There is a huge bank of literature on age and growth for many animals and plants, let alone marine species. It is even extensive just for the smaller subset of marine species which are taken in fisheries. I will only cover a few issues that informed my data collection and analyses for this turban snail study.
Early in my fisheries career, a basic approach for growth rates was to get a few hundred data points from a few sites, run them through the available computer program to calculate parameters of a growth equation, and use those parameters for management purposes (yield calculations, legal minimum length, etc,). The available analytical methods seemed inadequate to me, given the variability in my data, so my abalone data were not published for some time. It seemed to me that nearby or adjacent reefs were different enough that weighted average growth parameters for management of a whole fishery would be difficult to choose and could be dangerously wrong. There was more difference in growth between two sites at Eden, in southern NSW, than between a good growth site at Eden and a good growth site at Broughton Island, 600km north (near the northern range limit for abalone). Since then, several authors have identified particular problems if individuals have different, inherent growth characteristics and other studies have used sophisticated modelling to identify bias from inadequate sampling of ages and sizes. (see Maunder et al, 2016, Haddon et al 2008)
Processing my tagging data in the early 1980’s involved computer punch cards (“what are they?” you ask) and ONE iteration PER NIGHT on an external mainframe computer. There were no personal computers and no internet. Much later, after I moved into policy and management, colleagues re-analysed my data with a more sophisticated data-fitting model (GROTAG, see Francis 1988b) and more advanced computing, making possible HUNDREDS or THOUSANDS of iterations per MINUTE (see Worthington et al, 1995). It also turned out that morphometry was important: width was a better dimension to use than maximum length (Worthington et al 1995; Worthington and Andrew, 1997).
From all of that, I assumed for this study that lots of data points from multiple sites and times were important.
Prior to complex computing for non-linear parameter estimation, analysing growth data for fisheries purposes involved simple regression plots of growth rate against size, (Gulland-Holt plot) or size at a given length vs size a year later (Ford/Walford plot). The focus, historically, was to produce estimates of the two main parameters of a simple von Bertalanffy growth equation, Lt=Linf(1-e-Kt), where Lt is length at age t yrs, Linf is the theoretical average maximum length, and K is the Brody-Bertalanffy growth coefficient and indicates how concave the growth curve is (it is not a growth rate). In this simple version, the curve does not change shape, there is no inflexion point, and it passes through 0,0 (see Fig 1 below).
For tag recapture data, since age is not known, the simplest logical model using graphical or linear analysis might seem to be a regression of growth rate against size. As length at tagging increases, the expected growth increment would decrease. The average length at which no growth would be expected (Linfinity) would be the x axis intercept. An example follows.
The maximum expected monthly increment (at length zero which might be unrealistic) is 3.5mm and the Linfinity estimate is 106.4mm. While visually beguiling, there are MANY problems with such a naive and simplistic plot. It includes recaptures for some very short time periods where the measurement error might be greater than the actual increment (hence the outlier growth increments at 4mm/month). I did not find negative growth at larger sizes, but many studies have – these can be measurement error, data transcription error or shell damage.
It also assumes decreasing growth from settlement, whereas there is evidence in some gastropods for linear or increasing juvenile growth rates, then decreasing growth from the age or length at sexual maturity (e.g., Helidoniotis and Haddon,2013). However, even a complex logistic growth model to address this curvilinear issue still requires the same basic data of growth rate and length at tagging. So whatever analytical method I was to use, it didn’t change my data collection. It appears that juvenile growth might be linear, especially for Turbo militaris, but I will address juvenile and adult growth in later posts.
Most models assume that growth stops at some average, large size. This may not be true, and while a snail lives, it might grow (albeit slowly). The largest shell I have is 125mm, much larger than the average Linfinity values of around 80mm-90mm produced by the data. Abalone seem to be similar with the largest shells (up to 210mm) well above published Linfinity values.
The graph above also includes seasonal growth. For the same length, a monthly summer increment might be much larger than a monthly winter increment. The same applies to shorter period variability, for example if there is no growth during a storm, or if there is very fast growth based on short term (weeks to months) food abundance and a narrow window of opportunity for grazing.
The graph also includes males and females, and analysis of my data confirmed that growth is different for each sex, so such plots should be separated into male and female.
Finally, interpreting graphical analyses sort-of assumes that individual growth varies for environmental reasons around a K and Linfinity that apply fundamentally to all individuals in the population. As with humans, there must be snails that are genetically programmed with different K’s and Linfinity’s. The existence of such individual variability is the foundation of natural selection and evolution. This complicates the analysis considerably (Sainsbury 1980). It is difficult to identify whether individual growth differences are due to environmental or innate genetic factors (see Webber and Thorson, 2016)
In any case, all my linear plots of this form show high variance (low correlation coefficients). With a variable spawning season, possibly two spawnings per year, and highly variable growth, the difference in size between a fast growing one-year-old and a slow-growing two-year-old can be zero or negative. Without independent ageing methods, the relationship between age and size is difficult, if not impossible, to determine. A length at age curve can be produced from tagging data alone but it cannot be relied upon.
My tagging work was only designed to assess growth rates, not length at age. In my planning, I did hope that complementary length frequency data might provide insights about age. I have kept (but not anlysed) gonad samples and recaptured shells for possible future study. I can make them available to anyone who is interested.
The seasonal variability issue can be resolved by limiting recaptures to a full year which includes all four seasons equally. The Ford-Walford equation for this has the form L(t + 1) = a + bL(t) where L(t +1) and L(t) pertain to lengths separated by one year. From this, the theoretical vonBertalanffy 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.
A Ford/Walford plot of my early tagging data (from all sites) for snails at liberty for about one year appears below.
The variance reduces dramatically from the earlier plot of increment vs length, but the number of useful tag recaptures is also greatly reduced. While I have over 1,200 tag recaptures for Lunella and about half that for Turbo, less than 100 and 50 respectively are for about a year, and therefore suitable for a Ford/Walford plot, (and even those are from 11months to 13months rather than exactly 12 months). In the graph above K is estimated at 0.23 and Linfinity at 84mm. These are very different values to those I obtain with other models.
It is difficult to implement field studies in oceanic conditions to get numerous recaptures at multiple sites exactly one year apart. Gulland and Holt (1959) recognised a need to analyse tag data where the time-period varies and is NOT one year. Their approach was to plot growth increment per unit time against the mid-point of the growth interval [(Ltagging + Lrecapture)/2] which, under partly reasonable assumptions, produces a regression line which cuts the x-axis at L∞. For small, similar time intervals, the slope of the regression line will approximate -K. A Gulland-Holt plot for one of my eight sites is shown below (K=0.6 and Linfinity=82.2).
I see two advantages of the Ford/Walford approach to get initial or approximate estimates and a feel for the data. Firstly, it can be easily constructed using either annual length-frequency data or tagging data and they can thus be compared. It is much easier, practically, to get size samples one year apart than 12-month tag recaptures. (That same LF data, however, can also still be used for a Gulland-Holt plot and for more sophisticated nonlinear modelling.) Secondly, it includes all four seasons and equally so. Thus, if there is strong seasonal growth, a Gulland-Holt plot with many shorter-term recaptures will show much greater variability.
The advantage of both is that they present a simple 2-dimensional picture of the empirical data. A visual grasp of what is going on is harder to achieve for a non-linear, multi-parameter model. It can be done however (see Haddon et al 2008). The disadvantage of both the Ford/Walford and Gulland/Holt plots is that they are too simplistic. I will revisit these two linear methods in future posts.
There are logical problems with using only two parameters to describe a complex and variable biological process like growth. As Knight (1968), Francis (1988) and many others have noted, there are problems with both the K and Linfinity parameters. K is not a growth rate. It is a relative growth coefficient that indicates the steepness of the growth curve. Linfinity is an average, and depending upon whether tagging data or age data are used, it means different things (Francis 1988a). It can also be larger or smaller than the maximum size found in the animal being studied.
One of the complexities discussed by Francis and Sainsbury, among others, is estimating t0 which appears in a slightly more complex 3-parameter version of the von Bertalanffy equation. Essington and Walters (2001) note that the 3-parameter version is the specialised form. The original form has more parameters. t0 is the theoretical age at which length is zero if the three-parameter growth curve represents, on average, the whole life of an animal. Many researchers have simply assumed it was zero, but you only need to think of human growth (with infant growth spurts and puberty), and senility in animals in general, to understand that growth patterns may not be uniform over a lifetime.
After computerised, non-linear parameter estimation was widely available, data could be fitted to growth models with more than two or three parameters. The more parameters, the more realistic is the model, up to a point. “Overfitting” a data model can be a problem. Models with up to nine or ten parameters have been used but generally four to six parameters seem to have been pragmatically decided upon. There is plenty of reading for those interested in how to choose the best model.
Given this background, my aim was to tag and measure enough snails to minimise the “noise” (effects of statistical and individual variability) to be able to broadly explore biological differences between seasons, sexes, years, and sites, and to compare various analytical approaches. I have great difficulty thinking like a statistician or mathematician rather than as a biologist, so my discussions will focus more on the biology than the statistics.
A first consideration was “what to measure?” A model is a simplified representation of reality, and a famous quote is “all models are wrong, but some models are useful”. Generally, growth studies require a relationship between “size” and “age”, and most studies use maximum dimension rather than weight or volume as a proxy for “size”. For turban snails I used the maximum dimension which is technically “shell height” as taxonomists and malacologists describe it. I could have used “breadth”, especially considering research about it being preferable to length for abalone, and the fact that I have found many Lunella individuals that are thicker and have more cup-shaped margins. However, I decided there was more measurement error in that dimension, and it could be explored later. In much of the literature, and for fisheries enforcement purposes, the maximum dimension is called “length” and for simplicity I will use that term in these posts.
A third decision was whether to use change in length (dL= Lrecapture – Ltagging ) or radial increment (distance around the spiral shell from the margin at tagging to the margin at recapture). Since gastropods grow in a spiral, it might be argued that the spiral or radial increment is better. In the photo below, dL is represented by the difference in length of the two white lines (recapture length minus tagging length). Radial increment is the length of the curved yellow line and is always larger.
Once I had some recaptures, I compared the two and found that they were highly correlated (see figure below which shows these in relation to one site but is typical for all my sites).
I decided to stay with the change in length (dL) as my key variable, because, as with breadth, I felt there was more potential measurement error in the spiral/radial increment estimate.
For all the models, both simple and complex, the same basic data are required: length at time one, length at time two, and the dates of time one and two. In another post I describe the methods I used to get that data. I will be comparing the results from the different models in future posts.
References
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