Pauly et al.: Growth and related traits of Hippospongia lachne 
101 
This is assuming a regular spherical shape for the 
sponge. 
For the particular values obtained on the growth rate [of 
sponges] off the Piney Point area the formula, modified for 
ease in working, became [as follows]: 
ye ip MQ (2) 
Since the data gathered were for diameters from 2 to 
7 inches, the use of this formula for extrapolation of the 
curve beyond 7 inches was particularly useful. . . . The for- 
mula indicates that growth will almost completely stop 
when the sponge reaches a 12-inch diameter.” 
The caveat that must be mentioned here is that it is risky 
to extrapolate beyond the range of one’s supporting data. 
Storr (1964) included only data for wool sponges with diam- 
eters up to 7 in (18 cm). Therefore, his estimated asymptotic 
size of a diameter of 12 in (~30.5 cm) for wool sponges is 
tentative. Indeed, larger wool sponges exist, although from 
12 inches on, they usually take a more cylindrical shape, a 
theme to which we return later. 
Storr (1964) writes further, “The rate of growth in 
diameter indicated by [Equation 2] appears to be valid 
for the first 5 or 6 years. Beyond this point the growth 
rate must be assumed to be somewhat less than indi- 
cated by the formula, the increase in diameter gradually 
approaching a uniform rate as the sponge assumes the 
doughnut shape.” 
Using the software WebPlotDigitizer (vers. 4.5; Rohatgi, 
2021), we read the size (diameter in inches) and volume 
(in cubic inches) from figure 7b of Storr (1964) and trans- 
posed them (Table 1). We used these data to estimate the 
Table 1 
The diameter, volume, and age of sheepswool sponges 
(Hippospongia lachne), also known as wool sponges, tagged 
off Florida in the Gulf of Mexico, as read off the graph in 
figure 7b in Storr (1964). Ages in parentheses were extrap- 
olated by using the equation V=aD? (Equation 3), where 
V is volume, D is diameter, and a is estimated by solving 
a=V/D°. Volumes in parentheses were derived through 
Equation 2. 
Age Diameter Volume 
(years) (in®) 
8.6 
44.2 
88.8 
181.9 
269.4 
333.3 
388.7 
434.8 
480.8 
(609.8) 
(716.3) 
value of the parameter a of a diameter-to-volume relation- 
ship of the following form: 
V =aD’, (3) 
where V = the volume; 
D = the diameter; and 
a =the mean value estimated by solving a=V/D® 
and averaging the resulting estimates of a. 
The diameter-age pairs in Table 1 were then used to esti- 
mate the parameters of the VBGF (von Bertalanffy, 1938), 
as presented by Beverton and Holt (1957): 
ib, shi i=e<") (4) 
where L, = the mean size (here: diameter) at age ¢ of the 
animals in question; 
L.,= their asymptotic size (i.e., the mean size 
attained after an infinitely long time); 
K = a growth coefficient (here: year‘); and 
ty = the (usually negative) age they would have had 
at a size of zero if they had always grown in 
the manner predicted by the equation (which, 
in fish species, they usually have not; see, e.g., 
Pauly, 1998). 
Combining Equations 3 and 4 leads to a version of the 
VBGF that can express growth in volume: 
3 
%, = V., (1-8) (5) 
where V, = the predicted volume of wool sponge at age f¢; 
and 
V., =the mean volume attained after an infinitely 
long time. 
The different forms of the VBGF are derived by inte- 
grating the differential equation posited by Piitter (1920): 
dw/dt = HW% — kW, (6) 
where dw/dt = the growth rate in weight (here: volume), 
conceived as the difference between a process 
adding weight and a process reducing it; 
H =the rate of anabolism or the synthesis of 
new protein and other molecules (a process 
adding weight); 
W = weight, representing mass and volume; 
d = an exponent power <1; and 
k = the rate of catabolism or molecule denatur- 
ation (a process reducing weight). 
The parameter d takes a value of 2/3 when Equation 4 is 
integrated to produce the VBGF, and whose parameter 
K is k/3. Also note that stressful environmental factors, 
especially high temperatures, tend to increase k (and K) 
and therefore to reduce asymptotic sizes and longevity 
(Pauly, 2021a). 
Pauly (1981, 2019a, 2021a), based on von Bertalanffy 
(1951), interpreted Piitter’s (1920) equation such that— 
because oxygen is required for anabolism—the positive 
