L0(
  age = ages, Linf = 100, L0 = 25, k = 0.08,
  two_phase = TRUE, tmat = 12, late_model = "logistic"
)

dt <- data.table(
  age = rep(ages, 3),
  M   = c(M_single, M_gomp, M_logis),
  Model = rep(c("Single-phase", "Gompertz senescence", "Logistic senescence"),
              each = length(ages))
)

ggplot(dt, aes(x = age, y = M, color = Model)) +
  geom_line(linewidth = 1) +
  labs(x = "Age (years)", y = "Instantaneous Mortality (M)",
       title = "Chen-Watanabe Mortality with Senescence Modifications") +
  theme_bw() +
  theme(legend.position = "top")

Practical Implications

Using the Best Growth Model for Mortality

The growth-model-agnostic framework means researchers no longer face a choice between statistical fit and mortality estimation capability. In elasmobranch datasets with sparse adult observations, the VB model frequently produces biologically implausible \(L_\infty\) estimates (below observed maximum lengths or far exceeding plausible estimates of a physiological asymptotic length) and unstable \(k\) values driven by the strong \(L_\infty\)-\(k\) correlation (Katsanevakis & Maravelias, 2008). Gompertz and Logistic models often provide more reliable fits under these conditions.

With the reparameterization, the workflow is straightforward: fit all candidate growth models, select the best model by LOO-CV or other criteria, and pass that model’s posterior directly to get_stochastic_mortality(). The function handles the VB-equivalent \(k\) computation automatically.

When Growth Models Agree and Disagree

If all three growth models fit the data comparably, their CW mortality estimates will be similar — the biological milestones \((L_\infty, L_0, L_{mat}, t_{mat})\) are well-constrained and produce nearly identical \(k_{VB}^{equiv}\) values. Divergence between mortality estimates from different growth models indicates genuine model uncertainty: the data do not strongly favor one growth trajectory, and the implied mortality schedules differ as a result. In such cases, reporting sensitivity to growth model choice is informative.

Relationship to Other Mortality Models

The \(M(t) = M_\infty / G(t)\) formulation is specific to the Chen-Watanabe model. The Peterson-Wroblewski and Lorenzen mortality models in vitalBayes use different theoretical foundations (allometric scaling of metabolic rate with body weight) and do not share this \(G(t)\) structure. See vignette("mortality_estimation") for equations and practical guidance on all three mortality models.