Coexistence in North Sea fish communities: implications for growth and natural mortality
1 National Institute of Aquatic Resources, Technical University of Denmark, Charlottenlund Slot, DK–2920 Charlottenlund, Denmark
2 Norwegian College of Fishery Science, University of Tromsø, N–9000 Tromsø, Norway
3 DFO Science Advisory Secretariat, Department of Fisheries and Oceans, 200 Kent Street, Ottawa, Ontario, Canada K1A 0E6
4 Institute for Marine Resources and Ecosystem Studies, PO Box 68, 1970 AB IJmuiden, The Netherlands
Correspondence to H. Gislason: tel: +45 33963361; fax: +45 33963333; e-mail: hg{at}difres.dk
Gislason, H., Pope, J. G., Rice, J. C., and Daan, N. 2008. Coexistence in North Sea fish communities: implications for growth and natural mortality. – ICES Journal of Marine Science, 65: 514–530.For a fish community to persist over time, all species must be able on average to replace themselves on a one-for-one basis over their lifetime. We use this principle and a size-based equilibrium model where asymptotic length is used as a functional trait to investigate how natural mortality should scale with size within and across pelagic and demersal species of North Sea teleosts. The model predicts natural mortality to scale with body length raised to a power of –1.66 at current levels of exploitation. Additionally, natural mortality of demersal species should be proportional to asymptotic length raised to a power of 0.80, so generating a higher natural mortality at a given length for large species than for small ones. The model also suggests that the exponent in the scaling of the von Bertalanffy growth parameter K with asymptotic length should be more negative for pelagic than for demersal species. We test our results by analysing independent estimates of predation mortality, the scaling of maximum recruitment per unit of spawning-stock biomass with asymptotic length, and the general relationship between K and asymptotic length for demersal and pelagic families of fish. All tests are consistent with our modelling results.
Keywords: coexistence, fish communities, growth, life history, natural mortality, net reproductive rate, replacement, size-based model, size structure, stock–recruitment
Received 22 February 2007; accepted 5 January 2008.
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Introduction |
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 Top Introduction The modelParameter values and model...ResultsDiscussionAppendixReferences |
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Modelling exploited fish populations (Quinn and Deriso, 1999) often involves attempts to close the life cycle of the population by functionally linking numbers of recruits to numbers of spawners. This includes examining processes that regulate the number of recruits produced per unit of spawning-stock biomass (R/SSB) and processes regulating the spawning-stock biomass produced per recruit (SSB/R). In both cases, the underlying principle has been that for a population to persist, on average its members will have to replace themselves on a one-for-one basis over their lifetime. Failure to replace at least one-for-one will lead to population declines, which cannot continue indefinitely without extinction. Consistently replacing at a rate greater than one-for-one will produce population growth that cannot continue indefinitely for ecological and energetic reasons.
Approaches for equilibrium and temporary non-equilibrium modelling of single populations are well established (Murdoch, 1994), with density dependence as a routine property of at least some processes, including feeding, growth, and recruitment. However, when such models are placed in a community context, additional considerations emerge. Fish communities do persist over time in the sense that sympatric species coexist over geological timescales (Daan et al., 1990). For this to happen, not only must individual populations on average maintain replacement, but competitors, predators, and prey on average must coexist. This does not mean that the populations need ever reach a static equilibrium, nor that the dynamics of species need be tightly coupled. However, it does mean that no predator population can consistently prevent its prey populations from being able to replace themselves spawner-for-spawner, or consistently fail to find sufficient prey to support replacement of itself. Density dependence remains a necessary consideration at the community scale, although the ways it is expressed requires more directed study.
How populations solve the challenges of existence are reflected in their life-history strategies (Stearns, 1992), in solutions to the challenges of coexistence and coadaptation in their strategies of predation and predator avoidance (Holt and Polis, 1997), and in trade-offs among life-history processes (Kneitel and Chase, 2004). Many of the more abundant fish species of commercial interest are trophic generalists, occupying broad feeding niches (Garrison and Link, 2000; Trenkel et al., 2005). This raises questions about how multiple populations can coexist without the bulk of the biomass gradually becoming monopolized by a few species characterized by sets of life history, predation, and predator-avoidance strategies and parameters that make them more efficient at replacement, or more effective as a predator or at avoiding predation. Species coexistence in communities means that just as single-species models must close the loop for replacement from spawner to spawner (or egg to egg), multispecies models must close the life cycles of all populations in a community simultaneously, with spawner-for-spawner (or egg-for-egg) replacement on average. Otherwise, populations will gradually be lost and eventually the community will come to consist of only the few most successful species.
Size has proven a powerful property for understanding and modelling the processes involved in life-history strategies leading to replacement. Basic biological processes, such as metabolism, food intake, growth, reproduction, and mortality, all scale with body size (Kooijman, 2000; Brown et al., 2004; West and Brown, 2005). For fish populations, there have been empirical observations of relationships between von Bertalanffy growth parameters, natural mortality, and size or age at first maturity (Pauly, 1980; Beverton, 1987; Charnov, 1993; Jensen, 1996). These size-based relationships have been found among as well as within species, allowing size-dependent scaling to be combined with the theory of life-history strategies to understand fish community dynamics (Charnov et al., 2001; Denney et al., 2002; Goodwin et al., 2006). Using functional traits such as asymptotic length (L
) to provide scaling relationships across species in size-based models offers a promising avenue for identifying the trade-offs involved in the processes important for coexistence (McGill et al., 2006). The size-based multispecies models of Hall et al. (2006) and Pope et al. (2006) focused primarily on the role that size-based predation plays in fish community regulation. We confront the question of whether recruitment scales with asymptotic length (L) across the species in a given community. This question is fundamental for community persistence, by framing the requirement for, on average, one-for-one replacement in the setting of all the interacting populations in the community.
Two pieces of evidence suggest that strong size-dependent scaling in recruitment is necessary to allow species to coexist. First, empirically there is a large and systematic difference between large and small species in the maximum R/SSB (Denney et al., 2002). Those authors found that large, slow-growing species generated significantly lower maximum R/SSB than did smaller, fast-growing species. Across 54 northeast Atlantic stocks, maximum R/SSB scaled with maximum length raised to a power close to –3, indicating that a species with L = 100 cm produces 1000 times fewer R/SSB than would a species with L = 10 cm (Denney et al., 2002). Second, in developing multispecies size-based predation models for Georges Bank and the North Sea, respectively, Hall et al. (2006) and Pope et al. (2006) found that for their models to match observed values for abundance, small species needed to consistently generate a higher R/SSB than larger ones. Hall et al. (2006) found that the maximum R/SSB estimated from data collected by Myers et al. (1995) scaled to asymptotic length raised to a power of –2.3. At low population sizes, a species with L = 10 cm would need to produce 
200 times more R/SSB in their model than a species with L = 100 cm. Likewise, Pope et al. (2006) found it impossible to maintain both small and large species in their model simulations unless the R/SSB of the small species was three orders of magnitude higher. If on average R/SSB was constant over all species, over time only the largest species would remain in both model simulations.
These apparent differences in realized R/SSB between small and large teleosts have important implications for replacement of populations in a community. Consider two species for which their L differs by a factor of 10. If both species have the same SSB, spawn once annually, and produce the same number of eggs per gramme of female body mass, they should both produce approximately the same total number of eggs annually. However, because of the difference in R/SSB, these equal numbers of eggs will result in the smaller species producing 2–3 orders of magnitude more recruits. Because the biological processes producing this difference are acting on prerecruits that are of the same size, regardless of how big they will eventually grow, this difference begs for a plausible explanation.
To investigate possible causes of the difference in R/SSB, we construct a simple size-based model and use it to explore the conditions required for simultaneous replacement of different sizes of coexisting fish species. Our model uses the scaling of mortality, growth, maturation, and fecundity with L to calculate the net reproductive rate, R, for fish of different L in the equilibrium situation. The differences in survivorship may reflect differences in predator-avoidance strategy, so the model considers demersal and pelagic habitats separately, to reflect the different potential strategies for predator avoidance in the two types of habitat. The model is parameterized with data on fishing mortality and scaling factors for growth and reproduction representative for the North Sea fish community. We use the model to estimate how natural mortality should scale with body size and L to achieve replacement (i.e. R = 1) simultaneously for all species, irrespective of their L. We also use the model to explore what aspects of species life histories and behaviours may contribute to the estimated scaling of mortality with size. Winemiller and Rose (1992) classified fish species according to different life-history strategies. McCann (1998) used these strategies in a stage-structured model with density-dependent interactions, and was able to obtain coexistence for only two out of four strategies. Others have used differences in average body size to explain coexistence (e.g. Etienne and Olff, 2004; Scheffer and van Nes, 2006). However, before this study, the scaling of growth and reproduction with L and length has not been used to explore how natural mortality should scale with body size. Nor have the implications that a large difference in R/SSB for small and large species would have for recruitment and replacement been addressed.
Our model and analyses are limited by incomplete information on the reproduction and early life stages of marine teleosts. Most fish species produce eggs with a median diameter of 1 mm (Chambers, 1997; Appendix Table A2), and mature gonads account for 20% of female body weight. Nonetheless, the reproductive potential of stocks may vary systematically as a function of an individual's age and size, and changes in sex ratio, sexual maturity, and condition (Tomkiewicz et al., 2003). Likewise, estimates of the natural mortality of first-feeding post-larvae used as input to our model are uncertain, so we investigate the sensitivity of our results to changes in relative fecundity and to the natural mortality of the post-larvae. We check the model by comparing the output with estimates of natural mortality derived from multispecies virtual population analysis (MSVPA) using comprehensive stomach content data, with growth characteristics of pelagic and demersal families of fish from FishBase (Froese and Pauly, 2006), and with the scaling of maximum recruit production with L reported by Denney et al. (2002). We also investigate the possibilities for coexistence in a simpler model where no distinction is made between pelagic and demersal habitats. Finally, we discuss the implications of our findings in the context of general differences in life-history characteristics of demersal and pelagic species, and for management.
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The model |
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General description
The model describes a fish community composed of a number of teleost populations in steady state, i.e. where growth, mortality, and recruitment are constant over time. As in the size-based model of Pope et al. (2006), where additional justification for many of the formulations that follow are presented, we represent different fish species in the community by their L
, with one hypothetical species per 5 cm increment from the smallest to the largest species present, and use information on how growth, mortality, and reproduction scale with L to model the steady-state populations in each L group. However, in contrast to Pope et al. (2006), we also account for fish <5 cm and include eggs, larvae, and early juveniles. Further, we split the species into demersal and pelagic, to account for possible habitat-related differences in scaling factors.
Briefly, the model calculates the number of individuals for each L group ("species") in subsequent size classes, and estimates the total number of eggs produced by the mature females. The calculations start with the same number of eggs (E0) for all species. These eggs and the subsequent yolk-sac larvae suffer from mortality. The yolk-sac larvae become post-larvae when they start feeding at a fixed length, L0. The post-larvae grow and suffer mortality until they become juveniles at a length, Lmet. Juvenile demersal species settle and may experience a change in the level of their natural mortality at settling, whereas juvenile pelagic species continue to have a natural mortality determined exclusively by their individual length. Both pelagic and demersal species are subject to fishing mortality. Females mature at a size Lmat after which they start to produce eggs. Relative fecundity is used to calculate the total lifetime egg production (E1,L) of a cohort for each L group. The model is finally used to find the values of natural mortality (ML) producing a one-for-one replacement of eggs (R = 1) across the size range of fish species found in the fish community (i.e. a value of E1,L/E0 as close as possible to 1 for all L groups in the range). Length is measured in cm, body weight in g, age in years, and all mortality and growth rates are annual unless otherwise specified.
Egg and yolk-sac larval survival
Of the E0 eggs spawned, only some will hatch to become yolk-sac larvae, and only a fraction of the yolk-sac larvae will survive to become post-larvae. If the fractions surviving the egg (Ce) and yolk-sac larval stages (Cy) are known, the initial number of post-larvae at length L0, (NL0) can be calculated from
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| (1) |
Growth
Many fish species undergo major ontogenetic changes during their early development (Fuiman and Higgs, 1997), and parameters derived from adult fish often provide a poor description of larval growth. We follow Pepin (1991) and describe post-larval growth as an allometric function of larval length:
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Rearranging Equation (3) provides the age (tL) of a post-larva of given length L:
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The growth parameter K is assumed to scale with L. However, the scaling may be habitat-dependent and may hence differ for pelagic and demersal species:
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Mortality
The natural mortality of juvenile and adult fish is difficult to determine empirically (Quinn and Deriso, 1999), and fisheries scientists seem divided into those who consider it to be a species-specific constant depending on life-history parameters such as asymptotic length or K (Pauly, 1980; Jensen, 1996; Griffiths and Harrod, 2007) and those who consider it to be an allometric function of individual body size (McGurk, 1986; Lorenzen, 1996). We assume that natural mortality (ML) consists of a size-dependent part reflecting predation and a constant size-independent part generated by all other causes of natural death (M1), which is identical for all species and sizes. In accord with general size-spectrum theory (Peterson and Wroblewski, 1984; Andersen and Beyer, 2006), predation is assumed to scale with individual body size, using the same exponent (n) for all species and stages. However, to explain the relationship between L, K, and ML identified by Pauly (1980), we also allow for potential differences in the predation mortality-at-length on different species depending on their asymptotic size.
Conceptually, such a difference seems more likely for demersal species than for pelagic. Many piscivorous fish species are found in association with the seabed, and small demersal fish found at or close to the seabed are likely to run a considerable risk of being eaten (Myers and Cadigan, 1993; Van der Veer et al., 2000; Hixon and Jones, 2005; review by Juanes, 2007). Among the smaller species found on sandy or muddy seabeds, many are characterized by adaptations to reduce predation, such as spines, flattened body size, venom, and cryptic or burying behaviour, but such adaptations are largely lacking in the larger species. Species with these adaptations have been demonstrated to have a low natural mortality (Griffiths and Harrod, 2007). Hence, for demersal species, the assumption that ML may depend on asymptotic length once they have settled seems reasonable. We therefore describe their predation mortality as a power function of both individual and asymptotic lengths, with a constant rest mortality (M1) added to account for diseases and other non-size-related causes of death:
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For pelagic species, adaptations to reduce predation seem less likely to differ systematically among small and large species, so we assume that predation mortality depends on individual length, L, only:
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This equation is also used to describe mortality in the pelagic post-larval phase of demersal and pelagic fish alike, because empirical investigations have shown natural mortality to be a function of individual size (McGurk, 1986; Bailey and Houde, 1989; Pepin, 1991), whereas dependence on asymptotic length has not been identified so far. Natural mortality is therefore described as being habitat-dependent, using Equation (8) for demersal life stages and Equation (9) for all pelagic stages.
Fishing mortality-at-length (FL) is modelled by an equation borrowed from Pope et al. (2006):
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is the length in proportion of L at which 50% of the individual fish become selected by the fishery, 
a slope parameter, 
and 
the parameters expressing how fast fishing mortality changes between species in relation to their L, and F is the overall level of fishing mortality acting on all species. The nominator is intended to change species-specific fishing mortality linearly from small to large species, and the denominator ensures that the length at 50% of maximum exploitation of each species is reached when the individual fish have attained a fraction () of their L. Fishing mortality is thus modelled as a function of L, but with the usual sigmoid relationship between exploitation rate and length.
Numbers at length
For each L-group, the population is divided into a number of equally sized length intervals of width 
L. For post-larvae, the number of survivors at length L + L is calculated from the numbers surviving to length L:
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L was set to 0.25 cm.
Reproduction
To calculate the total lifetime egg production of each L group, the average number of individuals in each length interval is first calculated from
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Survivors recruit to the spawning stock when they reach (knife-edge) maturity-at-length (Lmat), determined from
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Relative fecundity (RE) is expressed as the number of eggs produced per gramme of female, and is assumed to scale with L:
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value, the number of eggs generated can be calculated from
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The ratio of parent egg production and offspring egg production, the net reproductive rate, can be expressed as
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Parameter estimation
Growth and maturity data are available for many fish species, and have previously been shown to scale with L (Cury and Pauly, 2000; Froese and Binohlan, 2000), whereas the parameters describing the relationship between M, L, and L are more difficult to estimate and are generally lacking. However, assuming that all species within a given range of L should be able to replace themselves, the model can estimate the combination of M parameters that would, given empirical estimates of all other parameters and as far possible, produce a one-for-one replacement. We use the model to calculate the net reproductive rate in the equilibrium situation for every 5 cm of asymptotic length, and estimate the missing mortality parameters by minimizing the standard deviation of R over the range of L values observed to coexist in the North Sea fish community, subject to the constraint that within this range, the net reproductive rate should, on average, equal 1. Because different calculations are necessary for demersal and pelagic species, the actual minimization is done by minimizing the sum of the standard deviation for demersals and pelagics:
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groups within the size range from the smallest (L,MIN) to the largest (L,MAX) value of L for demersal species, P the number of L groups for pelagics, and 
and 
the average net reproductive rates of demersal and pelagic species, respectively, over the size ranges observed.
The model was implemented in Excel, and Solver was used to find the parameters generating the minimum standard deviation of the net reproductive rate for different values of L, so maximizing the possibilities for coexistence of species of different asymptotic lengths. An overview of the variables and constants used is provided in Appendix Table A1.
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Parameter values and model setup |
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Estimates of L
, K, Lmat, and RE for North Sea teleosts from different sources are given in Appendix Table A2. Usually, the estimates used have been derived from fish sampled in the North Sea proper. When multiple von Bertalanffy growth estimates were available for a single species, we selected those where the value of the age at zero length was <1, and where L was within ±33% of the maximum length given in Froese and Pauly (2006), to avoid bias caused by questionable age readings or insufficient coverage of the length range. Overall, 45 pairs of L and K estimates were available, 35 estimates of L and Lmat, and 16 estimates of L and RE (Figure 1).
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Non-linear regression was then used to estimate the scaling parameters for growth, maturity, and relative fecundity (d, e, q, s, u, and v), and their approximate confidence intervals (Table 1). The relationships for growth and maturity against L
are highly significant for all species combined (p < 0.0001; n = 45 and n = 35, respectively), and for demersal species (p < 0.0001; n = 41 and n = 31, respectively), but barely significant for pelagic species (p = 0.03 and p = 0.04, respectively; n = 4 in both cases). The scaling parameters for the pelagic species do not differ significantly from the parameters estimated for demersal species. The relationship between RE and L is significant for all species combined (p = 0.0003, n = 16), but the exponent does not differ significantly from zero. To calculate the total egg production in the model, an average RE of 997 eggs per gramme female (s.d. = 758) is therefore used for all L groups.
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The survival of eggs and early larvae is temperature-dependent (Pepin, 1991). With average surface temperatures in the North Sea of 5°C in March and 6°C in April (ICES, 2006), the fractions surviving the egg and yolk-sac larval stages can be estimated as 0.435 and 0.074, respectively (Pepin, 1991), corresponding to an overall total survival of 3.2% to the post-larval stage. This stage is assumed to start when the larvae commence feeding at an average length of 0.5 cm (Russell, 1976). From then on, the larvae grow in accordance with the empirical model of Pepin (1991), with an exponent b equal to 0.54 and a constant a equal to 0.064 at 10°C, the average North Sea surface temperature in the period from April to July (ICES, 2006). The post-larvae are assumed to metamorphose at a size of 4 cm, corresponding to an age of 114 d, after which their growth starts to follow the von Bertalanffy growth equation, with a growth parameter K determined by their asymptotic length, L
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The estimated empirical relationship between K and L was kept for demersal species, but for pelagic species the scaling exponent (epel) was estimated within the model because of the limited number of four observations available from the North Sea. To generate values of K for pelagic species in accord with the observations, dpel was adjusted so that the K for a pelagic species with an L of 32.1 cm (the average value for the pelagic species included in the non-linear regression) remained the same (K32.1 = 0.38 year–1), irrespective of changes in epel:
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The parameters of Equation (10) describing fishing mortality as a function of individual and asymptotic lengths were taken from Pope et al. (2006). Other natural mortality, M1, was assumed to be 0.1 year–1 for all sizes. This value corresponds to the lowest value of natural mortality used for adult fish in North Sea stock assessments. The L ranges for demersal and pelagic species were set at 10–180 and 10–50 cm, respectively, in accord with the size range of the species included in Appendix Table A2. All calculations were constrained to produce replacement within these ranges. Hence, solutions were sought where 
These criteria were met in all runs.
Initial runs produced unlikely parameter estimates or failed to converge, suggesting that the model was over-parameterized. A constraint on natural mortality was therefore introduced. Newly hatched fish larvae are reported to suffer daily mortalities in the order of 10–50% per day (McGurk, 1986; Bailey and Houde, 1989; Pepin, 1991; Houde and Zastrow, 1993). According to Pepin (1991), the daily mortality of a first-feeding post-larva is some 15.1% d–1 at 10°C. Assuming natural mortality to be 15.1% d–1 or 59.7 year–1 in the first post-larval size class (
= 0.625 cm) allowed hpel to be calculated from
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The sensitivity of the model to changes in input parameters was examined by changing fecundity by a factor 100, 10, and 0.1, by changing ML0 by ±50%, and by halving M1. In addition to a key-run where scaling relationships were allowed to differ for the pelagic and the demersal habitats, runs with identical growth and mortality relationships and parameters were used to explore the consequences of simplifying the model. Using the scaling parameters for species combined (shown in the first column of Table 1), natural mortality was assumed to depend only on individual size, L, or to follow the formula of Pauly (1980) and hence to depend only on L.
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Results |
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Key-run
The overall exponent expressing the scaling of ML with length for both pelagic and demersal species was estimated at n = –1.66 (Table 2).
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For pelagic species, the minimum standard deviation of the net reproductive rate over the L
interval from 10 to 50 cm was generated with an estimated exponent in the K/L relationship of epel = –1.34. The model produced a dome-shaped replacement curve with a maximum in net reproductive rate for species with an asymptotic length of 30 cm and a difference of less than a factor of 4 in net reproductive rate within the pelagic L range of 10–50 cm (Figure 2a). The total natural mortality (ML) of a 25 cm fish was 0.23 year–1 (Figure 2b).
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For demersal species, the deviation in net reproductive rate over the L
range 10–180 cm was minimized when ML after settling was almost proportional to L (i = 0.8). The differences between maximum and minimum replacement within the length range considered were again small. R increased gradually with L to peak at a value of 85 cm, after which it declined (Figure 2c). A demersal fish with L = 85 cm had an R 4 times higher than a 20 cm species, and twice as high as a 180 cm species. The ML of a 5-cm newly settled fish was in the order of 0.4% d–1 (1.33 year–1) for L = 20 cm, but 1.5% d–1 (5.54 year–1) for L = 130 cm (Figure 2d). The natural mortality of a 50-cm fish with L = 75 cm was 0.18 year–1, whereas the value for an individual of the same length with L = 130 cm was 0.22 year–1.
Because of the reasonably constant replacement of fish of different size and the realistic values for natural mortality, this run was adopted as the key-run. For both demersal and pelagic species, the value of M/K at half the asymptotic size of a species was virtually independent of L, ranging between 1.1 and 1.5.
Sensitivity
Given the uncertainty in the relationship between SSB and viable egg production, as well as in larval mortality, Table 2 also provides estimates for key parameters under alternative assumptions about these factors. The scaling factors for both growth and mortality were much more sensitive to the value of post-larval mortality than to changes in fecundity: a tenfold change in fecundity produced a similar change in the parameter estimates as a twofold change in the initial natural mortality of the post-larvae. The exponent in the length–mortality relationship (n) changed substantially between the different runs, but remained within the interval –1.11 to –2.31. The exponent in the ML/L relationship of demersal species after settling (i) changed from 0.46 to 1.18 when the initial daily post-larval mortality was set at 7.5% and 22.5%, respectively. The K/L relationship of pelagic species changed comparatively less than the mortality parameters, and the exponent (epel) remained within the interval –1.17 to –1.38, irrespective of the large changes in fecundity and in post-larval mortality. Halving M1 resulted in a <2% change in any of the parameter values.
Using a simpler model with no difference between habitats
When natural mortality was assumed to scale with individual length, L, only, solutions minimizing the across-species difference in R (in the L range 10–180 cm, corresponding to the range found for common North Sea species) resulted in a scaling exponent n = –1.28, and values of adult total natural mortality in the order of 0.32 year–1 for a 50-cm fish and 0.19 year–1 for a 100-cm fish. These values of M are not unlike those currently used for demersal species in North Sea stock assessments. However, the replacement curve increased monotonically as a function of L, suggesting that coexistence of different-sized species of fish was unattainable (Figure 3). Comparing species with an L of 20 and 100 cm, the latter were predicted to produce a 927 x higher R than the former, a difference hardly conceivable to be conducive for coexistence.
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Pauly (1980) described natural mortality as a function of K, L
, and ambient temperature. Using Pauly's formula with an ambient temperature of 12°C to calculate natural mortality from the post-larval stage on, a steadily increasing replacement curve was obtained. With a constant M (independent of individual length), R for a species with L = 100 cm was 44 times larger with the model of Pauly (1980) than R for a species with L = 20 cm (Figure 3).
Checking the results from the key-run against independent data
Recent estimates of quarterly predation mortality from a North Sea MSVPA (1963–2003; excluding seals) were kindly provided by Morten Vinther (of DTU-Aqua). The average quarterly predation mortality (M2) for cod, haddock, whiting, Norway pout, herring, sprat, and sandeel was calculated across all years and regressed against average length (Figure 4) using the model ln (M2L,j) = ln(h) + iln(L,j) + nln(L), where j is the species, length (L) was calculated from MSVPA weights-at-age, and the other parameters correspond to Equation (8). The model explained 51% of the total variance of the data and was highly significant [r2 = 0.51, F = 49.7, Pr(F) < 0.001, n = 98]. Introducing separate slopes and intercepts for individual species did not produce parameter estimates that differed significantly from the overall.
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Fitting the model separately for demersal (cod, haddock, whiting, Norway pout, and sandeel) and pelagic species (herring and sprat), the parameter estimates for demersal species were all significantly different from zero and their confidence limits contained the theoretical values estimated by the size-based model (Table 3). The exponent i of the ML/L
relationship was here estimated as 0.71 compared with 0.80 in the key-run, the exponent n in the ML/L relationship was –2.11 rather than –1.66, and the parameter ln(h) was 1.45 vs. 0.54 in the key-run. For pelagic species, i was not significantly different from zero, but n was significantly different from zero (–0.99), and its confidence limits contain the –1.66 estimated in the key-run, and ln(h) (1.03) had confidence limits large enough to contain the 3.31 estimate from the key-run.
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Values of L
and K for selected families of demersal and pelagic teleosts were obtained from FishBase (Froese and Pauly, 2006). Only data from families for which >100 individual estimates were available, and with few multiple estimates for individual species, were included. Non-linear regression was used to estimate the parameters d and e (Table 4). Fitting a linear model with separate intercepts and slopes for demersal and pelagic fish to log-transformed values of L and K revealed a significant difference in slope between the two groups (F = 9.7, p < 0.0019), the slope being consistently more negative for pelagic species than for demersal. Except for the Sciaenidae, the exponent was between –0.85 and –0.63 for demersal families, and between –1.57 and –1.08 for pelagic families, conforming to the difference in growth patterns suggested by the model where the estimated exponent for the pelagic species was –1.34, different from the –0.64 estimated for demersal fish.
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Denney et al. (2002) found an inverse relationship between the natural logarithm of the maximum R/SSB and maximum body length. Re-analysing these data (in their Table A1) revealed that maximum recruitment scaled with maximum length raised to a power of –3.03 [r2 = 0.63, F = 87, p < 0.0001, n = 54, 95% confidence intervals (–2.38, –3.69)].
Many commercially exploited North Sea species are close to or at levels of spawning–stock biomass where recruitment is impaired. In accord with Denney et al. (2002) and others (Myers et al., 1999; Barrowman and Myers, 2000), we assumed that the effect of density dependence on recruitment would be negligible in this situation and that R/SSB was close to the maximum possible. However, the maximum R/SSB reported by Denney et al. (2002) were derived from VPA and represent cohort strength before the cohorts enter the exploited phase. In the size-based model, small species are subject to fishing in their first year of life, and the maximum number of 1-year-old recruits generated per unit of SSB was hence obtained at zero fishing mortality. Changing F from its current level to zero in the key-run without changing the other parameters generates a scaling of recruitment at age 1 with L with an exponent of –2.69 for demersal species and –2.96 for pelagic species. Both values fall well within the confidence intervals of the estimate derived from the data of Denney et al. (2002).
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Scaling relationships and coexistence
The model describes a teleost community where the species share a common set of habitat-dependent scaling relationships describing how growth, maturity, and mortality depend on length and asymptotic length. We assume that these scaling relationships have evolved in response to interactions with co-occurring species, to habitat characteristics, and to phylogenetic and physiological constraints. Most of the parameters in the model can be gathered from published data on growth, maturity, and fecundity, but assuming that both small and large species are able to replace themselves, the remaining parameters describing natural mortality and, for pelagic species, the parameters linking growth to asymptotic length have to be estimated within the model. This was done by minimizing the standard deviation of the net reproductive rate (R) across the range of asymptotic lengths found in the community, subject to the constraint that, on average, R should equal 1.0 across the range of asymptotic lengths.
We have used the model to explain a general scaling of SSB/R with L. This scaling is puzzling because of its magnitude—three orders of magnitude for a one-order change in asymptotic length—and important because of its potential consequences for species coexistence. The trivial explanation for the scaling of SSB/R that spawners of a species with large L necessarily accumulate more biomass per recruit than a species with small L must be placed in the fuller context of how many equally sized recruits of both species will be produced per kilogramme of spawner. M or K simultaneously has to scale in some way to produce a long-term expectation of one spawner per spawner (R = 1) for all species to persist in the community. The solutions obtained with the simpler models (with M depending only on individual length, and with M depending only on L) produce large differences in R among species. This demonstrates that there would be little opportunity for coexistence of large and small species of fish in a simple single-factor model where M is assumed to scale either with individual body length within species (i = 0) or with L across species (n = 0).
The scaling problem may be solved by allowing small species to produce more eggs than large species, by subjecting large species to a higher M than small species, or by letting small species grow more rapidly than large species. The regression of the number of eggs produced per gramme of female provided little evidence for a general scaling of relative fecundity with L, and a trial run where the estimates of u and v given in Table 1 were used to calculate relative fecundity did not produce results that differed significantly from the key-run. Relative fecundity may vary over time within species, but the variation seems to be relatively modest (Oskarsson et al., 2002; Yoneda and Wright, 2004; Thorsen et al., 2006). RE may also increase somewhat with size within species (Kamler, 2005), but this would only aggravate the problem by affecting large species more than small species. No evidence was found in the literature (e.g. Pepin, 1991; Houde and Zastrow, 1993) to support a scaling of larval growth and mortality with L. We also cannot find a plausible explanation as to why the mechanisms regulating growth and survival in the pelagic larval and early juvenile stages should work quite differently for species with different L, when their larval and early juvenile stages are of similar size. We therefore consider a scaling of the survival of pelagic prerecruits of the necessary magnitude to be unlikely and propose an alternative hypothesis where differences between pelagic and demersal habitats are used to explain how small and large species can coexist in the North Sea.
Demersal species
For demersal fish, the observed scaling of K with L implies that the rate of growth immediately after metamorphosis and settling differs between small and large species. In the von Bertalanffy growth equation, the rate of growth in length approaches a maximum of LxK as length moves towards zero. If L were exactly inversely proportional to K, all species would have the same maximum growth rate. When the exponent differs from –1, maximum growth rate is a function of L. For exponents <–1, the maximum growth rate of a large species is lower than the maximum growth rate of a small species. For exponents >–1, the opposite occurs. With an estimated exponent of –0.64 in the scaling of K with L, the initial slope of the growth curve is much steeper for large demersal species than for small. At a size of 4 cm, the growth rate of a demersal species with L = 130 cm is four times the growth rate of a species with L = 10 cm.
Securing the food necessary to maintain a fast growth rate may entail a fish being exposed to a higher M (Sogard, 1997), in particular during and immediately after major life-history transitions such as settling. Moreover, many of the small demersal species in the North Sea seem better protected against predation than large species fish of similar size (Figure 5). They typically possess anti-predator defences such as (venomous) spines, flat body shapes, or cryptic behaviour, or they live in habitats where they can hide from predation (e.g. sandeel and species associated with structurally complex habitats such as rocky seabeds). Developing predator defences or hiding may involve a metabolic cost or reduce their food intake and affect growth. For example, hiding by fresh-water fish reduces food intake while decreasing predation mortality (Mittelbach, 1986; Werner and Hall, 1988). In their analyses of empirical estimates of the natural mortality of 173 fish species, Griffiths and Harrod (2007) found species that were cryptic, hid in burrows, or had morphological defence against predators to have the lowest natural rates of mortality. We therefore hypothesize that the scaling of M with L reflects a trade-off between growth and mortality. Demersal fish with small L may have a lower maximum growth rate at the larval/juvenile transition than fish with a large L, specifically because they adjust their behaviour or morphology to reduce predation.
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Our model shows that the difference in R with L
can be largely solved for demersal species by making M after settling a function of individual length and L. Many authors have identified settling as an important life-history event and have found density-dependent predation in the post-settlement phase to be important (Myers and Cadigan, 1993; Van der Veer et al., 2000; Hixon and Webster, 2002; Almany and Webster, 2006). To our knowledge, whether mortality immediately after settling generally scales with L has so far not been investigated. Our results suggest that M after settling should be almost directly proportional to L. This proportionality obviously depends on the input parameters. Changes in larval mortality and relative fecundity alter the results substantially, but the values of the mortality parameters from the key-run are not significantly different from those generated by an independent analysis of the average predation mortalities estimated by North Sea MSVPA.
For demersal species, our estimate of the exponent in the relationship between M and size is in general accord with the values reported. M and body weight or length scale with an exponent close to –0.25 or –0.75, respectively (Peterson and Wroblewski, 1984; McGurk, 1986; Brown et al., 2004). At first glance, our results may seem to suggest that M should scale with length raised to a power of –1.66 rather than with length raised to –0.75 (Table 2). However, this apparent difference can be explained by the confounding effect of the L term for demersal species. Assuming that the values of M for adult fish used by the authors cited above were derived mainly from demersal fish of a length corresponding to a constant fraction (µ) of their L, their predation mortality should scale as h x (L)i x (µL)n, i.e. as i + n. We estimate i + n to equal –0.86 (Table 2), so the overall exponent relating M to weight is estimated to be –0.86/3 = –0.29, close to the –0.25 reported by others. Further, the exponent relating M to larval body weight is reported to be significantly more negative than it is for adults (McGurk, 1986; Houde, 1987). In our model, the exponent for the post-larvae is –1.66/3 = –0.55 on a weight basis, compared with the –0.29 estimated for adult fish. Finally, in an empirical analysis of 175 fish stocks, Pauly (1980) found that M at a given temperature was proportional to K0.654 x L–0.28. Inserting KL–0.64 (Table 2) in this relationship, we find that natural mortality should scale to asymptotic length raised to –0.70, not far from the value of –0.86 derived above.
Pelagic species
For pelagic species, the model predicts that K should scale with L raised to a power of –1.34. Although this result is based on very few species, an analysis of L and K from FishBase resulted in exponents for four pelagic families ranging from –1.08 for Engraulidae to –1.57 for Clupeidae, significantly lower than those for most demersal families (Table 4). Hence, the growth pattern appears generally different among the two habitats. Small pelagic species seem initially to grow faster than their larger counterparts, and this may explain how the former are able to sustain the higher levels of M associated with a smaller body size, by being exposed to size-dependent M for less time at each size. Where demersal species seem to solve the apparent difference in replacement of large and small species by modifying their growth and vulnerability to predation, pelagic species may overcome the problem by changing their growth rate with asymptotic size.
Ideally, we should have used estimates of L and K for pelagic and demersal species from different fish communities and analysed their relationship allowing for differences in intercepts and exponents between communities. However, for pelagic species, we could not find sufficient estimates of growth parameters from separate communities to perform such analyses. By analysing the data at a family level, we may have introduced additional variation caused by differences between fish communities in, for instance, ambient temperature and exploitation. Despite the likely inflation of overall variance caused by these differences, the difference in the slope of the K/L relationship between pelagic and demersal families of fish proved highly significant. Our analysis not only confirms the overall difference between pelagic and demersal species suggested by the model, but it shows that this difference is present also at a family level.
Scaling of maximum R/SSB
Our model can reproduce the scaling of maximum R/SSB with L reported by Denney et al. (2002). Calculating the maximum survival from egg to age 1 by assuming that only M operates during this period results in numbers of R/SSB for both demersal and pelagic species that scale with L raised to powers that are close to the estimate of –3.03 based on their data from the Northeast Atlantic. This scaling cannot be reproduced by a model where natural mortality depends only on individual length and where the combined scaling of L and K is used to model growth.
The importance of stabilizing and equalizing mechanisms for species coexistence
We base our conclusions on a simple model of coexistence in which populations are assumed to be in steady state. In line with common approaches in life-history theory (Stearns, 1992; Charnov, 1993), we attempt to capture the general patterns in the average life-history characteristics of demersal and pelagic teleosts. For the sake of simplicity, we used a model with an unstable equilibrium, providing no mechanism for recovery of species with a less than one-for-one replacement.
Species can coexist in variable environments if they respond differently to differing conditions and can persist through unfavourable periods or retract to favourable areas, and stage structure, resource partitioning, non-linear competition, and density-dependent biological interactions are able to promote coexistence (McCann and Yodzis, 1998; Chesson, 2000; Abrams and Holt, 2002; Bonsall and Hastings, 2004; Chase, 2005). We recognize that periods of decline or increase are common in fish populations, and that stabilizing mechanisms must be important for their long-term persistence. Still, when the species in a community manage to persist over long time-spans, their net rates of population growth are bound to be close to zero (Hubbell, 2001). This can only be realized if their growth, mortality, and reproductive characteristics on average generate spawner-for-spawner or egg-for-egg replacement, irrespective of the nature of the mechanisms involved. Although long-term persistence will be achieved by both fitness-equalizing and -stabilizing mechanisms (Chesson, 2000; Adler et al., 2007), there is no obvious reason why stabilizing mechanisms should result in a relationship between R/SSB and L, producing a three orders of magnitude difference in recruitment for a single order change in L. Rather, stabilizing mechanisms alone are unlikely to be responsible for more than a single-order change in replacement. In their analysis of >700 time-series of spawners and recruitment for different populations and species of fish, Myers et al. (1999) found that the maximum relative growth rate of fish populations is typically around 1–7 replacement spawners per spawner per year, suggesting that a factor of 10 is the likely maximum difference in average replacement that stabilizing mechanisms could effectively buffer against. The net reproductive rate is the proper fitness measure to use when populations are regulated by density-dependence early in life (Mylius and Diekmann, 1995). Still, by neglecting other stabilizing mechanisms and assuming populations to be in steady state, we cannot be completely sure that the model will generate life-history strategies that are evolutionarily stable. Therefore, investigating the consequences of our model assumptions in an evolutionary context is a clear target for further research.
Model limitations and applicability
Using a simple scaling model to capture variations in life-history parameters across a multitude of species makes it likely that some species rely on life-history features not adequately accounted for by the model to compensate for insufficient replacement. This is, for instance, reflected by the spread of the observations of L and K around a log-linear relationship (Figure 1a). It is therefore unrealistic to expect general scaling relationships to produce an exact one-for-one replacement at any one time. Additional trade-offs between growth, mortality, and reproductive investment should allow species with identical L to achieve a one-for-one replacement, e.g. if they counteract a reduction in growth rate with an increase in fecundity or a reduction in M (Gunderson, 1997). Species of similar L may be found on a continuum from sit-and-wait strategists to active swimmers hunting for food or use parental care to increase egg survival, as several of the smaller North Sea demersal species do.
Our model is primarily aimed at describing marine fish communities typical of temperate continental shelves with sandy and muddy seabeds, and it mimics the basic life-history features of determinate spawners with pelagic eggs. Although we applied information from North Sea species, the model could be applied easily to other shelf areas to test our prediction that for a given individual length, large demersal species after settling should suffer greater mortality than small demersal species. Size-based scaling of life-history parameters across species may also be used to compare replacement and to explore trade-offs among life histories in other types of communities, such as in deep-water fish communities where the light intensity shapes the trade-offs between feeding and predation (Strand et al., 2002; Fiksen et al., 2005), or in reef-fish communities where species diversity is linked to habitat complexity (Gratwicke and Speight, 2005). These may be particularly interesting systems for studying size-based trade-offs, because superimposed on the processes considered here, complex habitats offer the various species a multitude of possibilities for composing their own particular life-history strategy to secure their persistence, constrained by size as a factor in securing shelter and reduced predation mortality. However, where shelter favours certain sizes, it may not suffice to model predation mortality by an expression where a single exponent is used to relate predation mortality to length.
The surprisingly good fit to the estimated predation mortalities from the North Sea MSVPA calls for further investigations of our model's ability to fit existing estimates of M for various species and sizes of fish, and for a revision of the formula for estimating M derived by Pauly (1980). This formula ignores the within-species length dependence frequently reported for fish (Lorenzen, 1996; Sogard, 1997; Bax, 1998). It has been widely used to estimate M for fish populations in situations where research costs would prohibit empirical observations, but many of the empirical values used in its derivation refer to inferred values (such as 0.1 or 0.2 for many temperate commercial fish stocks) rather than independent estimates. Often, estimates of M derived from the formula have been used in stock assessments leading to management advice. Taking the length dependence of M into account is likely to change predictions of the effects of size-based management measures such as mesh-size regulations, as well as predictions of responses of fish communities interlinked by predation to ecosystem-based harvesting strategies.
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We thank Tom Fenchel, Ken Haste Andersen, Anna Rindorf, Marie-Joëlle Rochet, and three anonymous referees for helpful comments and advice on earlier versions of the manuscript. Funding to pay the Open Access publication charges for this article was provided by the National Institute of Aquatic Resources, Technical University of Denmark.
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