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THE SEA: THE CURRENT AND FUTURE OCEAN
Journal of Marine Research, 77, Supplement, 283–302, 2019
A metapopulation model for whale-fall specialists: The
largest whales are essential to prevent species extinctions
by Craig R. Smith,1,2Joe Roman,3and J. B. Nation4
ABSTRACT
The sunken carcasses of great whales (i.e., whale falls) provide an important deep-sea habitat for
more than 100 species that may be considered whale-fall specialists. Commercial whaling has reduced
the abundance and size of whales, and thus whale-fall habitats, as great whales were hunted and
removed from the oceans, often to near extinction. In this article, we use a metapopulation modeling
approach to explore the consequences of whaling to the abundance and persistence of whale-fall
habitats in the deep sea and to the potential for extinction of whale-fall specialists. Our modeling
indicates that the persistence of metapopulations of whale-fall specialists is linearly related to the
abundance of whales, and extremely sensitive (to the fourth power) to the mean size of whales. Thus,
whaling-induced declines in the mean size of whales are likely to have been as important as declines
in whale abundance to extinction pressure on whale-fall specialists. Our modeling also indicates
that commercial whaling, even under proposed sustainable yield scenarios, has the potential to yield
substantial extinction of whale-fall specialists. The loss of whale-fall habitat is likely to have had the
greatest impact on the diversity of whale-fall specialists in areas where whales have been hunted for
centuries, allowing extinctions to proceed to completion. The North Atlantic experienced dramatic
declines, and even extirpation, of many whale species before the 20th century; thus, extinctions of
whale-fall specialists are likely to have already occurred in this region. Whale depletions have occurred
more recently in the Southern Hemisphere and across most of the North Pacific; thus, these regions
may still have substantial “extinction debts,” and many extant whale-fall specialists may be destined
for extinction if whale populations do not recover in abundance and mean size over the next few
decades. Prior to the resumption of commercial whaling, or the loosening of protections to reduce
incidental take, the impacts of hunting on deep-sea whale-fall ecosystems, as well as differential
protection of the largest whales within and across species, should be carefully considered.
Keywords: Whaling, whale fall, whale-fall specialist, metapopulation model, habitat loss, extinction
1. Introduction
The bodies of great whales (baleen and sperm whales) form the largest detrital parcels
in the ocean, and their sunken carcasses (i.e., “whale falls”) can create organic-rich habitat
1. Department of Oceanography, University of Hawaii, Honolulu, HI 96822; orcid:0000-0002-3976-0889.
2. Corresponding author: e-mail: craigsmi@hawaii.edu
3. Gund Institute for Environment, Universityof Vermont, Burlington, VT 05405; orcid: 0000-0001-6515-4623.
4. Department of Mathematics, University of Hawaii, Honolulu, HI 96822; orcid: 0000-0001-5959-3304.
© 2019 Craig R. Smith, Joe Roman, and J. B. Nation.
283
284 The Sea: The Current and Future Ocean [77, Supplement
Figure 1. Carcass of a 30-ton gray whale that has been on the seafloor at 1674 m for 6.8 y in Santa
Cruz Basin, off the coast of California, USA. This whale fall is in the sulfophilic stage, as indicated
by the mats of yellow and white sulfur bacteria covering bones and nearby sediments, and the
white shells of vesicomyid bivalves visible in adjacent sediments. The carcass hosts many species
considered to be whale-fall specialists, including polychaete worms and molluscs that have been
found in abundance on whale falls and not in any other habitat.
islands at the seafloor (Smith 2006; Smith et al. 2015). In the food-poor deep sea, whale-
fall communities pass through a series of overlapping successional stages characterized in
turn by mobile scavengers, enrichment opportunists, and sulfophilic (chemoautotrophic)
assemblages (Baco and Smith 2003). The latter two stages are supported, at least in part, by
the large, persistent reservoirs of energy-rich lipids in the whale skeleton (Baco and Smith
2003; Smith et al. 2014, 2015). The enrichment-opportunist and sulfophilic stages combined
provide distinct “island” habitats for >100 faunal species found in abundance on deep-sea
whale falls and, thus far, in no other habitat (Smith et al. 2015). These species are likely to
be whale-fall specialists, requiring a sunken whale carcass (Figure 1) to complete their life
cycles and maintain their populations (Smith et al. 2015). Potential whale-fall specialists
include multiple species with chemoautotrophic endosymbionts such as bathymodiolin
mussels; a diversity of dorvilleid polychaete worms and gastropod limpets that graze mats
of sulfur bacteria (Wiklund et al. 2012; Smith et al. 2015); and heterotrophic species that
directly utilize lipids and collagen from the bones such as bone-eating polychaetes in the
genus Osedax and sipunculan worms.
Whaling by humans has depleted the abundance of great whales by 66%–90% compared
with prehistoric periods (Roman et al. 2014). Industrial whaling during the 20th century
led to even steeper declines of the largest whales, the blues and fins, which were the focus
of intense hunting in the Southern Hemisphere (Tulloch et al. 2018).
2019] Smith et al.: Metapopulation model for whale-fall specialists 285
A necessary consequence of the reduction of great-whale populations is a decrease in the
frequency of whale falls arriving at the deep-sea floor, resulting in habitat loss for species
dependent on whale falls (Smith 2006). Habitat loss is a major cause of species extinctions
across a broad range of ecosystems (Pimm et al. 1995); we thus expect loss of whale-fall
habitat to apply extinction pressure on species dependent on whale falls.
There are at least two major consequences of whaling for whale-fall specialists and
their habitat: (1) a vast reduction in the abundance of live whales and hence the potential
for whales to die and sink to the deep-sea floor to provide whale-fall habitats; and (2) a
reduction in the mean size of great-whale species by removing the largest species (e.g.,
blue whales) and the largest individuals within species. In this article, we use a variation of
the Levins (1969) metapopulation model to explore how changes in abundance and size of
whales could influence the likelihood of persistence of whale-fall specialists, in particular
species dispersing as planktonic larvae and living as adults on lipid-rich whale skeletons.
In brief, we find that the decline in the average size of great whales could be as important
as the decline in whale abundance in endangering whale-fall specialists. Extinctions of
whale-fall specialists because of whaling are likely to be most advanced (with extinction
debts realized) in regions such as the North Atlantic, where great-whale populations (e.g.,
those of gray, right, and bowhead whales) have been depleted or extinct for >100 years
(Notarbartolo di Sciara et al., 1998; Allen and Keay 2006).
a. Goal of the article
We use a metapopulation modeling approach to explore the consequences of whaling to
the abundance and persistence of whale-fall habitats, and thus to habitat loss and potential
extinction of whale-fall specialists. We focus on the sulfophilic stage of whale-fall suc-
cession, when reduced inorganic chemicals (e.g., sulfide) emanating from the whale bones
support diverse macrofaunal assemblages sustained by chemoautotrophy.
b. Metapopulation context for whale-fall communities
Deep-sea whale falls provide a classic metapopulation framework, with distinct habitat
islands supporting populations of specialized species linked by dispersal to other island
populations. Whale falls, particularly during the sulfophilic stage, are seafloor oases rich in
organic matter and reduced inorganic chemicals distributed across a vast deep-ocean floor
where organic matter and reduced chemicals are very limited (Baco and Smith 2003; Smith
et al. 2015). Whale-fall habitats are created abruptly after a carcass sinks to the seafloor.
The sulfophilic stage, during which reduced chemicals from anaerobic decomposition of
whale-bone lipids support a chemoautotrophically dependent fauna (Figure 1), can per-
sist on deep-sea whale skeletons for decades (Schuller, Kadko, and Smith 2004; Smith
et al. 2015). Whale falls are widely dispersed, typically separated by >10 km, and provide
seafloor habitat patches of approximately 10–100 m2in area (Smith et al. 2014). Whale-fall
286 The Sea: The Current and Future Ocean [77, Supplement
communities contain a diversity of macrofaunal species, including many apparent whale-
fall specialists, with limited adult mobility (Baco and Smith 2003; Smith et al. 2015); these
species require larval dispersal to colonize new whale falls. The lipids and collagen in whale
skeletons become depleted over time as they are consumed by sulfate-reducing microbes
and by metazoans such as Osedax (Treude et al. 2009; Smith et al. 2015). The sulfophilic
assemblage on the whale skeleton persists as long as bone lipids and other tissue are avail-
able; once these resources are depleted, the sulfophilic habitat island disappears (Smith et al.
2015). The persistence time of the sulfophilic stage has been shown to vary with carcass
size, and it may also vary with water depth and other environmental parameters (Smith
et al. 2015). Populations of whale-fall specialists can disappear from individual carcasses
because of declining resources related to bone lipids, but also from Allee effects, species
interactions such as predation and competition, and pathogen outbreaks. These characteris-
tics lead to a system of island habitats of finite duration, connected by larval dispersal, and
subject to population extinctions, making whale-fall populations well suited to study with
metapopulation and metacommunity models (Hanski and Gilpin 1997; Roman et al. 2014).
Such characteristics are also broadly analogous to hydrothermal vent communities in the
deep sea (Mullineaux et al. 2018).
2. A metapopulation model and extinction criterion
To analyze the criteria for global extinction of a whale-fall specialist, let us consider a
model that takes into account a number of related factors. The basic equation governing our
model is a variation of the classic Levins metapopulation equation (Levins 1969, 1970).
Our equation is
dO
dt =q(S −O) −ϕF. (1)
where trepresents time in years; Ois the number of whale-fall sites occupied by our
species; qis the probability that a given unoccupied site will become occupied in a given
year; Srepresents the total available number of nondepleted whale-fall sites (i.e., skeletons
containing sufficient lipids to support the sulfophilic stage); ϕrepresents the probability
that a site that is Tyears old is occupied by our particular species, where Tdenotes the
mean persistence time of an active (i.e., lipid-containing) whale fall as a habitat; and Fis
the annual number of new whale falls, which at equilibrium is also the number of whale-fall
sites that become lipid-depleted annually.
For future reference, note that at equilibrium the total number of whale-fall sites satisfies
S=F·T.
The last term of the differential equation has been modified from the standard model to
account for the fact that, because whale-fall habitats last for a relatively long period of time,
an older site that is becoming lipid-depleted is much more likely to be occupied than an
average site. On the other hand, we have omitted the term for habitat extinction resulting
from causes other than resource depletion. The main contributor to this term for whale-fall
2019] Smith et al.: Metapopulation model for whale-fall specialists 287
specialists would probably be predation or possibly pathogens (e.g., Van Dover et al. 2007),
and random habitat extinctions seem unlikely because of large population sizes in whale-fall
specialists (Baco and Smith 2003; Smith et al. 2015). An analysis of the model including
both terms (habitat loss attributable to resource exhaustion and other causes) is given in the
Appendix. The analysis is only slightly more complicated, and the results are qualitatively
similar.
For our metapopulation model, P=O/S is the fraction of whale-fall sites occupied by
a particular whale-fall species.
One goal is to describe the proportion of available sites that should be occupied by a
particular species prior to whaling to ensure survival after whaling, in terms of the population
of living whales and their mean length. To this end, we introduce those variables: Nis the
number of living (nonjuvenile) whales, and Ldenotes the mean length of a whale sinking
to the seafloor.
There are three parameters, whose roles will be explained subsequently: mis the annual
mortality rate for (nonjuvenile) whales, so that F=m·N;ais a constant such that T=a·L;
and dis a parameter for the colonization rate, used in equation (2).
Our analysis will consider potential equilibrium situations, pertaining to either prewhaling
or postwhaling. For that reason, we suppress the possible time dependence of some of these
variables.
There is an implicit assumption that bone-lipid depletion in adult whale skeletons ulti-
mately results from microbial processes deep within the bone matrix (Treude et al. 2009;
Smith et al. 2015); thus, whale-fall persistence time does not depend on whether the site
is occupied by a particular macrofaunal species. We also assume that interactions between
macrofaunal species at a whale fall do not cause local population extinctions; this assump-
tion may not apply to some members of the genus Osedax with overlapping niches and thus
a potential for competitive exclusion (Higgs et al. 2014). Because whale-bone resources
(lipids, collagen) are consumed by bacteria and bone borers such as Osedax, which degrade
the bone from the surface area inward (Schuller et al. 2004; Treude et al. 2009; Smith et al.
2015), we assume that the rate of bone degradation is proportional to bone surface area
(i.e., L2). The amount of degradable bone resource is proportional to bone volume (i.e.,
L3) (Schuller et al. 2004; Higgs, Little, and Glover 2010), so we assume that the persis-
tence time Tof a mature whale skeleton harboring whale-fall specialists is proportional to
L3/L2=L. Thus, for some constant a, we obtain T=aL.
Populations of whale-skeleton colonists are concentrated on the skeleton surface, so we
assume that the number of individuals on an occupied site is proportional to skeleton surface
area, and hence to L2. The total population size of a whale-skeleton specialist is proportional
to the number Oof occupied sites and to the number of individuals per site.
Let us further assume that q(t), the probability that an unoccupied whale-fall site will
become colonized within a given year, is proportional to the size of the larval pool produced
by the existing colonies. The number of larvae produced in turn depends on the total popu-
lation on all occupied sites, and hence on the product of Oand L2.Asq(t) is a probability,
288 The Sea: The Current and Future Ocean [77, Supplement
it has a maximum value of 1. Introducing a proportionality constant d, we can write
q=min(dL2O,1). (2)
The constant ddepends on the rate of production of larvae per unit area of colony, but also
on the spacing of the occupied whale falls (wider spacing yields greater larval wastage),
which will vary with basin size and the number of active whale-fall habitats. The constant
dmay be hard to estimate, but as long as we are comparing prewhaling and postwhaling
situations from the same basin, we do not need to know its value. Moreover, equation (2)
implicitly assumes that larvae produced by whale falls are instantaneously mixed so as to be
homogeneous (equally available) across the modeled region. This may not be fully realistic,
but it is necessary to assume in a model without spatial structure.
The crucial term ϕcan be determined by considering the probability ψ(t) that a given
whale fall is occupied by our modeled species after tyears. This satisfies the differential
equation
dψ
dt =q(1−ψ), (3)
ψ(0)=0.(4)
Regarding qas a constant (at equilibrium), the solution is
ψ(t) =1−e−qt,(5)
so that
ϕ=ψ(T ) =1−e−qT.(6)
Now we rewrite the original differential equation (1):
dO
dt =q(S −O) −ϕS
T.(7)
At equilibrium, we set dO /d t to zero, divide by S, and solve to obtain
P=O
S=1−ϕ
q·T.(8)
This is interesting, but ϕdepends on q, and qdepends on O, so the variables are not
independent. In principle, we would like to rewrite equation (8) using only the variables
O,L, and Nand then solve for Oto obtain an expression of the form O=f(L,N) for
the steady-state occupancy as a function of length and the number of whales. This is not
practical, so we take a different tack.
2019] Smith et al.: Metapopulation model for whale-fall specialists 289
Substituting the expression for ϕinto the right side of equation (8) yields
P=1−1−e−qT
qT (9)
=e−qT −1+qT
qT .(10)
Let us assume that q<1, so that q=dL2O. Then, substituting for Oand Sin the left
side,
q
dL2·F·T=qT
dL2FT2=qT
ma2dL4N=e−qT −1+qT
qT ,(11)
whence
1
ma2d=L4Ne−qT −1+qT
(qT )2.(12)
Because qT =adL3O, equation (12) implicitly gives the relation between O,L, and N,
but in a form that cannot be solved for O; moreover, it involves parameters m,a, and d,
which we can only roughly estimate. However, there is another way to use equation (12).
Let us use q0,L0,N0, and so on, for the prewhaling values, and q1,L1,N1, and so on, for
the postwhaling values. Because the left side (1/ma2d) of equation (12) is a constant, we
have that
L4
0N0e−q0T0−1+q0T0
(q0T0)2=L4
1N1e−q1T1−1+q1T1
(q1T1)2.(13)
To get the extinction criterion, we set q1=0, which involves taking a straightforward
calculus limit for the expression in parentheses:
L4
0N0e−q0T0−1+q0T
(q0T0)2=L4
1N1·1
2(14)
or
e−q0T0−1+q0T0
(q0T0)2=1
2L1
L04N1
N0(15)
at the threshold for extinction.
Given the values of L1/L0and N1/N0, equation (15) can be solved for q0T0numerically.
Comparing the left side of equation (15) with the formula for P0,
P0=e−q0T0−1+q0T0
q0T0
.(16)
290 The Sea: The Current and Future Ocean [77, Supplement
Table 1. Prewhaling occupancy rate of whale falls (P0) required for the postwhaling survival by a
whale-fall specialist in terms of abundance of live whales before (N0) and after (N1) whaling and
the mean length of whales before (L0) and after (L1) whaling.
N1/N0
0.10.20.30.40.50.60.70.80.91.0
0.11111111111
0.21111111111
0.31111111111
0.411110.99 0.99 0.99 0.99 0.99 0.99
L1/L00.5 1 0.99 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97
0.6 0.99 0.99 0.98 0.97 0.97 0.96 0.95 0.95 0.94 0.93
0.7 0.99 0.98 0.96 0.95 0.94 0.92 0.91 0.89 0.88 0.86
0.8 0.98 0.96 0.93 0.91 0.89 0.86 0.83 0.80 0.76 0.73
0.9 0.97 0.93 0.89 0.85 0.80 0.74 0.68 0.61 0.54 0.47
1.0 0.95 0.89 0.82 0.74 0.64 0.53 0.41 0.28 0.15 *
We multiply the right side of equation (15) by q0T0to obtain the value of the prewhaling
occupancy ratio P0needed for survival; that is,
P0>q0T0
2L1
L04N1
N0.(17)
Thus, we can use P0as a surrogate for the unknown colonization parameter dto determine
the prewhaling occupancy ratio that would be needed for the species to survive after whaling.
In the next section, we will apply this criterion to data for various ocean basins and globally.
3. Applications of the survival criterion
Table 1 uses the inequality (17) to estimate, for various values of L1/L0and N1/N0,
the minimum proportion of whale-fall habitats that a species must have occupied prior to
whaling to allow metapopulation survival after postwhaling reductions in whale abundance
and mean body size. Figure 2 shows the level curves for the function in Table 1.
Our model indicates that the long-term persistence of whale-fall species after whaling,
and hence the chances of whale-specialist extinctions, is (a) sensitive (linearly related) to
reductions in the number of living whales, and (b) extremely sensitive (to the fourth power)
to reductions in the mean size of whales.
If whale populations were reduced by only 10% (to 90% of prewhaling levels) but the
mean size of whales were reduced by 30% (i.e., L1/L0= 0.7) through removal of the largest
species and the largest individuals within a species, a prewhaling occupancy rate of nearly
90% would be required for long-term survival of a whale-fall specialist in the aftermath
of whaling. In fact, the largest individuals, as well as the largest species, of great whales
were typically the targets of commercial whaling. The mean body size of blue, fin, and
2019] Smith et al.: Metapopulation model for whale-fall specialists 291
Figure 2. Level curves for the function P0, showing the pre-whaling occupancy rate of whale falls
required for the post-whaling survival by a whale-fall specialist. P0is plotted in terms of the relative
decline in mean whale length (L1/L0) versus the relative decline in whale abundance (N1/N0)
resulting from whaling. For example, if a L1/L0versus N1/N0point falls on the green curve
marked P0=0.99, then a whale-fall specialist would need to have occupied at least 99% of the
available whale-fall sites prior to whaling to avoid extinction. The “+” signs mark our estimates
for the various ocean regions, based on Table 2. Note that P0is initially much more sensitive to
changes in whale length than to changes in whale abundance. The calculations and graph were
done using Derive 6 (Texas Instruments).
sei whales caught during the 20th century declined dramatically during industrial whaling,
with L1/L0apparently falling to 0.8–0.9 over a 70-year period (Clements et al. 2017). The
reduction in mean body size within species, plus the dramatic reductions in the abundance
of the largest species of whales, very likely reduced L1/L0across all great whale species
to well below 0.8. Considering that the largest species of whales, the blue whale, remains
heavily depleted, and that widespread commercial whaling was halted in the 1970s and
1980s, which is less than one life span ago for many whales (Bannister, 2018), the mean
body size of great whales likely remains substantially below prewhaling levels.
According to our metapopulation model, the drastic reduction in the number and size of
whales sinking to the deep-sea floor is likely to cause, or to already have caused, numerous
species extinctions of whale-fall specialists. Even maintaining great whale populations at the
International Whaling Commission sustainable yield level of 54% of prewhaling population
sizes may put whale specialists at risk of extinction, requiring prewhaling occupancy rates of
more than 90% of lipid-rich skeletons at the seafloor at the current length ratio L1/L0≈0.7
(and more than 75% for L1/L0=0.9).
292 The Sea: The Current and Future Ocean [77, Supplement
It is important to note, however, that whale-specialist extinctions resulting from reductions
in the abundance and size of whale carcasses would not occur instantaneously. Rather, they
would lag behind the reductions in whale populations as whale-fall habitats and associated
populations of specialists die out, causing an “extinction debt” (Kuussaari et al. 2009). The
duration of such a debt, or the length of time it will take for species richness on whale
falls to relax to a lower postwhaling equilibrium, is related to the persistence times of
the sulfophilic whale-fall habitats and to the life spans and recruitment dynamics of the
whale-fall specialists destined for extinction (Smith 2006; Kuussaari et al. 2009). Because
these whale-fall habitats can last for decades, the abundance of sulfophilic communities
should respond with a time lag of at least 30–40 years to whale depletion (Smith 2006),
and long-lived or self-recruiting whale specialists may persist even longer on large whale
falls that last for 70–80 years (Schuller et al. 2004). If great-whale populations are allowed
to recover in numbers and body size before extinction debts are fully paid, biodiversity
losses in whale-fall specialists could be mitigated (cf. Kuussaari et al. 2009). Because of
this time lag, the temporal and spatial patterns of whale-population decline and recovery
are likely to influence the realized levels of whale-specialist extinctions in different ocean
basins. In the North Atlantic, for example, some great-whale species or populations have
long been extinct; the North Atlantic gray whale and the eastern population of the North
Atlantic right whale were likely extirpated more than 150 years ago (Notarbartolo di Sciara
et al. 1998; Smith 2006). Ongoing commercial whaling for fin and minke whales, and
incidental mortality from shipping and fishing-gear entanglements, could continue to apply
extinction pressure to these whale-fall communities. In the North Pacific, the removal of
many great whales occurred as recently as the 1960s and 1970s, and some populations (e.g.,
the northeast Pacific gray whale) have substantially recovered in abundance (Smith 2006),
so extinctions might have been avoided. Declines in the diversity of whale-fall specialists
resulting from whaling may be relatively advanced in areas of the North Atlantic, but they
could be mitigated by whale-population recoveries in parts of the northeast Pacific. There is
reason to be concerned about the Southern Hemisphere, which suffered the world’s largest
removals of great whales from 1920 to 1960 (Smith 2006). Even with 100 years of future
protection from whaling, the Antarctic blue, fin, and southern right whales will likely remain
at <50% of preexploitation numbers because of slow population growth rates (Tulloch et al.
2018), yielding ample time for extinction debts to be realized.
Because the extinction effects of whaling are likely to be region specific, we now turn
to projections for particular ocean regions. For these projections, data from Doughty et al.
(2016) and Clements et al. (2017) were used to estimate the number of whales and their
mean length, respectively, for various ocean regions and for the global ocean (see the
Appendix). The whale species used for these calculations are blue, bowhead, Bryde’s, fin,
gray, humpback, minke, right, sei, and sperm whales. The column labeled P0in Table 2 is
derived according to the condition set in equation (17).
An obvious conclusion from these projections is that extinction pressure is more extreme
in the Southern Ocean, even without considering extinction debt, than in the northern oceans.
2019] Smith et al.: Metapopulation model for whale-fall specialists 293
Table2. Prewhaling occupancy rate of whale falls (P0) needed to ensure long-term survival of a whale-
fall specialist after whaling for different ocean regions. Lengths before (L0) and after (L1) whaling
are given in meters, and population sizes before (N0) and after (N1) whaling are in thousands.
Basin L0L1N0N1P0
North Atlantic 15.9 12.3 875 369 0.91
North Pacific 17.5 14.9 712 296 0.87
Southern Hemisphere 17.0 10.1 2,461 819 0.98
Global 16.8 11.6 4,049 1,484 0.96
This is because the largest species of whales remain at very low abundances in the Southern
Hemisphere; that is, Southern Ocean blue and fin whales are still at a few percent of estimated
prewhaling levels (https://iwc.int/status). These two species were also among the first to
reach commercial extinction in the Southern Hemisphere, as whalers focused on the largest
individuals and species first, before depleting the smaller humpback and sperm whales, and
eventually resorting to the diminutive minke whale (Rocha, Clapham, and Ivaschenko 2015).
To make matters worse for deep-sea communities, many whale carcasses were completely
removed from the ocean during the 20th century, when they were processed aboard factory
ships rather than being flensed alongside wooden ships, where skeletons and organs would
often be left to sink (Smith 2006). Given that most blue and fin whales were removed from
the Southern Ocean between 1920 and 1960 (i.e., more than 60 years ago), it is likely that
whale-specialist extinction debts are well on their way to being paid, with extinctions likely
to reach completion before blue and fin whale populations recover to prewhaling levels.
The situation in the North Pacific is not as dire because P0is lower (i.e., 0.87 vs. 0.98 in
the Southern Hemisphere), whale removal was more recent (occurring after 1955; Smith
2006), and some whale populations (e.g., gray and humpback whales) are recovering. The
diversity of whale specialists may thus be substantially closer to prewhaling levels in the
northeast Pacific, especially within the gray whale’s range.
Our global results should be interpreted with caution, as they combine basins that may
have distinct species and metapopulations of whale-fall specialists (e.g., Sumida et al.
2016). They also do not consider the geographic and temporal variation of commercial
whaling. Whale species along the European coast, such as the North Atlantic right whale,
were extirpated centuries ago, and whale-fall specialists in that area are likely to have
disappeared along with their habitat. This habitat loss might have had a particular effect on
whale-fall species in the Mediterranean, which has a high degree of endemism (Coll et al.
2010). Despite these potential limitations, our results emphasize that whaling may have
severe consequences on the survival prospects for whale-fall specialists globally, as well as
in major ocean regions.
4. Conclusions and insights
Our metapopulation model for whale-fall specialists suggests that occupancy rates of
deep-sea carcasses prior to whaling would have to have been extremely high (0.87 to 0.98)
294 The Sea: The Current and Future Ocean [77, Supplement
for these specialists to persist in many parts of our whale-depleted oceans. Our modeling
also indicates, surprisingly, that the extinction potential for whale specialists may be more
sensitive to reductions in the mean size of whales than to declines in population abundance.
Some parts of the oceans, especially the Southern Hemisphere, are predicted to be especially
vulnerable to whale-specialist extinctions because very large populations of the biggest
whales were heavily depleted and remain at extremely low abundance levels; furthermore,
these depletions occurred more than 60 years ago, so extinction debts may be well on their
way to being paid. For other regions, such as the North Pacific, extinction pressure on
whale-fall specialists does not appear be as large or as long-standing because some whale
populations are in advanced stages of recovery.
The model yields some new and intriguing insights. First, in regions where whale abun-
dance and mean size have long been reduced, such as the North Atlantic and very likely
the Southern Ocean, the diversity of whale-fall specialists is expected to be low relative
to prewhaling conditions. Whale-fall occupancy rates by specialists may also be relatively
low in these regions because of reduced connectivity caused by lower larval production
and higher carcass spacing. Second, as the abundance and mean size of great whales return
to prewhaling levels, occupancy rates of whale falls by specialists should eventually be
high (>80%); otherwise, these species would be unlikely to have survived the prolonged
bottleneck in whale-fall habitat availability. Unfortunately, we cannot test these predictions
because whale falls are too poorly sampled to generate reliable estimates of whale-fall
occupancy rates (Smith et al. 2017). A third insight is that the evolution and persistence
of whale-fall specialists are likely to have been highly dependent on the evolution and
widespread occurrence of very large whales; thus, regions with an abundance of very large
whale species over evolutionary timescales, such as the Southern Ocean, are likely to have
developed the greatest diversity of whale-fall specialists. Finally, conservation of whale-
fall specialists is likely to be highly dependent on conservation of the largest whales, both
within and across species, because of nonlinear effects of whale length on whale-fall habitat
persistence times and larval production. By inference, the carcasses of juvenile whales may
be relatively unimportant in preventing extinctions from whaling because of this nonlinear
scaling with whale size. The observed absence or very short duration of the sulfophilic
stage on juvenile whale falls (Baco and Smith 2003; Smith et al. 2015) is consistent with
this inference. Thus, as whale populations recover, any discussion of the resumption of
commercial whaling should address the impact of hunting on deep-sea whale-fall ecosys-
tems and especially consider differential protection of the largest whales within and across
species.
APPENDIX
In the model presented in Section 2, site extinction was assumed to be attributable to
depletion of resources, and the usual term corresponding to random site extinction was
omitted from the metapopulation equation. The first part of the Appendix considers the
2019] Smith et al.: Metapopulation model for whale-fall specialists 295
case when both terms are included. As we shall see, the adjustments required are relatively
minor, and the results are qualitatively the same.
The calculations for Table 2 are based on estimates for historical and current whale
populations and lengths. The second part of the Appendix reproduces the spreadsheet we
used for those calculations and gives the sources for those estimates.
a. A generalized model and extinction criterion
For a metapopulation model that allows site extinction attributable to both resource deple-
tion and random other factors, we replace equation (1) with
dO
dt =q(S −O) −εO−ϕF, (A1)
where εrepresents the annual probability that an occupied site will cease to be occupied for
reasons other than resource depletion, and the other variables are the same as in Section 2.
The three terms on the right side of equation (A1) represent (i) the site colonization rate
q(S −O), (ii) the random site extinction rate −εO, and (iii) the rate for site extinction
attributable to resource depletion −ϕF.
The standard Levins model contains the first two terms. The model in Section 2 assumes
the second term is negligible and contains only the first and third terms. Now we consider
equation (A1) with all three terms on the right side.
Again note that at equilibrium the total number of whale-fall sites satisfies S=F·T, and
let P=O/S be the fraction of the nondepleted sites occupied by our particular species.
As before, Nis the number of living (nonjuvenile) whales, and Ldenotes the mean length
of a whale carcass. Likewise, the parameters m,a, and dare used as previously, and the
probability qsatisfies
q=min(dL2O,1). (A2)
The crucial term ϕin equation (A1) is determined by considering the probability ψ(t)
that a given whale-fall is occupied by our species after tyears. The differential equation for
dψ/dt must be modified to include a term for random site extinction:
dψ
dt =q(1−ψ)−εψ,(A3)
ψ(0)=0.(A4)
Regarding qas a constant (at equilibrium), the solution is
ψ(t) =q
q+ε(1−e−(q+ε)t ), (A5)
so that
ϕ=ψ(T ) =q
q+ε(1−e−(q+ε)T ). (A6)
296 The Sea: The Current and Future Ocean [77, Supplement
Now we rewrite the original differential equation (A1):
dO
dt =q(S −O) −εO−ϕS
T.(A7)
At equilibrium, we set dO /d t to zero, divide by S, and solve to obtain
P=O
S=1
q+εq−ϕ
T.(A8)
Substituting the expression for ϕinto the right side of equation (A8), and simplifying yields
P=1
(q +ε)T qT −q
q+ε(1−e−(q+ε)T )(A9)
=q
(q +ε)2Te−(q+ε)T −1+(q +ε)T .(A10)
Still mimicking Section 2, let us assume that q<1, so that q=dL2O, then substitute for
Oand Sin the left side:
q
dL2·FT =qT
dL2FT2=qT
ma2dL4N=q
(q +ε)2Te−(q+ε)T −1+(q +ε)T ,(A11)
whence
1
ma2d=L4Ne−(q +ε)T −1+(q +ε)T
(q +ε)2T2.(A12)
This is the same as equation (12) in the main text, with qreplaced by q+ε. Note, however,
that the equations (A9) and (A10) for Pare different by a factor of q/q +ε.
Again we use q0,L
0,N
0, and so on, for the prewhaling values, and q1,L
1,N
1, and so on,
for the postwhaling values. Because the left side (1/ma2d) of equation (A12) is a constant,
we have that
L4
0N0e−(q0+ε)T0−1+(q0+ε)T0
(q0+ε)2T2
0=L4
1N1e−(q1+ε)T1−1+(q1+ε)T1
(q1+ε)2T2
1.(A13)
To get the extinction criterion, set q1=0, which gives
L4
0N0e−(q0+ε)T0−1+(q0+ε)T0
(q0+ε)2T2
0=L4
1N1e−εT1−1+εT1
ε2T2
1(A14)
or
e−(q0+ε)T0−1+(q0+ε)T0
(q0+ε)2T2
0
=L1
L04N1
N0e−εT1−1+εT1
ε2T2
1(A15)
at the threshold for extinction.
2019] Smith et al.: Metapopulation model for whale-fall specialists 297
In order to use equation (A15), we need estimates for the values for L1/L0,N1/N0,ε,
and T1. The value of T0can then be obtained from the assumption T1/T0=L1/L0. Next,
we evaluate the right side of equation (A15); note that the third term is a function of εT1,
which has a limit of 1/2 when εT1=0 and decreases slowly toward 0 as εT1→∞.Ifthe
right side evaluates to R, then numerically solve
e−x−1+x
x2=R(A16)
for x=(q0+ε)T0. Because εand T0are known, this determines q0. Comparing equations
(A10) and (A15), we see that
P0=q0T0R(A17)
at the point of extinction. Thus, the criterion for survival becomes
P0>q
0T0R. (A18)
Again, P0is the prewhaling occupancy ratio that would be needed in order for the species
to survive postwhaling.
Let us work through an example of the calculations. Consider the case L1/L0=0.85,
N1/N0=0.40, ε=0.10, and T1=30, with the first two corresponding roughly to the
global figures. From T1/T0=L1/L0, we obtain T0.
=35. Plugging these numbers into the
right side of equation (A15) gives R=0.0474. Solving equation (A16) numerically, we
get x=(q0+ε)T0=20, whence q0T0=16.5. The extinction criterion then becomes
P0>16.5×0.0474 .
=0.78.(A19)
Note that this is somewhat lower than the global extinction criterion P0>0.96 given in
Table 2, mainly by the factor q0
q0+ε
.
=0.471
0.571
.
=0.82. This may seem paradoxical at first,
but random site extinctions also have a large effect on prewhaling occupancy. Indeed, by
equation (A8), the maximum possible occupancy (with q=ϕ=1) is
Pmax =1
1+ε1−1
T.(A20)
b. Populations and mean lengths of great whales
The calculations for Table 2 in the article are based on estimates for historical and cur-
rent whale populations and lengths. Table 3 reproduces the spreadsheet we used for those
calculations. Present and historical population sizes are from Doughty et al. (2016) with
several updates.
Population sizes for the Antarctic minke whale are from Ruegg et al. (2010) and the
International Whaling Commission (2013a). Population sizes for the southern right whale
298 The Sea: The Current and Future Ocean [77, Supplement
Table 3. Estimates for historical and current whale populations and lengths. Lengths are given in
meters, and populations in thousands.
Basin Species L0L1N0N1
North Atlantic
Blue
(Balaenoptera musculus)
27 22 7.5 0.4
Bowhead
(Balaena mysticetus)
20 20 80 8
Common minke
(Balaenoptera acutorostrata)
7 7 211 157
Fin
(Balaenoptera physalus)
23 20 73 56
Humpback
(Megaptera novaeangliae)
16 16 112 20
Right
(Eubalaena glacialis)
16 16 14 0.5
Sei
(Balaenoptera borealis)
16 14 10.6 7
Sperm
(Physeter microcephalus)
18.5 14.5 367 120
Means/totals 15.9 12.3 875.1 368.9
L1/L0=0.775 N1/N0=0.422
North Pacific
Blue
(Balaenoptera musculus)
27 22 6 3
Bowhead
(Balaena mysticetus)
20 20 30 18
Bryde’s
(Balaenoptera brydei)
15 15 52 41
Common minke
(Balaenoptera acutorostrata)
774732
Fin
(Balaenoptera physalus)
23 20 65 31
Gray
(Eschrichtius robustus)
15 15 25 16
Humpback
(Megaptera novaeangliae)
16 16 20 20
Right
(Eubalaena japonica)
16 16 32 0.4
(Continued)
2019] Smith et al.: Metapopulation model for whale-fall specialists 299
Table 3. Continued
Basin Species L0L1N0N1
Sei
(Balaenoptera borealis)
16 14 68.4 14.7
Sperm
(Physeter microcephalus)
18.5 14.5 367 120
Means/totals 17.5 14.9 712.4 296.1
L1/L0=0.851 N1/N0=0.416
Southern Hemisphere
Antarctic minke
(Balaenoptera bonaerensis)
7 7 670 515
Blue
(Balaenoptera musculus)
27 22 290 2
Bryde’s
(Balaenoptera brydei)
15 15 94 91
Fin
(Balaenoptera physalus)
23 20 625 23
Humpback
(Megaptera novaeangliae)
16 16 170 30
Right
(Eubalaena australis)
16 16 78 13.6
Sei
(Balaenoptera borealis)
16 14 167 27.4
Sperm
(Physeter microcephalus)
18.5 14.5 367 120
17.0 10.1 2,461 822
L1/L0=0.595 N1/N0=0.334
Global
L0L1N0N1
16.8 11.6 4,048.5 1,487
L1/L0=0.689 N1/N0=0.367
are from the International Whaling Commission (2013b). Population sizes for bowhead
whales in the North Atlantic are from Allen and Keay (2006), Cooke and Reeves (2018),
and Vacquié-Garcia et al. (2017). Population sizes for bowhead whales in the North Pacific
are from Cooke and Reeves (2018), Givens et al. (2016), and Shpak et al. (2017). Population
sizes for blue whales in the Southern Hemisphere are from Branch, Matsuoka, and Miyashita
(2004) and Branch et al. (2007).
300 The Sea: The Current and Future Ocean [77, Supplement
The global distribution of sperm whales is based on Whitehead (2002). He estimated that
33% of the global population would be found in the North Atlantic, and we assumed a 50%
split of the remainder between the North Pacific and Southern Hemisphere.
The mean lengths of blue, fin, sei and sperm whales before and after whaling are based
on Clements et al. (2017).
Acknowledgments. C. R. Smith was partially supported by National Science Foundation (NSF)
grant OCE 1155703, and J. Roman was partially funded by a Fulbright-NSF Arctic Research Grant
in Iceland. We thank James McCarthy for his inspiration and mentorship in examining the role of
whales in marine ecosystems, and Mark Altabet for encouraging us to submit this article.
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Received: 11 January 2019; revised: 10 May 2019.
