ICES Journal of
Marine Science
ICES Journal of Marine Science (2014), 71(8), 2008– 2011. doi:10.1093/icesjms/fsu152
Contribution to the Special Issue: ‘Commemorating 100 years since Hjort’s 1914 treatise on
fluctuations in the great fisheries of northern Europe’
Food for Thought
The graceful sigmoid: Johan Hjort’s contribution to the theory
of rational fishing
Sidney J. Holt*
Voc Palazzetta 68, Paciano 06060, Italy
*Corresponding author: e-mail: [email protected]
Holt, S. J. The graceful sigmoid: Johan Hjort’s contribution to the theory of rational fishing. – ICES Journal of Marine Science, 71:
2008 –2011.
Received 12 August 2013; revised 14 August 2014; accepted 14 August 2014; advance access publication 8 September 2014.
This historical essay describes the theory behind, and implications of, models of optimum yield from an exploited animal population (in particular
for whaling) formulated by Johan Hjort and his colleagues, Per Ottestad and Gunnar Jahn, in the 1930s. The essay places the evolution of fishery
science during the 1930s – 1940s into context.
Keywords: cod, growth overfishing, herring, maximum sustainable yield, Michael Graham, optimum yield, recruitment overfishing, size and age
composition, surplus production model, whaling.
Johan Hjort is widely known and honoured for his immense contributions to oceanography and fisheries ecology. He is less well known
for his seminal role in promoting the application of science to
whaling and the conservation of the great whales. I hope to
correct the balance.
The first formal international agreement on the conduct of
whaling, including a number of regulations intended to limit or
control commercial whaling, was negotiated in 1931 under the auspices of the League of Nations in accordance with an initiative
launched in 1926 by the International Council for the Exploration
of the Sea (ICES). The first important move in August 1929 under
the ICES initiative was the appointment of an International
Statistics Committee (ISC) chaired by Gunnar Jahn, Director of the
Norwegian Statistical Bureau, with Sigurd Risting, Secretary of the
Association of Whaling Companies (AWC), and Johan Hjort as
members. Risting was effectively the founder of the Bureau of
International Whaling Statistics. He had been collecting whaling statistics on his own since the end of the 19th century, completing them,
as far as was possible, back to the beginning of “modern whaling” in
1868 (i.e. using steam catcher boats, bow-mounted cannons, and a
harpoon with a grenade head). In addition, as Secretary of the
AWC, he had as early as 1919 asked the Directors of the whaling companies to ensure that the lengths of all whales killed in the Antarctic
were measured, including the sizes of foetuses in the pregnant females.
Hjort served on the ISC until 1939 when he resigned. During the
1930s, Hjort was the scientific brain behind the construction of the
best international database ever assembled on whale fisheries.
Interest in whaling was not a side line. The catching and processing
of the largest baleen whales—blue, fin, humpback, and sei—was by
far the biggest Norwegian fishery in terms of sheer quantity and economical importance of all Norwegian fisheries during the inter-war
years. This fact has been obscure simply because whale catches were
always recorded as numbers, not weights, and the significance of the
records of quantities and prices of products from the whales was not
always noticed.
The whaling database was remarkable, especially because it
recorded not only catches but the whaling effort, biological information about whales, the details of the whaling fleets, the quantities and
values of products and some other economic data, particularly
about whale oil. The information was collected by national inspectors on whaling factory ships. Thus, by the beginning of World War
II, Hjort and his colleagues had a fairly good idea of the states of the
populations of the whales that migrated to and from their Antarctic
feeding grounds, which were targeted by the whaling industry. They
based their assessments on changes in relative abundance (derived
from catch-per-unit-effort data), size compositions, proportions
of mature and immature animals, sex ratios, longitudinal and
limited latitudinal movements within the Southern Ocean, the
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The graceful sigmoid
timing of migrations and growth of the whales during the feeding
season.
Hjort’s work concerning whales and whaling was not limited to
the interpretation of statistics. The Norwegians and British were
already cooperating in an extensive tagging programme with the
standardized Discovery tags that were retrieved from the blubber
during the flensing process on board the factory ships, and from
the boilers used to extract the oil, also on board. The recording
tag returns was also the responsibility of the national inspectors, occasionally helped by scientists who participated in the long, arduous
voyages. The other research interest was in methods of estimating
the ages of dead whales, which Hjort knew, from Norwegian fisheries research on herring and cod scales, was of critical importance if
science-based management measures were to be taken.
British researchers—mainly—were using the number of corpora
lutea in the ovaries of killed mature females to get a grip on age.
There was one corpora lutea per pregnancy, but these numbers
could not, of course, be converted to absolute ages without knowledge of the frequency of pregnancy (thought to be one every two
years) and the age at sexual maturity. However, if the time
between pregnancies could be estimated accurately, then they
could be used as indicators of age composition of catches and,
hence, total mortality rates. Hjort and his colleagues in Norway
were focused on determining the ages of young animals (up to
3 –4 years) from grooves (striations) on the baleen plates.
Hjort knew from studies of the ages of herring and cod that large
variations in annual catches could be attributed mainly to natural variations in recruitment. And ageing studies had also shown that fish
grow fastest when they are small, and that growth continues, but at
a decreasing pace, throughout their lives. This gave strong support
to the fisheries inspectors of the late 19th century, such as Ernest
Holt and Frank Buckland, who had argued that more fish should
be given the chance to grow bigger and become more valuable by reducing the intensity of fishing operations. Age determination provided the means by which this theory could be quantified and tested.
In 1933, Hjort published a seminal paper on what later became
called “The Theory of Fishing”. It was co-authored by Gunnar
Jahn and Per Ottestad and published in a special issue of a scientific
journal largely devoted to studies of whales (Hjort et al., 1933a).
That issue, entitled Essays on Population, contained three other
papers: one by Hjort et al. (1933b) on Norwegian pelagic whaling
in the Antarctic; another entitled “Introductory Remarks” by
Hjort on whales and whaling (Hjort, 1933); a paper by Ottestad
on “The Mathematics of Growth” (Ottestad, 1933) and one by Alf
Klem on experiments on the growth of cultivated yeast populations
(Klem, 1933), a far cry from whales in size, but a useful subject for
population modelling.
Hjort et al. (1933a) developed the notion of an “optimal catch” at
some intermediate intensity of exploitation. They did this mainly in
relation to the exploitation by Norwegians of the fin whales found
off the coast of Iceland, but they also looked at the fisheries for
spawning herring and cod landed in Norway and the English trawl
fishery for plaice in the Southern North Sea. Hjort and his
co-authors used age-distribution data for the cod and herring—
from scale rings. For the whales, they used a proxy for age—the proportions of juvenile (0– 1 group), immature (2- to 3-year-old
group) and mature individuals in the catches. They took a similar
approach to the study of plaice, referring to the market categories
“large” and “small”—i.e. older and younger fish.
Eventually, Peter Purves, at the British Museum in London,
resolved the ageing problem in baleen whales by reading rings in
2009
the waxy ear plugs, although the frequency of these rings—one or
two per year—remained unclear until the late 1960s. An incorrect
assumption that the calving interval was only 1 year distorted assessments of whale stocks until then (Purves, 1955; Gabriele et al., 2010).
Scientists and authorities have usually discussed whales in terms
of population numbers, not weight (biomass). There is a good
reason for this: they are not easily weighed. Even length measurement when alive is difficult and unreliable. However, they have
always been valued commercially in terms of the weight or, with
respect to oil, the volume of products. The focus on numbers
instead of weigh has important implications in terms of managing
whaling and recovery of depleted populations. In the 1970s, IWC
scientist proposed regulations to restore depleted stocks in terms
of weight, but the whaling industry resisted. Presumably, they
knew that a depleted stock, if protected to allow recovery, takes
much longer to reach an optimal sustainable weight than merely
to recover numerically, especially considering that whales may live
for a century and, like fish, they have indeterminate body growth.
Very soon after the Hjort and Ottestad work, Michael Graham
published his “Modern Theory of Exploiting a Fishery” (Graham,
1935). In 1946, Graham told me that at that time Hjort and he
were corresponding to each other and with Gunnar Rollefsen in
Bergen, and also meeting each other in the ICES context.
Graham’s “theory”, which he applied to North Sea cod and plaice,
was similar to Hjort’s but with a certain difference. Hjort’s
sigmoid curve was derived from the interaction between mortality—through change in the size/age composition of the catch—
and reproduction; the birth rate. Graham’s, on the other hand,
was based on the interaction between a mortality rate and the rate
of growth in body size. Reproduction, through a relatively constant
annual recruitment, played no part in Graham’s theory, essentially
because no relationship between the number of spawners and the resultant recruitment could be found. Conversely, it is not surprising
that whales demonstrate a strong stock–recruit relationship, considering their much lower reproduction rate and the long period
of maternal care for their young offspring.
Graham viewed the sigmoid population growth curve as possessing a sort of magic Graham (1939), and he once told me that he conveyed this idea to Hjort and to Rollefsen. In his remarkable book,
“The Fish Gate”, Graham credits Hjort with the opinion that
“there is something here that has very wide application, in Nature
and in human endeavor” (Graham, 1943, p. 172). Like D’Arcy
Thompson before him (Thompson, 1917), Graham saw the shape
of this relationship as typical of natural biological processes as
opposed to the human-generated shapes of circles and straight
lines found in mechanical devices and constructions. William
Hogarth had described the sigmoid in his The Analysis of Beauty
(1753) as “the serpentine curve that swings both ways”. With
similar thoughts, Graham saw the so-called Gothic arches in architecture as essentially “organic”. Aesthetics aside, the original inventors saw the “magic” of their new arches in their extraordinary
strength. However, it may be that the significant feature of the
sigmoid function is the existence of an inflexion and, hence, an
intermediate reference point.
A decade later, Milner Schaefer used Verhulkt’s symmetrical logistic population growth curve as the basis of his very different
theory of fishing (Schaefer, 1953, 1954, 1955a,b), now usually referred to as the Surplus Production model—a term borrowed
from Karl Marx’s “Surplus Value” theory of political economy—
in which it is assumed that sustained biological productivity, and
hence potential catch, is a simple linear—or, later, as developed by
2010
Schaefer’s associates Pella and Tomlinson (1969), a more flexible
power function of population biomass, regardless of the size and
age composition of that biomass. Schaefer had to ignore age composition because there was at the time no way of determining the
age of an individual tuna (he was studying the yellowfin in the
Eastern Tropical Pacific, although he had published similar ideas
with respect to the Pacific halibut 1954).
Some recent authors (notably Quinn and Deriso, 1999, following
Smith, 1994) have erroneously conflated the Graham/Hjort ideawith
that of Schaefer, referring to the logistic curve—which Schaefer also
supposed could equally apply to population growth in weight as to
the growth in number—as the “Schaefer/Graham” procedure
(Graham considered the curve of population growth in weight as
being sigmoid but did not apply the logistic expression to it).
The matter of the importance of the inflexion of the sigmoid as a
guide to fisheries management was implicitly a subject of considerable controversy at the United Nations’ (UN) Technical Conference
on the Law of the Sea held at the Headquarters of the Food and
Agriculture Organization in Rome in 1955 and it had political repercussions in the first UN Law of the Sea Conference, held in Geneva in
1958. Maximum surplus production (which is equivalent to
Maximum Sustainable Yield, MSY) that corresponded to the inflection point according to Schaefer’s model was advocated as a global
objective for fisheries. In private correspondence (of which copies
are in my files and the archive of Schaefer’s papers), Graham vigorously rejected Schaefer’s approach to both assessment and management and dismissed his request for co-authorship. In controversial
arguments about “surplus production” models, a basic fact was generally overlooked: the Schaefer version of the logistic model was
derived from an assumption about density dependence, but there
was no explicit density dependence in either Hjort’s model (mortality rate vis-à-vis a constant-specific rate of reproduction), or in
Graham’s interaction between a constant mortality rate and a constant sigmoid curve of body weight against age. Body growth of
whales came into the Hjort et al. story only by virtue of size being
an indicator of approaching sexual maturity.
Neither Hjort nor Graham knew in the 1930s about the paper by
Baranov (1918) that provided a formal algebraic description of a
population model—but only in Russian and at a politically inconvenient time of civil war, revolution, and foreign invasions aimed
at St Petersburg, where Baranov was based. A translation in
English by the Canadian scientist W. E. Ricker was circulated
outside the USSR only towards the end of the Second World War.
Unfortunately, Baranov incorporated a linear body growth equation
in his formula instead of a sigmoid one with an upper asymptote
(such as those by L. v. Bertalanffy and by B. Gompertz—see
Quinn II and Deriso) which was a serious flaw in an otherwise innovative approach. Ricker (1940) and Hulme et al. (1947) later
made similar errors, especially Ricker who assumed exponential
body growth. The result is a distortion of stock weight by the inclusion of a very small number of enormously old animals in catches:
Ricker and Hulme et al. corrected this by simply truncating their
age compositions. Thompson (1937; 1950) and Thompson and
Bell (1934), made no such mistake; they applied an empirical
series of yearly fish weights in their rather similar calculations, but
eventually Ray Beverton and my use of the von Bertalanffy expression became almost universally applied in fish stock assessments.
All fishing, including whaling, is selective with respect to the size
and age of animals taken, and it has long been known that exploitation changes the size and age composition of the stock. Hjort
and Graham were well aware of this and their approaches permitted
S.J. Holt
some account to be taken of that phenomenon and the consequences of changes in selectivity. Schaefer and his followers, applying the surplus production theory, were unable to do that. The
models by Beverton and me in the 1950s generalized this matter.
The International Convention for the Regulation of Whaling,
1931 (The Geneva Convention), negotiated under the auspices of
the League of Nations, prohibited “the taking or killing of calves,
suckling whales, immature whales and females accompanied by
calves”. In reality, gunners could not tell whether a whale was
mature or immature before they had killed it. As time went on,
and the stocks declined and the average size—hence average age—
of animals in them diminished, the relative frequencies of mature
and immature animals in catches changed. Hjort and his colleagues
were, therefore, concerned about what Cushing (1972) later defined
as recruitment overfishing.
Graham and his collaborators were, on the other hand, concerned essentially with the consequences of the interaction
between mortality and body growth, and Cushing’s growth overfishing. They mostly favoured increasing the mesh sizes of the codends
of otter trawls. There was little if any evidence at the time (despite
much argument and speculation) that fishing was affecting the
numbers of annual recruits of teleost fish to the major stocks. The
underlying implicit assumption was that there is extremely strong
density-dependent mortality (effectively infinite) in abundant prerecruit life history phases.
At the beginning of the 20th century Ernest Holt and others had
advocated allowing fish the chance to spawn at least once; this idea is
to be found in several official reports—especially in the UK
Parliament—during the second half of the 19th century, but this
idea did not come from an expectation that effective reproduction
was being affected; the idea was essentially to allow fish to grow
bigger than they otherwise would. Fish generally attain maturity
when they reach between 25 and 37% of their theoretical asymptotic weight, near the inflexion somewhere along one of the sigmoid
Generalised Logistic curves postulated by Richards (1959). For the
now commonly used von Bertalanffy (essentially a cubic) version,
this would be at 0.3 of the asymptote, and for the Gompertz or
Fox equation, an exponential version, this is 1/e ¼ 0.37 of the
asymptote.
A consequence of the combination of exponential mortality and
sigmoid growth is that the total weight of a cohort (a single year class
of fish) usually increases as it gets older, then declines as continuing
mortality “overcomes” a slowing growth rate. The point at which
this happens is called the critical age and size, and the maximum
total weight the critical weight. I think it was Herrington (1943)
and Nesbit (1943) who first noticed the significance of the critical
weight, which he called the optimum catch. It was discussed by
Beverton and Holt (1957, p. 374; although in our jointly authored
publication this section was actually written by Beverton). The critical weight of the oldest cohort in the population is, strictly speaking,
the MSY from the stock, but it is unobtainable because it could only
be secured by catching simultaneously all the surviving members of
the cohort, which would require infinite fishing effort.
Herrington’s optimum was, in fact, the upper end of the
“eumetric” ridge along an isopleth diagram such as Beverton and
I constructed, with horizontal axes of fishing rate and selectivity:
the ridge itself maps the “local” MSYs corresponding to all possible
selectivities.
The term “optimum” itself could now be better applied to designate best compromises between two or more alternative measures,
policies or parameters, including such as catch rate/profitability,
The graceful sigmoid
desirable size of fish in the catch, degree of precaution against
human error or natural change.
It is of interest that Herrington was science leader of the US delegation in the peace negotiations with Japan in 1947–1951. He
pressed for adoption of the “abstention principle” to be applied
when a coastal state claimed to be taking the MSY from adjacent
waters. Clearly, the USA’s interest in embedding Schaefer’s MSY
concept into international fisheries management persisted.
The need for serious attention to the consequences of selectivity
and changes in it became evident when, in the post-WWII years, industrial fishing for “reduction” to oil and meal, in which the size of
fish does not matter much, became profitable and technically feasible in many circumstances. Local MSYs, and the fishing intensity
needed to take them, differ enormously between an industrial
fishery and a fishery for mature fish utilizing the same stock.
Hence, it was no accident that the first collapse of a large valuable
fishery—that for anchoveta off the coast of Peru in the 1950s—
was triggered by a virtually infinite foreign market, with minimal
production costs (no buildings needed to house reduction plants
in the coastal climate of Peru) and a fishing gear that would take
baby fish, perhaps aided by changing ocean conditions. Several
other collapses were to follow, for similar reasons—and I am not
talking about the peculiar diadromous salmons and eels (Myers,
1949), or even the grey mullet which spawns in the sea and feeds
in freshwater.
A level of exploitation effort required to obtain any MSYdepends
critically on the chosen selectivity. “Modern whaling” has, however,
practically always been a combination of securing a product suitable
for direct human consumption, and production of “reduced” substances such as protein-rich meal and fats and oils. This consideration has never, however, played a significant role in decisions
about minimum acceptable sizes of whales to be caught.
Unfortunately, Hjort died in October 1948 and, therefore, did
not live to see his, Graham’s, E. S. Russell’s (Russell, 1931, 1942),
Thompson’s and Baranov’s ideas come into bloom in the 1950s.
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Handling editor: Howard Browman