sharks of the open ocean || shark productivity and reproductive protection, and a comparison with...

Chapter 26

Shark Productivity and Reproductive Protection, and a Comparison with Teleosts

David W. Au, Susan E. Smith and Christina Show

Abstract

Intrinsic rates of increase at the population size for maximum sustainable yield (MSY), that is, rebound potentials, are calculated for 27 sharks and 10 large pelagic teleosts after determining the mortalities producing MSY. Those mortalities, Z � 1.5M and Z � 2.0M, respectively, were derived by linking stock–recruitment and abundance-per-recruit rela-tionships via Lotka’s demographic equation. Rebound potentials ranged from 1% to 14% per year for sharks and from 8% to 34% for billfi shes and tunas. Small coastal sharks have productivities similar to those of some large teleosts and could recover from depletion within a decade. The least productive sharks would require as long as four decades. Pelagic sharks have mostly intermediate productivities. Most sharks would suffer population collapse with mortalities around three times the rate of natural mortality. Protecting the fi rst two to three mature ages of the most productive sharks, but up to the fi rst 10 mature ages of the least productive, would ensure enough reproduction to prevent population collapse.

Key words: collapse threshold, rebound potential, reproductive protection, shark productivity, population collapse.

Introduction

Demographic analysis is a useful tool for studying shark populations, especially for comparing productivities among species or predicting responses to fi shing based on life-history traits and reproductive potential. Here we review the “intrinsic rebound potential” that Smith et al. (1998) calculated to measure the productivity of sharks, and evaluate how protecting the reproductive potential incorporated in that measure can guard against popu-lation collapse. We will show how the mortality rate that produces maximum sustainable yield (MSY) can be determined and will compare productivities among different sharks, and some teleost species. We will also consider the likely bounds of these estimates, the mortality levels at which populations collapse, and the years required for depleted popula-tions to recover.

Sharks of the Open Ocean: Biology, Fisheries and Conservation. Edited by M. D. Camhi, E. K. Pikitch and E. A. Babcock

© 2008 Blackwell Publishing Ltd. ISBN: 978-0632-05995-9

Shark Productivity and Reproductive Protection 299

Methods

Estimating productivity

Intrinsic rebound potential measures the productivity of a population as it hypothetically rebounds from the size producing MSY, and hence estimates the productivity producing that yield (Au and Smith, 1997; Smith et al., 1998). It is a solution of Lotka’s (1907) demo-graphic equation, utilizing the concept that through density-dependent compensation every adult mortality rate up to some maximum can be sustained. This allows direct estimation of pre-adult survival, and thus circumvents a major diffi culty of conventional demographics.

To review, the Euler–Lotka (Lotka’s) equation in discrete form is

l mxrx

xx

w

e�

�

�α

∑ 1

(26.1)

where α � female age at fi rst reproduction, w � last reproducing age, lx � survival to age x, mx � fecundity at age x (female offspring), and r � intrinsic rate of increase. After sub-stituting lαe�M(x�α) for lx (where lα � survival to maturity and M � instantaneous rate of natural mortality) and average fecundity b for mx, completing the summation then gives

e e e� � � � � � �� ( ) ( )( )[ ]M r r M r wl bαα α1 11

(26.2)

which is equivalent to Leslie’s (1966) expression when w is suffi ciently large. The net pre-adult survival lα � lα,Z that makes an increased mortality Z (�M � fi shing mortality F) sustainable (r � 0) is determined from Equation (26.2) set with M � Z and r � 0.

If fi shing mortality is now removed (Z becomes M), the population under survival lα,Z will rebound at a certain productivity rate rZ (with the stable-aged distribution achieved), again as found from Equation (26.2) accordingly specifi ed (the “rebound” transforms catch mortality into its population growth equivalent).

Parameters of rebound potential

Age α and mortality M

Average age at fi rst maturity α and natural mortality M were taken from the literature, where M was often calculated from maximum age w: ln M � 1.44–0.982 ln w (Hoenig, 1983; see Table 26.1). That equation is a variant of the exact relationship analytically derived by Xiao (2001).

Fecundity

In determining lα,Z, average fecundity is used directly, but for rZ its effect is as the ratio of fecundity during the rebound phase (when average fecundity is higher from higher adult sur-vival) to fecundity during the fi shed phase (large females culled).* So by assuming a b-ratio, rebound potential can be estimated even for species whose fecundity schedules are poorly

*This comes from assuming compensation in l�,Z alone makes r � 0 possible; however, allowing the compensation to be within the product (l�,Zb), instead, would still give the same resulting rZ.

300 Sharks of the Open Ocean

Table 26.1 Intrinsic rebound potentials (rZ) and doubling times (TD) for (a) 10 sharks (with Z � 1.5M) and (b) 10 pelagic teleosts (with Z � 2.0M) at two fecundity b-ratios (1.00, 1.25).

Parameters* b-ratio 1.00 b-ratio 1.25

α (year) w (year) b (F/F) M (year�1) rZ (year�1) TD (year) rZ (year�1) TD (year)

(a) Sharks

Gray smoothhound (Mustelus californicus, Triakidae)2 12 1.6 0.368 0.079 5.8 0.141 3.3

Bonnethead (Sphyrna tiburo, Sphyrnidae)3 12 4.5 0.368 0.062 7.5 0.110 4.2

Sharpnose, Atlantic (Rhizoprionodon terraenovae, Carcharhinidae)4 10 2.5 0.440 0.050 9.2 0.092 5.0

Common thresher (Alopias vulpinus, Alopiidae)5 19 2.0 0.234 0.040 11.6 0.069 6.7

Blue (Prionace glauca, Carcharhinidae)6 20 11.6 0.223 0.035 13.2 0.060 7.7

Mako (Isurus oxyrinchus, Lamnidae)7 28 4.0 0.160 0.029 15.9 0.049 9.4

White (Carcharodon carcharias, Lamnidae)9 36 3.5 0.126 0.023 20.1 0.038 12.2

Leopard (Triakis semifasciata, Triakidae)13 30 6.0 0.150 0.018 25.7 0.031 14.9

Bull (Carcharhinus leucas, Carcharhinidae)15 27 1.8 0.166 0.015 30.8 0.027 17.1

Spiny dogfi sh, British Columbia (Squalus acanthias, Squalidae)25 70 3.6 0.065 0.010 46.2 0.016 28.9

(b) Billfi shes and Tunas

Skipjack tuna (Katsuwonus pelamis, Scombridae)1 5 1.500 0.160 2.9 0.344 1.3

Yellowfi n tuna (Thunnus albacares, Scombridae)2.5 8 0.900 0.109 4.2 0.182 2.5

Bigeye tuna (Thunnus obesus, Scombridae)3 10 0.400 0.104 4.4 0.156 3.0

Sailfi sh (Istiophorus platypterus, Istiophoridae)3 8 0.530 0.104 4.4 0.160 2.9

Striped marlin (Tetrapturus audax, Istiophoridae)4 9 0.470 0.082 5.6 0.126 3.7

Blue marlin (Makaira nigricans, Istiophoridae)4 11 0.380 0.084 5.5 0.125 3.7

Albacore tuna (Thunnus alalunga, Scombridae)4.5 12 0.300 0.074 6.2 0.110 4.2

Bluefi n tuna, northern (Thunnus orientalis, Scombridae)5 20 0.250 0.071 6.5 0.101 4.6

Swordfi sh (Xiphias gladius, Xiphiidae)5 20 0.210 0.067 6.9 0.096 4.8

Bluefi n tuna, southern (Thunnus maccoyii, Scombridae)6 20 0.250 0.062 7.5 0.089 5.2

*Life-history parameters for sharks are as listed by Smith et al. (1998). α: female age at fi rst reproduction; w: last reproducing age; b: fecundity as female pups per female (F/F); and M: instantaneous rate of natural mortality. Parameters for tunas are as listed by Shomura et al. (1995); for billfi shes as listed by Au (1998).


known. The b-ratios 1.00 and 1.25 were used, for up to a 25% increase in fecundity as per Smith et al. (1998). (Rebound potentials for pelagic sharks, using population parameters updated from those used here, are presented in Smith et al. (2008). Actual annual rates of increase are estimated for pelagic sharks by Cortés (2008), also with updated parameters, including age survival estimates.)

Population size for MSY

Rebound potential is determined for the stock size (S) producing MSY, that is, the SMSY resulting from ZMSY. The linkage to mortality requires a stock–recruitment (S–R) relation-ship, and we use the normalized Beverton–Holt (BH) S–R model that is appropriate for most marine fi shes (Kimura, 1988). In this form, both R and S are in the same relative units. Also, the locus of MSY (�maximum surplus R) forms a diagonal from the upper left

(a)

0.00.0

0.2

0.4

Frac

tio

nal

rec

ruit

men

t (R

)

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.2

R/S

S–R

MSY

0.4Relative stock size (S) Mortality Z (in M-multiples)

Frac

tio

nal

sto

ck (

S Z/S

M)

0.6 0.8 1.0

0.0 0.2 0.4

Stock and recruitment Stock and mortality

0.6 0.8 1.0 1 2 3 4 5 6

1 2 3 4 5 6

(b)

0.00.0

0.2

0.4

Frac

tio

nal

rec

ruit

men

t (R

)

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.2

MSY

0.4Relative stock size (S) Mortality Z (in M-multiples)

Frac

tio

nal

sto

ck (

S Z/S

M)

0.6 0.8 1.0

0.0 0.2 0.4

Stock and recruitment Stock and mortality

0.6 0.8 1.0 1 2 3 4 5 6

1 2 3 4 5 6

Fig. 26.1 The procedure, shown diagrammatically, for estimating the mortality ZMSY from an estimate of relative stock size SMSY (arrows indicate translation process for (a) sharks and (b) teleosts). The stock–recruitment (S–R) curve and recruitment–stock (R/S) slope line (left panels) are labeled in the upper left diagram of the normalized BH relationship. SZ/SM curves (right panels) are composites for the species included.


corner of the S–R diagram to the point (0.5, 0.5) on its 1:1 diagonal (Fig. 26.1, left panels). We defi ne R as individuals reaching age α.

SMSY is likely to lie within the range 0.2–0.5 of the unfi shed population size (S0) (Shepherd, 1982). In contrast to certain groundfi shes whose stock-independent recruitment gives MSYs at 0.2–0.3S0 (Clark, 1993), the BH S–R model would predict the SMSY for sharks to approach the limiting 0.5S0 (giving zero surplus R) because of their proportional S–R rela-tionship (Holden, 1974). But Thompson (1992) found from theoretical study that SMSY of species like sharks should be from about 0.30 to a maximum 0.37S0, the latter when recruit-ment is directly proportional to parental stock size. Because others (e.g., Restrepo et al., 1998) have suggested SMSY � 0.5S0 for such fi shes (which would require some other S–R model), we use Thompson’s theoretical upper limit, rounded to 0.4S0, as the lower bound of SMSY for sharks, the upper bound being the limiting 0.5S0 (but see Cortés, 2008, for still higher SMSY from a relationship to per capita increase per generation). We use SMSY � 0.4S0 as the most optimistic estimate of the resilience of sharks. Billfi shes and tunas should have smaller SMSY, for they are fast-growing, fecund, and productive, with M relatively high (Table 26.1) and R and S largely independent (e.g., IATTC, 1999, Fig. 43). We therefore place their SMSY at an intermediate 0.3–0.4S0 (between sharks and groundfi shes).

Mortality Z for MSY

This mortality is derived by linking the above estimates of SMSY to abundance-per-recruit as determined by mortality Z. It can be shown that diagonals drawn from the origin on the S–R diagram represent particular solutions to the demographic equation. Thus if a popu-lation produces MSY at stock size SMSY, its S–R curve must intersect the diagram’s MSY locus where it is defi ned by that SMSY, where the diagonal of slope RMSY/SMSY defi ning that condition (now with r � 0) also intersects the locus (Fig. 26.1, left panels; fi rst bend of up-arrows).

The inverse of that RMSY/SMSY slope equates to standing adult stock size per recruit, which is SZ/SM, the spawning stock per recruit (SSR) ratio [(1�e�Zλ)/Z]/[(1�e�Mλ)/M] (where λ � adult life span) (cf. Goodyear’s (1993) spawning potential ratio); it provides for fi nding ZMSY. SSRs were calculated for different types of sharks (small coastal, pelagic, medium-to-large coastal) and for the large pelagic teleosts (swordfi sh, marlins, tunas) and depicted as curves determined by Z (Fig. 26.1, right panels). By graphical solution, SMSY is transformed to ZMSY via the RMSY/SMSY diagonal on the S–R diagram and its demographic equivalent, SSR.

Reproductive protection against collapse and time for recovery

To determine the value of protecting reproducing adults, we calculated minimum repro-ductive output for sustaining maximal levels of exploitation. This minimum is obtained at the collapse threshold where productivity is highest and the sustainable mortality maxi-mal at Zτ (Mace and Sissenwine, 1993). An age at fi shery entry tc (α � tc � w) in an expanded Equation (26.2) set at that threshold defi nes this minimum output that prevents collapse while allowing mortality Z�τ � Zτ upon the older, still fi shed ages. The maxi-mum tc is tc max, beyond which there can be 100% exploitation because then the protected unfi shed age classes alone sustain the population. Two steps lead to solutions: (1) Set r in Equation (26.2) to its maximum value rτ (twice the rZ for MSY, as in the logistic model,


to allow the largest compensation), and then solve for the maximum pre-adult survival (lα,Z)max; (2) use (lα,Z)max in the expanded Equation (26.2) and solve for combinations of collapse threshold tc and Z�τ, and for tc max, with r � 0 (for just sustainable fi shing) and b-ratio 1.00 (to be conservative). The Z�τ are expressed as rates of exploitation E�τ [� (F/Z�τ)(1 � exp(�Z�τ))] varying from 0 to 1.00.

To evaluate recovery times after near collapse, we determined doubling times TD for populations depleted to 50% of their MSY-producing sizes. Their rZ, which is then 50% larger (1.5 times) than at MSY (again from the logistic model), is assumed to not change during recovery. Thus we calculated TD� [ln(2)]/1.5rZ as the recovery time for return to the MSY condition.

Results

The mortality corresponding to MSY

Our estimate of ZMSY is 1.5M for sharks and 2.0M for the pelagic teleosts, from trans-lating estimates of SMSY (see arrow fl ow in Fig. 26.1). The minimal SMSY for sharks, estimated as 0.40S0, produced an intersection of the MSY locus at (0.40, 0.60) on the S–R plot (Fig. 26.1(a), left panel), which defi ned a recruitment–stock (R/S) diagonal of slope 0.60/0.40 through that point. This slope’s inverse value, 0.67, is the abundance-per-recruit ratio SSR, as read off the top of the diagram at the diagonal’s intersection. Value 0.67 maps onto the SSR (or SZ/SM) versus Z plot via the 1:1 diagonal (Fig. 26.1(a), right panel), and its intersection with the curve defi nes its corresponding Z (in multiples of M). Thus the ZMSY for sharks is about 1.5M, an upper bound as derived here. Similarly, it is seen that the median billfi sh/tuna ZMSY is about 2M, which is the traditionally used value for teleosts (Fig. 26.1(b)).

Productivities: sharks and teleosts

The Z-standardized (Z � 1.5M, 2.0M) rebound potentials (Table 26.1) are strongly deter-mined by age at maturity α and are plotted accordingly for comparison (Fig. 26.2, with 17 other sharks recalculated from Smith et al., 1998). Most pelagic sharks have r1.5M produc-tivities in the 0.04–0.06 year�1 range (e.g., thresher, Alopias vulpinus, Alopiidae), while some small, short-lived coastal sharks, with productivities greater than 0.07 and up to 0.14 year�1, may be as productive per capita as some billfi shes and tunas. Short-lived tropi-cal tunas (e.g., yellowfi n, Thunnus albacares, Scombridae) have r2M values at least three times greater than the r1.5M of the medium-to-large coastal sharks and approximately twice that of the most productive, small coastal sharks. For perspective, Murphy’s (1967) classic r � 0.338 for the California sardine (Sardinops sagax, Clupeidae) is shown, a productiv-ity level apparently achievable by the skipjack tuna (Katsuwonus pelamis, Scombridae).

How these rZ values are specifi cally affected by the Z and b-ratios assumed in their deter-mination helps establish their likely bounds (Fig. 26.3). Thus considering likely ranges and interactions of those parameters (higher Z/lower b-ratio of small species; lower Z/higher b-ratio of large species), productivities from about 0.01 to 0.14 year�1 would seem likely among sharks as a group. Note that the rZ of the low-productivity sharks (e.g., spiny dogfi sh,


Squalus acanthias, Squalidae) are relatively unaffected by the ZMSY and b-ratio chosen (their response curves are nearly fl at).

Reproductive protection against collapse and recovery times from depletion

Without protection of reproducing females, E�τ � Eτ, the basic collapse threshold exploi-tation. The equivalent Zτ, as M multiples, averaged 2.2M and 4.1M among the sharks of Fig. 26.2, for b-ratio 1.00 and 1.25, respectively. The median of these collapse threshold mortalities, about 3M, is to be compared with the Z � 1.5M estimated for MSY.

By allowing females a few reproductive seasons before exploitation begins, there can be substantial protection against population collapse, at least for the more productive sharks (Table 26.2). Thus giving females three years of pupping protection (column “3”) would be enough to ensure against collapse of the more productive small coastal species (E�τ becoming 1.00, i.e., the exploited ages become completely expendable (age tc max being surpassed)), and would substantially protect pelagic species like the thresher (E�τ becoming 0.85). This level of reproductive protection is the least needed to save populations when collapse-producing exploitation cannot be prevented.

0.50

2838

2930 37

353631

3432

3312

3

45

671011 13

1425

2627

2423

222120

1819

1715

16129

8

0.40

0.30

0.20

0.10

0.08

0.06

Intr

insi

c re

bo

un

d p

ote

nti

al (

r z)

0.04

0.02

0.01

0 5 10 15

Age at maturity (a)

20 25

28. Skipjack tuna29. Yellowfin tuna30. Bigeye tuna31. Albacore32. North bluefin tuna33. South bluefin tuna34. Swordfish35. Blue marlin36. Striped marlin37. Sailfish38. Sardine

15. Tiger16. White17. Angel18. Lemon19. Spiny dogfish (northwestern Atlantic)20. School/soupfin21. Leopard22. Sandbar23. Scalloped hammerhead24. Bull25. Sevengill26. Dusky27. Spiny dogfish (BC)

1. Gray smoothhound2. Brown smoothhound3. Bonnethead4. Sharpnose5. Common thresher6. Oceanic whitetip7. Blue8. Blacktip9. Gray reef10. Sand tiger11. Mako12. Whitetip reef13. Galapagos14. Silky

Fig. 26.2 The relationship of rebound potential rZ to age at maturity α for selected sharks (numbered 1–27) and teleosts (numbered 28–38). The rZ are represented as ranges delimited by rZ calculated with b-ratio 1.25 and 1.00, respectively (with Z�1.5M for sharks and 2.0M for the pelagic teleosts). The estimate for sardine productivity is from Murphy (1967) (solid circle). Ten of the sharks shown are from Table 26.1; the other 17 are recalculated from Smith et al. (1998), using Z�1.5M. Note that the y-axis is log scale.


0.20

0.10

Pro

du

ctiv

ity

r z

0.15

0.05

0.001.0 1.1 1.2 1.3

b-ratio � 1.00

1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.20

0.10

Pro

du

ctiv

ity

r z

0.15

0.05

0.001.0 1.1 1.2 1.3

b-ratio � 1.25

1.4 1.5Total mortality (as Z/M )

1.6 1.7 1.8 1.9 2.0

Gray smoothhound

Gray smoothhound

Bonnethead

Bonnethead

Sharpnose

Sharpnose

Common thresher

Common thresher

Mako

Mako

White

White

Bull

Bull

Spiny dogfish (BC)

Spiny dogfish (BC)

Fig. 26.3 Responses of the rebound potential statistic rZ among eight representative sharks to total mortality Z (as M multiples) and to unadjusted (1.00b) and adjusted (1.25b) fecundity.

Table 26.2 Increase in maximum (collapse threshold) exploitation rates (E�τ) for 10 sharks according to the number of times adult females are allowed to reproduce before being fi shed (through raising fi shery-entry age above maturity age). Extending the protection past age tc max makes the remaining age classes completely expend-able (E�τ �1.00), thus ensuring against collapse under any exploitation rate (age in years and E�τ per year).

Species Sustainable exploitation rate E�τwhen there are 1, 2, …, 10 pupping seasons before fi rst exploitation

α-age 1 2 3 4 5 6 7 8 9 10 tc max

Gray smoothhound 2 0.31 0.74 1.00 3.2Bonnethead 3 0.32 0.76 1.00 4.1Sharpnose, Atlantic 4 0.36 1.00 4.9Common thresher 5 0.22 0.38 0.85 1.00 7.1Blue 6 0.21 0.35 0.74 1.00 8.2Mako 7 0.16 0.23 0.36 0.70 1.00 10.3White 9 0.13 0.17 0.24 0.36 0.67 1.00 13.4Leopard 13 0.15 0.21 0.32 0.56 1.00 16.6Bull 15 0.16 0.24 0.38 0.76 1.00 18.2Spiny dogfi sh, BC 25 0.07 0.08 0.10 0.11 0.14 0.17 0.22 0.31 0.48 1.00 34.0


Recovery (doubling) times for depleted shark populations ranged from 3 to 46 years (Table 26.1). Considering TD under both b-ratios, it is clear that the small coastal sharks with rebound potentials around 0.05 or higher (smoothhounds) should recover within a decade (like the teleosts), as might some of the oceanic-pelagic species (e.g., blue, Prionace glauca, Carcharhinidae). However, sharks with potentials less than 0.03, mainly the medium-to-large coastal species, would require two to four decades to return to the MSY condition.

Discussion

Sharks should be managed with emphasis on maintaining healthy reserves of reproducing adults, because catches in excess of annual production rates (�er � 1) are easily taken. Raising fi shery-entry ages to balance higher exploitation E� is a way to obtain the repro-ductive protection. Thus Holden (1968) determined the lengths at fi shery entry for keeping recruitment constant (r � 0) among spiny dogfi sh under different mortalities, and Smith and Abramson (1990) calculated replenishment (net reproductive) rates from combinations of age at entry and mortality for showing when yield per recruit for the leopard shark (Triakis semifasciata, Triakidae) would be unsustainable. Similarly, Au and Smith (1997) adjusted yield per recruit for that shark for reduction in recruitment from exploitation, though in using the rZ-derived productivity–stock rather than the recruitment–stock relationship, they overestimated the decline by 14% at SMSY.

Our calculations of the maximum exploitation rates made sustainable by raising age at entry make clear the value of the fi rst few mature age classes in providing necessary reproductive output. Even the most productive sharks require at least three years of pro-tected reproduction to ensure against collapse from extreme exploitation. Such protection might often not be practical, but still is part of the more general concept of protecting reproductive value in populations (MacArthur, 1960). As shown by demographic elastic-ity analysis of long-lived animals (Cortés, 1999, 2002; Heppell et al., 1999), population growth is enhanced most by protecting those juveniles, subadults, and young adults that are still relatively abundant and with high reproductive potential (having survived the high-mortality years). Protecting pupping females should be a high priority for sustain-able management of pelagic sharks, since any pregnant female is most reproductively valuable at the start of the pupping season, having survived all mortality sources up to that time and parturition being imminent.

Acknowledgments

For constructive comments, we thank Drs. George Watters, Gregor Cailliet, Enric Cortés, Paul Crone, Malcolm Francis, and Yongshun Xiao.

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