In posts 1 to10 I have now covered most of the preliminaries for a research report such as aims, methods, sites, assumptions and confounding issues to consider. It is time to report some results (although much information is already in these earlier posts). The first of my six research questions is “How much does growth rate vary with size?”
The concept is notionally simple and straightforward but in practice not so. Very few animals have linear growth over their whole lifespan (think of infantile growth spurts and puberty in humans then shrinkage with advanced age, or episodic growth of crustaceans, insects and reptiles when they moult.) The growth literature for marine animals is replete with papers about sigmoid or stepped growth (see Maunder et al, 2015, for an introduction).
Note that my research question relates to size, since age cannot be determined reliably from tag-recapture data alone. In a future blog post, after I address confounding factors, I expect to cover the reliability of converting growth and length data for my two species to age-length curves.
Also note that I am using maximum dimension (“length”) as a proxy for size. My length dimension is actually shell “height” for a molluscan taxonomist.
On this issue, I need to record a caution about my data – all turban snails are not the same “shape”. I have noticed some Lunella torquata specimens where growth has not just been in the usual shallow spiral. Some snails have a deeper “bowl” shape than others. These apparently slow-growing snails seemed bulkier than most others, wider and heavier for the same length. The same thing occurs in abalone. I measured length in my research work from 1974 to 1985 but with detailed analysis it turned out that width was a more useful measure of size for fisheries management (Worthington et al, 1995). I will discuss this more in a later blog post.
In Turbo militaris, many specimens have spurs or spines on their shells while others are smooth. If a spur happens to be forming at the edge on the maximum dimension the length will be several millimetres more than the same size snail with no spurs. In my data, this would be accommodated statistically in the individual variability component of my results. Seinor and Purcell et al (2025) discussed this and measured maximum width, with and without spurs.
Weight could be a better measure for doing fisheries yield work, but it would be messy to measure on abundant field samples and not useful for fisheries enforcement. Generally, weight can be estimated from a simple equation such as weight=a*lengthB where a and B are constants, which if calculated for local populations can be instructive in relation to life history parameters.
I used 3 common linear methods. In the following post (Blog 12) I will report on 3 non-linear methods.
11.1 Simple linear regression.
The simplest analysis is a linear plot of growth rate (mm/month) compared to length at tagging. All my data from all sites, sexes and seasons are shown below in Figures 1a,1b for Lunella torquata and Figures 2a,2b for Turbo militaris.
Fig 1a. Monthly growth rate from tagging and release of Lunella torquata for all times-at-liberty (dT’s). n=1166 (both sexes, all sites, all seasons). X-axis is length in mm.
Fig 1b. Monthly growth rate after tagging, release and recapture of Lunella torquata for times-at-liberty (dT’s) >= 60 days. N=1046.
For Lunella, growth rate declines linearly with higher lengths-at-tagging but note there are several outliers in Fig 1a, particularly those above 4mm/month. Just over 42% of the variability is explained by size differences. Removing short-term recaptures (<60days) increases the slope and the maximum theoretical monthly increment. It also increases the variability explained by nearly 10%. For the same data in Gulland-Holt plots (mm/month vs midL rather than mm/month vs L1), Fisher’s r to z transformation of the correlations yielded a z-value of -4.623 and a probability of less than 0.01 so excluding short term recaptures changes the correlation at a statistically significant level.
Blog posts 3 and 10 mention the other factors that can affect this simple regression: sex, season, site, measurement error, and individual variability. These will be discussed more in future blog posts. For now, note that the growth rate is an average over the length range for each time at liberty but is plotted above against the length at tagging. For long times at liberty, it is misleading to allocate the growth rate to the length at tagging since for most of the growth period, the snail will be larger than L1.
For Turbo militaris, a similar pattern appears, but the trend is curvilinear – most points at very small (around 20mm) and very large (above 70mm) sizes are below the line of best fit.
Fig 2a. Monthly growth rate after tagging and release of Turbo militaris for both sexes, all seasons and all sites, but with times-at-liberty (dT’s) >= 60 days. Note the domination of the dataset by snails <60mm at tagging. N=982.
Fig 2b. Monthly growth rate after tagging and release of Turbo militaris including both sexes, all seasons and all sites, but with times-at-liberty (dT’s) < 60 days. N=107
Figure 2b illustrates well the problem with using very short time intervals. In this case, the slope changes from negative to positive. This is further evidence for removing short term recaptures. Even with these data removed, only 28% of the variability is explained by the simple linear regression in Fig 2a, indicating that other factors such as sex, site, season or year are probably important, or the relationship is just not linear for all sizes regardless of those factors.
For both species, from about 30mm, growth rate generally decreases with size (Figure 3 below) and hence the von Bertalanffy growth curve probably represents growth well enough for broad fisheries management purposes. But there is a hint of linear or increasing growth rate with size for small juveniles, so the inverse-logistic curve may be more appropriate as outlined by Haddon et al (2008) (see next blog post on non-linear models). This is more pronounced for Turbo militaris, where nearly all the juvenile recaptures were from Mahon Pool. Since this is a man-made coastal swimming pool carved out of a rock platform (see blog post on Research Sites) and small juveniles were found in very high densities in narrow crevices, these growth rates might not reflect the situation for juveniles on normal ocean reefs. At such high densities, food may be limiting.
Fig 3. Simplified trend of mean growth rate vs size for Lunella torquata and Turbo militaris in 5mm size classes showing possible non-linearity around juvenile sizes. X-axis is size class in mm. Note this is a clear picture of the difference in growth rate for the two species.
Linearity can also be examined using a Savitzky and Golay (1964) filter or Locally Estimated Scatterplot Smoothing (LOESS fit). The figures below show different patterns for the two species.
Fig 4. LOESS fit for Lunella torquata recaptures (growth rate vs length at tagging, n=1046).
Fig 5. LOESS fit for Turbo militaris recaptures (growth rate vs length at tagging, n=982).
In both Figures 4 and 5, there is a deviation from linearity at smaller sizes (moreso for Turbo), and also at larger sizes (moreso for Lunella). This suggests the von Bertalanffy curve may not be the best model (Haddon, 2021).
Another way of viewing linearity uses residuals analysis from a basic general linear model. Figures 6 and 7 show these results for the two species.
Fig 6. General linear model residuals for growth rate/tagged length regression for Lunella torquata recaptures.
Fig 7. General linear model residuals for growth rate/tagged length regression for Turbo militaris recaptures (n=982)
The dominance of residuals above the line beyond 75mm for Lunella suggests a linearity problem. So does the number of residuals below the line less than 35mm and greater than 60mm for Turbo. The narrowing of the range with greater lengths suggests the variance is not constant across all sizes (heteroscedasticity) for both species. This indicates regression with log-transformed data might be more appropriate, but for the time being, I am not pursuing that because non-linear analyses might address the issue. Francis (1988b) and Haddon (2021) discuss fitting data using non-constant variances.
11.2 Gulland-Holt plots
The simple linear plots of my data based on length-at-tagging are interesting and descriptive, but key fisheries management questions require information about size at age. To answer these, tag-recapture data are usually converted to length-at-age data using the von Bertalanffy growth model. The first way to estimate the particular parameters required is a Gulland-Holt plot, which is a variation on the above figures. The key assumption of the G-H model is that periods at liberty are relatively equal and relatively short.
A Gulland-Holt plot regresses growth rate against the midpoint size of the time-at-liberty interval, i.e., against (L1+L2)/2 rather than against L1 (which, as outlined above, is misleading). This method provides a first estimate of K and Linfinity from the von Bertalanffy growth equation but has some issues (see blog post 10). For Lunella, the plot (below) looks like many in the literature.
Fig 8. Gulland-Holt plot for Lunella torquata for all sites, sexes and seasons but times at liberty not less than 60 days.
For Turbo militaris, the curvilinear pattern evident in Figs 2a and 5 above still exists, so Linfinity is over-estimated.
Fig 9. Gulland-Holt plot for Turbo militaris for all sites, sexes and seasons but times at liberty not less than 60 days.
LOESS fits for both species using mid-length rather than L1 show the same patterns as Figures 4 and 5 above which use L1, except that the inflexion point moves along the curve. For T militaris the curvature with growth rate vs mid length is even greater than with growth rate vs L1.
Figures 2a, 3, 5 and 7 suggest that juveniles should not be included in linear plots for Turbo militaris so an extra filter should be applied to the data to exclude them. But what is a juvenile? What cutoff length should be used for the filter?
My data (shown below) for Turbo militaris recaptures where sex was coded as (J)uvenile or (U)nknown suggest sexual maturity is around 35-40mm.
Fig 10. Percentage of Turbo militaris recaptures with codes J and U for sex following recapture and dissection. This is the reverse of the sigmoid curve shown in Chapter 5.6 (Maturity) in Haddon,2021.
For Figure 10 data, Code J was allocated where there was no colouration in the gonad (it was not based on lack of oocytes or spermatids). Code U was allocated where the gonadal material could not be extracted from the shell with the rest of the foot and viscera after boiling, or in some cases where there was no colouration in the gonad. The point of inflection between 35mm and 40mm suggests sexual maturity at about that size. Seinor et al (2025) found the length at sexual maturity for more northern populations of Turbo militaris to be 37mm at age 0.7 years.
Biologically, it is reasonable to assume there is a change in growth at puberty so I am using that size.
Applying a filter of L1>37.5mm (Fig 11 below) results in a much more reasonable Linfinity value and doubles the variance explained by the regression from 23% to 48%.
Fig 11 Gulland-Holt plot for Turbo militaris with juveniles excluded. (i.e. L1>37.5mm)
The same juvenile bias does not appear as dramatically in the Lunella data, partly because I have far fewer data with L1 between 15mm and 40mm. Defining “juvenile” on the same basis as for T. militaris, the cutoff length is about 30mm. Applying this filter to Lunella data did not change the K, Linfinity or R2 values.
“Forced” Gulland-Holt plots
In my previous blog post I discussed the option of using a “forced” Gulland-Holt plot where Linfinity is fixed and only K is estimated. I calculated a K value for each recapture, using the formula K/year=(mm/month)/(FixedLinfinity-midL)*12 (from De Graal & Prein, 2005).
For all Lunella recaptures over 60 days (n=1046), and fixing Linfinity at 90.6mm, the forced K is 0.47. Compare this with K=0.56 and Linf= 86.3 from the standard plot (Figure 8 above). This is a large difference in K but not so large for Linfinity.
For all Turbo recaptures over 60 days (n=983), fixing Linfinity at 112mm, the mean forced K is 0.45. Compare this with K=0.3 and Linf=142.6 from the standard plot (Figure 9 above). Both K and Linfinity values are very different. However, compared to the G-H plot truncated with no juveniles (Fig 11), there is little difference.
Effects of time at liberty on G-H plot estimates of Linfinity
In the previous blog post about models and filters, I presented the following data on this issue for Lunella.
Linfinity Males | Linfinity Females | Number of male and female recaptures used. | |
All Times at liberty | 92.2 | 85.7 | 837 |
dT’s < 60 days | 155.8 | 125.7 | 99 |
dT’s >= 60 days | 90.5 | 82.9 | 738 |
dT’s >= 60 days using Grotag package in R (2024 final data) | 93.4 | 87.1 | 462/375 |
dT’s>=60d Kienzle et al 2022 (early data) | 93.7 | 88.2 | 272/184 |
2 to 3 months | 87.5 | 83.9 | 74/57 |
3 to 6 months | 88.2 | 82.4 | 150/117 |
6 to 9 months | 91.8 | 88.0 | 107/69 |
9 to 12 months | 91.5 | 78.3 | 41/38 |
dT’s >12 months | 116 | 85.4 | 52/31 |
Table 1. Estimated Linfinity for tagged and recaptured Lunella torquata from all sites for different times at liberty. Estimates are from Gulland-Holt plots like Figure 8 except for the two middle rows which used Francis’ GROTAG model. Note the generally increasing trend in the last 5 rows.
But what about Turbo militaris? The trend for Linfinity estimates is shown in the table below. The increasing estimate as mean dT increases is less clear and three of the seven pairs are similar, but some sample sizes are small.
Linfinity Males | Linfinity Females | Number of male and female recaptures used. | |
All Times at liberty | 115.3 | 114.7 | 166/127 |
dT’s < 60 days | 275.5 | 100.1 | 15/6 |
dT’s >= 60 days | 112.7 | 116.0 | 151/121 |
2 to 3 months | 99.5 | 97.8 | 30/21 |
3 to 6 months | 126.5 | 158.6 | 68/64 |
6 to 9 months | 113.1 | 117.8 | 29/21 |
9 to 12 months | 133.8 | 108.0 | 21/12 |
dT’s >12 months | n/a | n/a | 1/5 |
Table 2. Estimated Linfinity for tagged and recaptured Turbo militaris from all sites for different times at liberty. Estimates are from truncated Gulland-Holt plots (i.e. juveniles removed, L1 >38mm.) n/a=not applicable (insufficient data)
11.3 Ford/Walford plots
As outlined in earlier blog posts, another common linear method for obtaining von Bertalanffy parameters is the Ford/Walford plot which uses data (Lt vs Lt+1) for times-at-liberty of only 12 months. (Ford,1933; Walford,1946). The fitted equation is Lt+1=a*Lt+b. The theoretical von Bertalanffy asymptotic length (L∞) is calculated as [a / (1-b)] and the Brody-Bertalanffy growth coefficient (K) = – logeb. Walford (1946) called this a transformation of the usual growth curve and noted that it applied to averaged growth rates of many species, but not all. He also noted that individual growth rates varied from the line more than averages, as other following scientists have expanded upon. The plot can be derived from tagging data, length frequency data, or from age data.
With ageing data, ages are exactly one year apart, but for tag-recapture data, the number of tag recaptures after exactly one year is difficult to control. So, the problem is to select how close to 12 months the time-at-liberty should be.
As the following table shows, the correlation coefficients for the data plots decline as more data points are added (but not by much). As I did these analyses, I had a wry smile thinking about the published growth papers I have reviewed with less than 50 recaptures or aged individuals in total.
Table 3a. K and Linfinity values for Lunella torquata from Ford/Walford plots using tag recaptures close to 12 months (within 7 days, and then within 2 weeks, 3 weeks, 1 month.
So, which interval to use? For Lunella I settled on 11-13months since it yielded a better sample size of n=75.
Figure 12a. Ford/Walford plot of Lunella torquata recaptures with time-at-liberty between 11 and 13 months.
For Turbo militaris the variation is similar but because they were tagged much more opportunistically (except for Mahon Pool) I have far fewer data points to work with, and most come from Mahon Pool which is not a typical environment. Many were juveniles (see Blog 10) so these T. militaris von Bertalanffy parameter values should be treated with extreme caution.
I used only three periods for “1 year”: 365days +/- 7days , +/- 21days, and +/- 30 days.
| n | K | Linfinity | R squared |
1 year +/- 7 days | 18 | .397 | 126.7 | .8891 |
1yr +/- 21 days | 35 | .493 | 106.1 | .8742 |
1 yr +/- 30days | 58 | .404 | 114.8 | .8814 |
Table 4. K and Linfinity values for Turbo militaris from Ford/Walford plots using tag recaptures close to 12 months (within 7 days, and then within 3 weeks, 1 month).
Fig 12b. Ford/Walford plot for Turbo militaris using times at liberty between 11 and 13 months. Tighter restraint gives a smaller sample size but similar R-squared values.
All the different values presented in this post are confusing, but this is not surprising given the material in previous posts. In particular, the different meanings of K and Linfinity and the high correlation between them (Francis, 1988a) are problematic. So, after I run the data for both species through seasonal growth analysis and partition the data by sex (next few blog posts) and apply non-linear models, (Fabens, GROTAG, and inverse logistic) I will produce a more useful comparative table of K’s and Linfinity’s and make some comments.
For now, it is clear that growth rate for both species generally decreases with size in a more or less linear fashion. Whether growth ceases at some point (asymptotic) or continues until death (indeterminate) is still an open question that will require further analysis of the growth of very large individuals, particularly for Turbo militaris. If growth is indeterminate, there would be no simple analytical solution for length-at-age – i.e. producing an accurate, reliable age/length curve would be difficult.
References
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