Case study 2: von Bertalanffy growth with lizard size data

Overview

Our second demo introduces size-dependent growth based on the von Bertalanffy function where Smax is the asymptotic maximum size and β controls the growth rate. We have implemented the analytic solution which is independent of age at the starting size Y(0) and instead uses the first size as the initial condition. The key behaviour of the von Bertalanffy model is a high growth rate at small sizes that declines linearly as the size approaches Smax. This manifests as growth slowing as a creature matures with a hard finite limit on the eventual size. We restrict β and Smax to be positive, and Smax to be larger than the observed sizes. As a result the growth rate is non-negative.

Load dependencies

# remotes::install_github("traitecoevo/hmde")
# install.packages(c("dplyr", "ggplot2"))

library(hmde)
library(dplyr)
library(ggplot2)

Visualise model

In the following code we plot an example of the growth function and the solution to get a feel for the behaviour.

#Analytic solution in function form
solution <- function(t, pars = list(y_0, beta, S_max)){
  return(
    pars$S_max + (y_0 - pars$S_max)*exp(-t * pars$beta)
  )
}

#Parameters
beta <- 0.35 #Growth rate
y_0 <- 1 #Starting size
S_max <- 20 #Asymptotic max size
time <- c(0,30) 
pars_list <- list(y_0 = y_0,
                  beta = beta,
                  S_max = S_max)
y_final <- solution(time[2], pars_list)

#Plot of growth function
ggplot() +
  xlim(y_0, y_final) +
  ylim(0, beta*(S_max-y_0)*1.1) +
  labs(x = "Y(t)", y = "f", title = "von Berralanffy growth") +
  theme_classic() +
  theme(axis.text=element_text(size=16),
        axis.title=element_text(size=18,face="bold")) +
  geom_function(fun=hmde_model_des("vb_single_ind"), 
                args=list(pars = list(S_max, beta)),
                colour="green4", linewidth=1,
                xlim=c(y_0, y_final))


#Size over time
ggplot() +
  geom_function(fun=solution, 
                args=list(pars = pars_list),
                colour="green4", linewidth=1,
                xlim=c(time)) +
  xlim(time) +
  ylim(0, y_final*1.05) +
  labs(x = "Time", y = "Y(t)", title = "von Bertalanffy growth") +
  theme_classic() +
  theme(axis.text=element_text(size=16),
        axis.title=element_text(size=18,face="bold"))

The von Bertalanffy model is commonly used in fishery management (Flinn and Midway 2021), but has also been used in reptile studies such as Edmonds et al. (2021) and Zhao et al. (2020).

Lizard size data

Our data is sourced from Kar, Nakagawa, and Noble (2023) which measured mass and snout-vent-length (SVL) of delicate skinks – – under experimental conditions to examine the effect of temperature on development. We are going to use the SVL metric for size.

We took a simple random sample without replacement of 50 individuals with at least 5 observations each. The von Bertalanffy model can be fit to shorter observation lengths, but fewer than 3 observations is not advised as there are two growth parameters per individual.

Implementation

The workflow for the second example is the same as the first, with the change in model name and data object.

lizard_vb_fit <- hmde_model("vb_multi_ind") |>
  hmde_assign_data(data = Lizard_Size_Data)  |>
  hmde_run(chains = 4, cores = 1, iter = 2000)
#> 
#> SAMPLING FOR MODEL 'vb_multi_ind' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 0.00013 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.3 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> 
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#> Chain 4:
#> Warning: There were 214 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: There were 4 chains where the estimated Bayesian Fraction of Missing Information was low. See
#> https://mc-stan.org/misc/warnings.html#bfmi-low
#> Warning: Examine the pairs() plot to diagnose sampling problems
#> Warning: The largest R-hat is 1.06, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess

lizard_estimates <- hmde_extract_estimates(model = "vb_multi_ind",
                                           fit = lizard_vb_fit,
                                           input_measurement_data = Lizard_Size_Data)

As before, we can compare the observed sizes over time to those predicted by the model.

measurement_data_transformed <- lizard_estimates$measurement_data %>%
  group_by(ind_id) %>%
  mutate(
    delta_y_obs = y_obs - lag(y_obs),
    obs_interval = time - lag(time),
    obs_growth_rate = delta_y_obs/obs_interval,
    delta_y_est = y_hat - lag(y_hat),
    est_growth_rate = delta_y_est/obs_interval
  ) %>%
  ungroup()


#Distributions of estimated growth and size
hist(measurement_data_transformed$y_hat, 
     main = "Estimated size distribution",
     xlab = "Size (cm)")

hist(measurement_data_transformed$delta_y_est, 
     main = "Estimated growth increments",
     xlab = "Growth increment (cm)")

hist(measurement_data_transformed$est_growth_rate, 
     main = "Estimated annualised growth rate distribution",
     xlab = "Growth rate (cm/yr)")


#Quantitative R^2
cor(measurement_data_transformed$y_obs, measurement_data_transformed$y_hat)^2
#> [1] 0.7445606
r_sq_est <- cor(lizard_estimates$measurement_data$y_obs,
                lizard_estimates$measurement_data$y_hat)^2
r_sq <- paste0("R^2 = ", 
               signif(r_sq_est,
                      digits = 3))

ggplot(data = lizard_estimates$measurement_data, 
       aes(x = y_obs, y = y_hat)) +
  geom_point(shape = 16, size = 1, colour = "green4") +
  xlab("Y obs.") +
  ylab("Y est.") +
  geom_abline(slope = 1, linetype = "dashed") +
  annotate("text", x = 25, y = 18, 
           label = r_sq) +
  theme_classic()


#Plots of size over time for a sample of 5 individuals
hmde_plot_obs_est_inds(n_ind_to_plot = 5,
                       measurement_data = lizard_estimates$measurement_data) +
  theme(legend.position = "inside",
        legend.position.inside = c(0.05, 0.9))

We have two parameters at the individual level and are interested in both their separate distributions, and if we see evidence of a relationship between them. We can also use the individual parameter estimates and estimated sizes to plot the growth function pieces.

#1-dimensional parameter distributions
ggplot(lizard_estimates$individual_data, 
       aes(ind_max_size_mean)) +
  geom_histogram(bins = 10,
                 colour = "black",
                 fill = "lightblue") +
  labs(x="S_max estimate") +
  theme_classic()


ggplot(lizard_estimates$individual_data, 
       aes(ind_growth_rate_mean)) +
  geom_histogram(bins = 10,
                 colour = "black",
                 fill = "lightblue") +
  labs(x="beta estimate") +
  theme_classic()


#2-dimensional parameter distribution
ggplot(data = lizard_estimates$individual_data, 
       aes(x = ind_max_size_mean, y = ind_growth_rate_mean)) +
  geom_point(shape = 16, size = 1, colour = "green4") +
  xlab("Individual max sizes (mm)") +
  ylab("Individual growth rate parameters") +
  theme_classic()


#Correlation of parameters
cor(lizard_estimates$individual_data$ind_max_size_mean,
    lizard_estimates$individual_data$ind_growth_rate_mean,
    method = "spearman")
#> [1] 0.5492917

#Plot function pieces over estimated sizes.
hmde_plot_de_pieces(model = "vb_multi_ind",
                    individual_data = lizard_estimates$individual_data,
                    measurement_data = lizard_estimates$measurement_data)

At the hyper-parameter level for the whole population we have centre and spread parameters for the log-normal distributions of Smax and β. As before, we can look at these as species-level features.

pars_CI_names <- c(
  "mean log max size",
  "mean max size in mm",
  "log max size standard deviation",
  "mean log growth par",
  "mean growth par mm/yr",
  "log growth par standard deviation"
)

#Vector that picks out which pars to be exponentiated
exp_vec <- c(FALSE, TRUE, FALSE, 
             FALSE, TRUE, FALSE)

#Print mean estimates and CIs
for(i in 1:nrow(lizard_estimates$population_data)){
  if(!exp_vec[i]){
    lizard_estimates$population_data$mean[i] 
    print(paste0("95% CI for ", 
                 pars_CI_names[i],
                 ": (",
                 lizard_estimates$population_data$CI_lower[i],
                 ", ",
                 lizard_estimates$population_data$CI_upper[i],
                 ")"))
  } else {
    exp(lizard_estimates$population_data$mean[i])
    print(paste0("95% CI for ",
                 pars_CI_names[i], 
                 ": (",
                 exp(lizard_estimates$population_data$CI_lower[i]),
                 ", ",
                 exp(lizard_estimates$population_data$CI_upper[i]),
                 ")"))
  }
}
#> [1] "95% CI for mean log max size: (3.17741289411034, 3.21522360518108)"
#> [1] "95% CI for mean max size in mm: (1.01330099028629, 1.04963332499416)"
#> [1] "95% CI for log max size standard deviation: (-4.1646016809886, -3.76533742448243)"
#> [1] "95% CI for mean log growth par: (0.0214900162311675, 0.236131961060446)"

References

Edmonds, Devin, Michael J Dreslik, Jeffrey E Lovich, Thomas P Wilson, and Carl H Ernst. 2021. “Growing as Slow as a Turtle: Unexpected Maturational Differences in a Small, Long-Lived Species.” PloS One 16 (11): e0259978.
Flinn, Shane A, and Stephen R Midway. 2021. “Trends in Growth Modeling in Fisheries Science.” Fishes 6 (1): 1.
Kar, Fonti, Shinichi Nakagawa, and Daniel W Noble. 2023. “Heritability and Developmental Plasticity of Growth in an Oviparous Lizard.” OSF. https://doi.org/10.17605/OSF.IO/HJKXD.
Zhao, Wei, Yangyang Zhao, Rui Guo, Yue Qi, Xiaoning Wang, and Na Li. 2020. “Age and Annual Growth Rate Cause Spatial Variation in Body Size in Phrynocephalus Przewalskii (Agamid).” Ecology and Evolution 10 (24): 14189–95.