Elvar H. Hallfredsson and John G. Pope

Transcription

1 1313 Modelling the growth, mortality, and predation interactions of cod juveniles and capelin larvae in the Barents Sea using a novel proto-moment population dynamics model Elvar H. Hallfredsson and John G. Pope Hallfredsson, E. H., and Pope, J. G Modelling the growth, mortality, and predation interactions of cod juveniles and capelin larvae in the Barents Sea using a novel proto-moment population dynamics model. ICES Journal of Marine Science, 64: Proto-moments of a fish population, the sums of products of powers of length with numbers at length, relate both to the traditional statistical measures, the mean, the variance, the skewness, and the kurtosis of their size distribution, and to the biologically important measures of the abundance and the biomass of the population. Population models based on this approach are constructed as matrix delay-difference equations. They model moments of the length distributions rather than the age distributions, and express population dynamic problems in an analytically tractable form. Here, a modification of this approach is explored for a case involving a predator prey relationship among young-of-the-year fish. The modelled species are juvenile cod (the predator) and capelin larvae (the prey) in the Barents Sea. Their population dynamics are modelled by the proto-moment method, but using two different approaches for the derivation of the predation mortality term. The first approach, a published matrix-based formulation, is formed purely in terms of the proto-moments. The second approach, developed here, converts the proto-moments back to consistent size distributions to calculate the rates of predation mortality on each proto-moment. The latter model produces a realistic development in time for the predator and prey length distributions, density in numbers, and biomass for their first summer. It also provides estimates of growth for both species along with estimates of the predation mortality that cod generate on capelin in this time period. The estimates permitted the accuracy of the matrix-based approach to be investigated, better understood, and improved. Keywords: capelin, cod, delay-difference, multispecies, predation, proto-moments, 0-group. Received 25 September 2006; accepted 6 July E. H. Hallfredsson and J. G. Pope: Norwegian College of Fishery Science, University of Tromsø, N-9037 Tromsø, Norway. Correspondence to E. H. Hallfredsson: tel: þ ; fax: þ ; elvar.hallfredsson@nfh.uit.no Introduction Describing populations using proto-moments is a method that may be useful either for modelling subpopulations in a spatially disaggregated approach or possibly as a simple aggregated model. Proto-moments are described by Pope (2003), who extends the ideas of Deriso (1980) and Fournier and Doonan (1987). The basic idea of proto-moments is to capture the size structure of the populations in terms of simple statistical properties, the sum of numbers times length to the powers 0 4, rather than in terms of the numbers of animals at each length or at each age, the typical practice in fisheries models. These simple statistics measure such biologically useful properties as the abundance and biomass of the population, and the mean length and weight of its members. Moreover, these measures can be more simply updated through time than equivalent age- or size-based models using information and data on the growth and mortality experienced by the population, and they require fewer data than equivalent age-based dynamic pool models but can provide almost equivalent outputs. They directly provide the succinct summaries (such as biomass) given by bulk biomass models, but also allow biological knowledge and complexities to be included in the analysis. These characteristics make them potentially powerful tools both for performing typical stock assessment functions (Pope, 2003), such as making retrospective analyses, or making short- or long-term predictions, and for process studies as here. However, their practical utility is still at an evaluation stage. In short, proto-moments are a way of describing the essential stock attributes of numbers, biomass, and size distribution. These are all described by the parsimonious state vector composed of the product of stock numbers by the 0th to the 4th powers of length. Updates of the state vector can be expressed in a recursive form and be updated by simple matrix algebra. Therefore, the proto-moment approach is a member of the delay-difference family of models whose use is advocated by Hilborn and Walters (1992). The proto-moment vector also provides the basis for calculating the traditional statistical descriptors of mean, variance, skewness, and excess (kurtosis). The proto-moment approach is described by Pope (2003) as a method for modelling adult ages of fish using a yearly time-step. Subsequently, Pope successfully used the approach in practical applications (as yet unpublished) in this mode for emulating age-based assessments of North Sea flatfish and investigating # 2007 International Council for the Exploration of the Sea. Published by Oxford Journals. All rights reserved. For Permissions, please journals.permissions@oxfordjournals.org

2 1314 E. H. Hallfredsson and J. G. Pope interactions between cod (Gadus morhua) and grey seals (Halichoerus grypus). The existing examples of the use of the method are of this type (i.e. yearly time-step). However, the sparse state vector and linear update of the approach should make it a suitable candidate model for describing the development of early life stages of fish on a much shorter time-scale. Protomoments have not been used previously for this purpose. The first summer of life of boreal fish is the most difficult part of their life history to study and to understand, because the fish are often in the process of advection from spawning areas to juvenile feeding areas. Their distribution changes in a fashion that would require frequent, extensive, and expensive surveys to comprehend, and both growth and mortality are fast/high and variable. Empirical studies at this stage fall awkwardly between the local patch-based intensive approaches appropriate to investigating larval fish, and the annual systematic surveys that are useful in understanding fish in their latter years of life. Given these difficulties, it is natural to try to fill the gap in understanding by models rather than surveys. Therefore, the primary objective of this work is to apply the proto-moment approach to a simple case study of modelling young-of-the-year fish in a multispecies context, to assess its usefulness and associated problems. We aim to test whether it can be adapted for use with a typical young fish dataset. As this is the first application of the approach, part of the analysis is dedicated to the estimation of the rate of predation mortality using a proto-moment-based model, and to checking the accuracy of these estimates. The case study involves the predation interaction between young-of-the-year capelin (Mallotus villosus) and cod in the Barents Sea between June and August. Cohorts of capelin larvae and cod juveniles overlap in time and space in the southern Barents Sea in summer (Olsen, 1968; Bjørke and Sundby, 1987; Gundersen, 1995; Gjøsæter, 1998; Helle and Pennington, 1999), and predation on capelin larvae by 0-group cod juveniles has been recorded in the Barents Sea (K. Helle, pers. comm., Hallfredsson, 2006) and in Icelandic waters (Thorisson, 1989). During June/July of 2002 and 2003, detailed process surveys were made of the cod capelin interactions. Both species are also recorded in results of the international 0-group survey conducted in the Barents Sea in late summer and early autumn (Anon., 2002b, 2003b). Interpreting the data from these surveys in population model terms is the second objective of this work. Extrapolation of the results from the June/July 2002 and 2003 surveys on to the period of the international 0-group survey, which provides size composition data in autumn, gives an opportunity to test the model over a period when both the average sizes and the relative size of predator and prey change rapidly (Table 1). Such an extrapolation also provides insights into how the cod might affect capelin numbers and size distribution. At present the model ignores the potential effects of prey availability on cod biomass. Moreover, this cod capelin model cannot itself be predictive of how capelin biomass and size distributions change, because there are other predators. In particular, the predation effect of herring (Clupea harengus) on capelin larvae is thought to be dominant at times when young herring are abundant in the Barents Sea (Hamre, 1994; Huse and Toresen, 2000; Godiksen et al., 2006; Hallfredsson, 2006). Extrapolation of this June/July predation behaviour should give some insight on the impact of cod on capelin between June/July and autumn. However, the main motivation is to test the model over an extended period. Table 1. Summary of the survey data. Year Survey Start day 24 June 10 August 23 June 27 July End day 5 July 8 September 7 July 2 October Density of juvenile cod (numbers m 22 ) Density of capelin larvae (numbers m 22 ) Number of stomachs analysed Number of capelin larvae in cod stomachs Number of measurable capelin larvae in cod stomachs Average cod length (mm) Average capelin length (mm) Average length of capelin in cod stomachs (mm) Densities for survey 1 each year are the average for all stations and are based on GULF III samples for capelin larvae and acoustic estimates for 0-group cod. Material and methods Mathematical model The population model of Pope (2003) tracks abundance and size distribution by updating a state vector C p,t of a population (p) at time t whose ith element c p (i) is given by c p ðiþ ¼ X all L p N Lp ;t Li p ; where N Lp,t are the number of individuals in the population at time t in the discrete length class L p for i = 0 to 4. The c p (i) have direct biological significance because c p (0) may be directly interpreted as population abundance and, where growth is isometric, c p (3) is proportional to population biomass. This is taken to be the case for cod, but for capelin larvae, c p (4) better relates to biomass, because at that life stage, capelin weight is approximately related to the 4th power of length [wet weight (g) = (total length (mm)] 4.241, range 9 38 mm, n = 24, r 2 = 0.98 (EHH, unpublished data). If the power is forced to be 4 exactly, the equation becomes wet weight (g) = [total length (mm)] 4, r 2 = Hence, c p (4) is used to estimate the biomass of capelin larvae (in g) in this study]. Additionally, proto-moments may be converted to give the first four statistical attributes of the size distribution of a population. For example (omitting the population suffix), the mean length = c(1)/c(0), and the variance of length = [c(2)2c(1) c(1)/ c(0)]/c(0). Standard measures of the skewness and the kurtosis of the length distribution may also be constructed. ð1þ

3 Modelling growth, mortality, and predation interactions of cod juveniles and capelin larvae 1315 For convenience, the c p (i) are called the proto-moments of the population. Proto-moments provide a convenient and parsimonious way of tracking the main size attributes of a population. Their use requires that population losses to mortality can be described as the sum of the number at length dying multiplied by length to the powers 0 4, i.e. in an equivalent form to Equation (1). A further requirement is that growth in length over a time-step (t to t + 1) can be adequately described by a linear recursive equation of the form L p;tþ1 ¼ a þ bl p;t : This allows von Bertalanffy growth, exponential growth, and linear growth to be modelled using suitable choices of the constants a and b in Equation (2). For example, b = 1 gives linear growth (but note that generally this equation describes growth of all the cohorts in the population, which are summed into the C, but in the example of this paper, it applies only to young-of-the-year from a single cohort of capelin and one of cod). If this condition is satisfied, then C is updated by a growth matrix G so that C tþ1 ¼ G ðc t lossesþþgains; where the elements g(i, j)of G are the coefficients of the jth powers of L p, t when Equation (2) is raised to the power i. Thus, gði; jþ ¼0; if j. i; i! gði; jþ ¼ a ði jþ b ð jþ ; if j i; ði jþ! j! for i = 0 to 4, j = 0 to 4. Equation (4) may also be adapted to the situation where individual growth follows Equation (2), but where a or b vary between individuals. In Equation (3), any losses such as catch or predation, and any gains such as recruitment or immigration, are expressed in an equivalent proto-moment-like form to Equation (1). In the current case, of predation by 0-group cod on 0-group capelin, the losses of the capelin proto-moments to predation are described as a vector D t, with elements d cap (i). d cap ðiþ ¼ X all L cod X all L cap D Lcap; L cod;t L i cap ; where i ¼ 0 to 4 and D Lcap, L cod,t are the numbers of 0-group capelin deaths at length L cap attributable to predation by 0-group cod of length L cod in day t. Here, all recruitment is taken to have occurred before day one, and there are no catches (other than insignificant amounts by research vessels). Hence, for capelin, Equation (3) reduces to C cap;tþ1 ¼ G ðc cap;t expð M1Þ D t Þ; where M1 is the instantaneous rate of non-cod predation natural death per unit time. This rate of mortality is taken as equal on all sizes of 0-group capelin and therefore acts through a scalar multiplier of the C cap,t. The equivalent equation for C cod,t+1 is similar to Equation (6), but predation deaths (D t ) are set to zero. ð2þ ð3þ ð4þ ð5þ ð6þ The mathematical formulations of predation losses are often quite complex and also can require knowledge of the abundance of other available prey species. One of the simpler forms is based on the assumption that a unit biomass of predators generates a rate of predation mortality on prey, with an intensity that varies as the ratio of predator to prey weight or length. This formulation assumes that the rate of predation mortality is unaffected by prey abundance, and hence that the abundance of alternative prey can be ignored, so this approach ignores the possibility of predator satiation. The distribution of predation mortality is typically taken as a log-normal function of the ratio of predator to prey weight or prey length with log-mean m and standard deviation s, and a constant multiplier r to scale the intensity of mortality (ICES, 1984, 1986; Nilssen et al., 1994). If total mortality is low per time-step, then D Lcap,Lcod,t may be written as D Lcap;L cod ;t ¼N Lcap;t r exp 0:5 N Lcod ;t L 3 cod;t : lnðlcod =L cap Þ mþ s 2 ::: Equations (5) and (7) that together define d cap (i) appear complicated. However, Pope (2003) indicates that many processes of interest can be reasonably approximated using polynomials of length, extending the approach that Deriso and Parma (1988) applied to selection curves. Such polynomials, when multiplied by abundance at length and summed over all lengths, are translated into linear combinations of the proto-moments. In particular, Pope (2003) suggests that d cap (i) might be approximated as a bilinear form of the proto-moments for each predation loss term. In matrix algebra, a bilinear form is the scalar that results from pre-multiplying a square matrix by a row vector and postmultiplying it by a different column vector; in the current case, the row vector being the transpose of the capelin proto-moments, the column vector the cod proto-moments, and the matrix being a 5 5 array of constants that define the approximation. Thus, dðiþ C0 cap;t FðiÞC cod;t ; where a suitable 5 5 matrix of constants F(i) is chosen for each i to approximate the amount of c cap (i) eaten. In the course of our investigations, we initially found that the Pope (2003) bilinear approximations for predation deaths sometimes gave biased results. Hence, as an alternative approach, the capelin and cod population proto-moments are reconverted to numbers in each discrete length class at each time-step. These numbers by length class are then used directly in Equation (7) to provide D Lcap, L cod,t and therefore the d cap (i) and the vector D t for Equation (6). These reconverted size distributions are also useful for interpreting the proto-moments, and the estimates of d cap (i) based on them are used to investigate the bias in Equation (8) and hence how F(i) in Equation (8) might be estimated better. In general, using the five proto-moments to reconstitute a size distribution with.5 length groups cannot give a unique solution. To achieve a unique solution requires additional constraints. Hence, in addition to constraining each reconstituted length distribution to have the correct proto-moment values, numbers at length were also constrained to be non-negative, and their ð7þ ð8þ

4 1316 E. H. Hallfredsson and J. G. Pope distribution was constrained to a smooth shape. The smooth shape was achieved by minimizing the third differences of the distribution of numbers per length class. Minimizing third differences is an approximation to minimizing, at each point, the second derivative of the curve with respect to length. Therefore, this minimization attempts to fit a smooth curve which locally fits as closely as possible to a quadratic function. The modelled proto-moment rates of predation mortality, estimated using the reconstituted size distribution approach described earlier, are compared with those estimated by the Pope (2003) approximation. Trials of the model defined by Equations (1) (8) were initially run on an Excel spreadsheet. In those runs, the bilinear approximations of the F(i) of Equation (8) were made for the whole size range of cod and capelin encountered between the first day of the June surveys and the last day of the August/ September surveys. However, the results proved unrealistic because of large biases in Equation (8) when F(i) was estimated in this fashion, so the reconstituted length distribution approach was adapted to estimate d cap (i). Additionally, to make comparisons between the results of the two methods of estimating predation, a more focused but more time-consuming approach of estimating F(i) at each time-step was adopted. To introduce these changes, the model was rewritten in MATLAB. In this formulation, approximations to F(i) were initially made over the 95% confidence intervals of the capelin and cod length ranges at each time-step. It was subsequently found that approximations of the F(i) made over points describing each 5% increase in the cumulative reconstructed size distribution gave more accurate results. Studies to find ways to make this approximation both accurate and parsimonious (i.e. based only on pure proto-moment results) are ongoing, because this would be useful in circumstances where computer time and storage capacity were at a premium. In most normal applications, however, the ability to reconstruct size distributions from protomoments is not a heavy burden and is itself a useful development of a technique, because it enables estimated distributions to be compared with observed size distributions in a more comprehensible fashion. Data The data used were from surveys in the southern Barents Sea, made with RV Johan Ruud, 24 June 5 July in 2002, and RV Jan Mayen, 23 June 7 July in 2003 (hereafter called survey 1 of each year). Capelin show considerable variance in both temporal and spatial location of spawning (Gjøsæter, 1998). In 2002, spawning was only found east of 278E (Anon., 2002a) and the area offshore of the eastern capelin spawning grounds was surveyed twice, with 29 and 32 stations per coverage. In 2003, capelin spawned mainly west of 268E (Anon., 2003a), and with eastward dispersal of larvae, the area from 248E to 328E was covered with 40 stations. In both years, the distance between stations were mostly 15 nautical miles in a north south direction and 20 nautical miles in an east west direction. The data inputs included length distributions for capelin larvae sampled by a GULF III highspeed plankton sampler, and length distributions for juvenile cod collected by a pelagic trawl, as well as information from stomach content analysis of juvenile cod. The GULF III had a mesh size of 375 mm and the pelagic trawl a m opening and a codend liner of 4-mm mesh (Hallfredsson, 2006). The length Figure 1. Length distribution for (a) capelin larvae from the Gulf III samples, (b) capelin larvae in cod stomachs, and (c) cod from the trawl samples. All distributions are scaled to numbers per 1 m 2 surface area (redrawn from Hallfredsson 2006). distributions of the surveys used to initiate the model were weighted by densities at stations and standardized to m 2 (Figure 1). Various growth models were utilized and their plausibility judged, by comparing the model results for length distributions of capelin larvae and juvenile cod with their distributions from the joint Russian/Norwegian 0-group surveys in the Barents Sea from 10 August to 8 September 2002, and from 27 July to 2

5 Modelling growth, mortality, and predation interactions of cod juveniles and capelin larvae 1317 October 2003 (Anon, 2002b, 2003b) (hereafter referred to as survey 2 of each year). Table 1 gives a summary of the information from the surveys used for the modelling work. Model inputs Model runs were made with linear growth increments in both years. For 2002, additional model runs were made with an exponential growth pattern and different values for s in the predation function [Equation (7)]. The 2002 survey data were used for this purpose because they are more coherent than the 2003 data. Alternative growth rates were explored with the intention of finding rates and models that led to the June population size distributions growing to match the size distribution seen in August. Using the linear growth model with increments for cod of mm d 21 and for capelin of mm d 21, and with s =1 gave the best fit to the June and August data and was the parameter choice for the key run used as a baseline to compare results from runs based on other parameter choices. These key-run estimates were at the high end of literature values. It was therefore also deemed appropriate to make runs with literature values, corresponding to the slower linear growth using increments set to 0.4 mm d 21 for capelin and 0.6 mm d 21 for cod (Gjøsæter, 1998; Ottersen and Loeng, 2000). Fitting the model to the data Growth matrices G [Equation (4)] were constructed with the growth model, and parameters were chosen. To initiate the model, population proto-moments for day 1 of survey 1 (24 June 2002, 23 June 2003) were initially guessed. The estimates r and m of the predation function were also initially set at assumed values. Populations were updated by the model using a time-step of 1 d up to the end of survey 2 (8 September 2002, 2 October 2003). The initial assumptions of day 1 population protomoments and of r and m were then modified, until the average modelled population closely matched those observed in survey 1 (Figure 1a), and until the proto-moments of the estimates of capelin in cod stomach per m 2 matched those observed. This utilized all the available data from survey 1 and, as a result, no degrees of freedom remained for further statistical fitting. Results Input data The input data adopted are shown in Figure 1. Note in particular that capelin were less abundant but somewhat larger in 2003 than in 2002, and that fewer but larger capelin larvae were found in cod stomachs in 2003 than in Detailed results from the key run The model proto-moment outputs of the key run for 2002 are shown in Figure 2. These are initially scaled on a per m 2 basis, but clearly as time progresses, so will the area that the fish cover increase, so these results should be seen as modelling a group of capelin and cod whose overlap remained the same as initially seen in the survey. In particular, note that the 0th proto-moment of each figure depicts abundance and the 4th capelin and 3rd cod proto-moments are proportional to the biomass of the respective species. Collectively, they provide a succinct statistical summary of the distribution of 0-group capelin in the sea, 0-group capelin found in 0-group cod stomachs, and 0-group cod in the sea. Interpretations of these proto-moments from the key run of 2002 as mean growth and weekly reconstituted size distributions Figure 2. Proto-moments in 2002 (key run) for (a) capelin, (b) capelin in cod stomachs, and (c) cod. PM0 PM4 refers to proto moments 0 4. These are scaled so that all proto-moments fit in a given panel. are depicted on Figure 3. Note that for capelin in the sea, this gives a bimodal distribution, as seen in survey 1, and that it becomes more pronounced with increasing time. A similar bimodal distribution is seen for capelin in cod stomachs, whereas cod had a unimodal length distribution throughout the modelled time period.

6 1318 E. H. Hallfredsson and J. G. Pope Comparison of the 2002 key run with runs made with other assumptions of growth and predation mortality Table 2 shows the inputs that were varied between two alternative models of growth (slow and exponential), two alternative choices of the s of the predation mortality model [Equation (7)], and the results of a comparable key run made with the 2003 data. Table 2 also list comparative output values for those model runs. Note that the predation mortality (M2) during survey 1 was similar for all runs, because all runs were tuned to reproduce the feeding levels observed. Figure 5 compares the respective trajectories of the 0th (depicting abundance) and the 4th proto-moment (proportional to biomass) of each run for capelin. Note that variation in s has a profound effect on biomass estimates, but that numbers are less affected. Figure 3. Mean length in 2002 (key run) with relative length distributions every 7th day for (a) capelin, (b) capelin in cod stomachs, and (c) cod. Figure 4 depicts a comparison of the modelled outputs of size distribution, averaged over the pertinent periods, with the equivalent input data for the 2002 key run. The length distribution and stomach contents results for survey 1 were compared on a per m 2 basis, but the capelin and cod size distributions in autumn were compared with the results of survey 2 on a per mille basis. This was because abundance was given as an index, so the absolute densities per m 2 were not estimated in survey 2, and also because the initial per m 2 populations could be expected to diffuse into wider areas with time. The model was able to simulate the observed length distributions. Comparisons of approaches to estimating the rate of predation mortality Results of different approaches to estimating the Pope (2003) bilinear form approximation for predation mortality rate (based on an assumption that both species size distributions might be approximated by normal distributions, with mean and variance given by the proto-moments) are compared with the results obtained from the reconstituted size distribution method used in the current model. Figure 6a shows, for each time-step and each proto-moment, the ratio of the rate of predation mortality obtained by the bilinear form approximation to the value calculated by the reconstituted size distribution approach. If both methods gave the same results, the lines would show a constant ratio of 1.0. The bilinear form approximation shows bias of up to 15%, which varies in a similar but lagged sinusoidal form for all five proto-moments, suggesting that the discrepancies are attributable to some lawful cause that in future studies might be predictable. Therefore, this issue was investigated further. Discrepancies were far smaller by estimating the bilinear form approximation over a length range defined by the percentile from 5 to 100% (in steps of 5%) of relative cumulative cod and capelin proto-moment distributions. Figure 6b shows the results of this improved fit, with all discrepancies,2%, and most,0.5%. Discussion Utility of the proto-moment approach for modelling young-of-the-year fish The young-of-the-year capelin cod interaction model provides a useful test of the utility of the proto-moment approach (Pope, 2003) to modelling fish populations at this early life stage. The matrix-based structure of the recursive model made it easy to adapt the approach to the particular circumstances of these two populations during their period of rapid growth from larvae in June to 0-group fish in autumn. The one aspect of the model that had to be modified from Pope s (2003) formulation was the treatment of predation. The initial bilinear form approximation to the predation function proposed by Pope (2003) was rather biased. In the case of cod preying on capelin, the accumulation of biases proved to be sufficient to undermine the workings of the model. However, the alternative approach adopted, using reconstructed size distributions to calculate the quantity of each capelin proto-moment eaten by cod, worked well and might be used to correct the accuracy of a bilinear form approximation,

7 Modelling growth, mortality, and predation interactions of cod juveniles and capelin larvae 1319 Figure 4. Modelled compared with survey length distributions in 2002 (key run) for (a1) capelin survey 1, (a2) capelin survey 2, (b) capelin in cod stomachs survey 1, (c1) cod survey 1, and (c2) cod survey 2. Note that the scales on the y-axis for survey 1 are frequency in numbers per m 2, whereas for survey 2, the same scales are frequency (%). in circumstances where its mathematical tractability makes its use desirable. To provide a simple example, the case study adopted to test the model was restricted to a two-species one-box model. However, the inherently simple structure of the model would make it a strong candidate for use in multiple-species, multiple-box models that would be required to address coupled hydrographic biological problems. Clearly, this could be facilitated if simple and reliable bilinear form approximations were to be found for the predation function, because these are inherently faster in operation than the reconstructed size distribution approach used here. The latter method requires the use of a relatively timeconsuming minimization routine at each time-step for each species, whereas the former might be based on prespecified sets of the F(i)s of Equation (8), provided these could be bias-corrected. Model results as interpretation of survey data The instantaneous mortality rates of d 21 (0.9% d 21 ) estimated by the model during survey 1 in both 2002 and 2003 (Table 2) are of the same order as those estimated by Hallfredsson (2006), where the predation mortality rate was estimated to be ca. 1% d 21 over the same time periods for both years, using data from the same surveys with another approach, and taking digestion rates into account. This supports the plausibility of the model approach. We are unaware of other estimates of these rates. The value of m of estimated in the 2002 key run (Table 2) seems plausible. It indicates that the optimum size of capelin larvae for cod is 0.36 of a cod s length. This accords reasonably with what is known of marine predator prey length preferences (Scharf et al., 2000). However, the value of m of (Table 2)

8 1320 E. H. Hallfredsson and J. G. Pope Table 2. Results from the model runs. Growth Exponential Linear increments Year a b 2003 Run specification Fast growth Low s High s Slow growth Linear growth increments for capelin (mm d 21 ) Exponential growth factor for capelin Linear growth increments for cod (mm d 21 ) Exponential growth factor for cod s in predation mortality function r in predation mortality function (mm 23 d 21 ) 1.36E E E E E E207 m in predation mortality function Daily Z for capelin on survey Daily Z for capelin for all days Daily M2 for capelin on survey Daily M2 for capelin for all days Average length of capelin on survey 1 (s.d.) 9.5 (2.3) 9.4 (1.9) 9.4 (1.9) 9.4 (1.9) 9.5 (2.4) 14.6 (2.2) Average length of capelin on survey 2 (s.d.) 47.5 (5.6) 46.2 (2.6) 46.2 (2.6) 46.1 (2.5) 32.8 (3.0) 40.4 (1.0) Average length of cod on survey 1 (s.d.) 30.3 (5.8) 30.3 (5.7) 30.3 (5.7) 30.3 (5.7) 30.4 (5.9) 28.4 (4.8) Average length of cod on survey 2 (s.d.) 74.7 (6.8) 75.1 (5.7) 75.1 (5.7) 75.1 (5.7) 64.0 (5.9) 65.8 (4.8) Density of capelin on survey 1 per m Notional density of capelin on survey 2 per m Density of cod on survey 1 per m Notional density of cod on survey 2 per m Biomass of capelin on survey 1 (g m 22 ) Notional biomass of capelin on survey 2 (g m 22 ) Biomass of cod on survey 1 (g m 22 ) Notional biomass of cod on survey 2 (g m 22 ) Cod predation mortality (M2) and other mortality (Z) are instantaneous rates of mortality per day. a Key run. b Literature values of growth. estimated in 2003 does not seem feasible because it would imply that the optimum size of capelin larvae for cod is 0.99 of the cod s length. This implies a preferred capelin weight of 25% of cod weight, which seems too high. Moreover, if this is the preferred size ratio, then the upper confidence interval for capelin length in cod stomachs would be almost twice the length of the cod. It seems far more likely that there was a proportionately greater abundance of larger capelin larvae in 2003 than in 2002 and that this was not detected by the Gulf III sampler. Clearly, there are some reservations with respect to these results. The first is that bimodal length distributions for capelin larvae in the stomachs in 2003 (Figure 1b) might indicate that the selection of the GULF III sampler has an upper limit and that it is not efficient at catching larvae.22 mm. Certainly the observations of some larvae in cod stomachs in 2003 that were larger than any seen in the GULF III samples that year suggests some escapement from the sampling gear. This possibility is supported by larger capelin larvae being frequently observed attached to the pelagic trawl liner. However, it is not possible to correct for escapement because the pelagic trawl is not a calibrated sampling gear for capelin larvae, and hence it is not possible to provide estimates of the abundance of any larger capelin larvae present. While in principle the cod stomachs themselves might allow an estimate to be made of these larger capelin larvae, in practice such estimates would require prior knowledge of the m and s values of Equation (7). Moreover, in 2003 when the abundance of larger capelin larvae seems to have been greater, there are severe limitations on the available data on lengths of capelin larvae in cod stomachs, only 24 capelin larvae being suitable to measure length that year. A second reservation concerns the use of the observed length distribution of capelin larvae in cod stomachs as an estimate of the length distribution of the capelin larvae eaten. Although the total numbers consumed were corrected for differential digestion rates by size, the size distribution was not so corrected. This approach was adopted because it is difficult to convert the relatively small numbers of measurable larvae found in the stomachs into length distributions weighted for abundance. This was mainly a problem with the 2003 data. With the 2002 data, more capelin larvae from the stomachs were measured, and their length distribution was closer to those from the GULF III samples (Figures 1a and 1b). Note that the 2002 data are used for the key run, and therefore that the main analyses of the model are not much affected by the above-mentioned modification. However, this practical data preparation problem might, as a further refinement, be dealt with in the model by adding a known digestion function when comparing the size distribution of capelin obtained from the sampled stomachs with model predictions of the size distribution of capelin consumed.

9 Modelling growth, mortality, and predation interactions of cod juveniles and capelin larvae 1321 Figure 5. (a) Densities in numbers per day for capelin as estimated by proto-moment 0. (b) Proportional development in biomass as estimated by proto-moment 4. All 2002 model runs plus one for Capelin cod predation interactions throughout the model time period The results of the model provide several insights into the possible interaction between cod and capelin in their first year of life. First, the model indicates that predation by cod has a significant effect on the survival of capelin larvae during the period between survey 1 and survey 2 each year. Figures 2 and 5 describe the development of the two species in terms of proto-moments. Over the 76 days simulated in 2002, capelin numbers declined by about an order of magnitude (0th proto-moment), but biomass (taken as the 4th proto-moment) increased until about day 70, before declining slightly. In a model run with the growth parameters set as in the key run 2002, but without predation (results not shown), capelin density only declined to 337 m 22 between surveys. Comparing this result with the result of the key run illustrates that predation by cod accounted for 86% of the reduction in abundance seen in the key run, whereas 14% was caused by the basal rate (daily rate = Z 2 M2 = 0.01 d 21 ; Table 2) of natural mortality assumed. However, capelin abundance in the sea will also be reduced by herring predation, so it is the size of the daily predation mortality rate of cod on capelin that is the truer measure of the impact of cod. Certainly, the average predation Figure 6. (a) Daily estimates of the ratio of the rate of natural mortality on each proto-moment obtained by the bilinear form approximation to the value calculated by the reconstituted size distribution approach. Here, the approximation is calculated at the percentiles of the species size distributions assuming both to follow normal distributions, with means and standard deviation suggested by the proto-moments. (b) Daily estimates of the ratio of the rate of natural mortality on each proto-moment obtained by the bilinear form approximation to the value calculated by the reconstituted size distribution approach. Here, the bilinear form approximations are calculated at the percentiles of the species cumulative proto-moment distributions obtained via the reconstructed size distributions. mortality rates of and d 21 for the whole modelled period in 2002 and 2003 (Table 2), respectively, are substantial. However, the values should be viewed as maximum estimates because of the simplifying assumptions adopted in the model. Second, the model indicates that cod numbers decline by about half over the period, but that biomass (3rd proto-moment) increases by approximately an order of magnitude. The numbers of capelin eaten by cod initially increase as cod biomass increases, but then decline, reflecting the decline in capelin abundance. The other moments are less easily interpreted, but may be interpreted as showing growth and relative size distributions (Figure 3). Third, the fact that capelin show a bimodal size distribution and that the upper mode grows in relative importance as time progresses as a result of differential predation mortality by size, illustrates that cod predation can change the size distribution of capelin. A similar but smaller change can be seen in the estimates of capelin in cod stomachs. Furthermore, the rate of predation mortality (results not shown) acting on each proto-moment starts at a value of 0.01 d 21 on day 1, then increases sharply to d 21 by day 25, and thereafter increases more slowly to

10 1322 E. H. Hallfredsson and J. G. Pope d 21 by day 76. After day 10, the rate of predation mortality is somewhat lower on the higher proto-moments. This indicates that, after day 10, cod predation is more focused on the smaller fish of a cohort. Consequently, cod could have a role in shaping the capelin size distribution and their apparent growth rate. How are the results affected by the assumptions? All models make assumptions and simplifications, and some may modify the results substantially. This model makes a first assumption that capelin and cod remain in the same spatial association throughout the time period studied, as was seen in survey 1. However, although the model developed here is a simple singlebox one, the state vector of proto-moments that describes the population abundance, biomass, and size distribution in a single box may be readily fractionated by size and components, combined with the results of other boxes. Hence, the model might easily be adapted to a multiple-box model. Such a multiple-box model might better account for the advection and diffusion of these stocks between the nursery area inhabited by capelin larvae at the time of the June/July surveys (survey 1) and the broader scale distributions of the two species at the time of the international 0-group surveys in autumn (survey 2) (Anon., 2002b, 2003b). Second, several aspects of the predation process were excluded to keep the model tractable and understandable. An important assumption is that predation is only caused by cod. In practice, capelin larvae are also subject to predation by other predators (Huse and Toresen, 1996, 2000; Godiksen et al., 2006). Juvenile herring in particular can appear in great abundance in the area in some years (2 3 million tonnes), and they are considered the most efficient predator on capelin larvae in those years (Hamre, 1994; Gjøsæter and Bogstad, 1998; Huse and Toresen, 2000; Hallfredsson, 2006). Moreover, alternative prey, particularly copepods and krill, are important in the diet of juvenile cod (Thorisson, 1989; Helle and Pennington, 1999; Hallfredsson, 2006), and high densities of these taxa might moderate the mortality that cod generate on capelin larvae. Third, within the model, assumptions are made with respect to the nature and parameters of the growth function. These assumptions affect the estimates of predation. The run with slower growth of capelin and cod shows higher average rates of mortality over the modelled period (Table 2). This result must be attributable to slower growth prolonging the period where the prevailing prey predator length ratios encourage predation. A further problem is that while the r and m terms used in the predation equation [Equation (7)] may be estimated from the June data, the s term is chosen as a sensible but arbitrary value, because past experience suggests that this term is difficult to estimate with accuracy. This is a potential source of error, because results from model runs with high and low values of s suggest (Figure 5, Table 2) that a higher s results in greater predation mortality over the period June August. This is because, with a higher s, more of the capelin size distribution will lie within the preferred predation window of cod for a longer period. The bilinear form approximation to predation mortality rate As indicated in the Results section, some problems exist with applying the bilinear form approximation proposed by Pope (2003). A viable alternative approach is proposed here. However, Pope s (2003) approximation is potentially valuable because it allows the biological interaction between two species to be described using matrix algebra techniques alone. The young-of-the-year interaction between cod and capelin is particularly valuable for testing the bilinear form approximation, because the size distribution of fish of both species is quite simple. Therefore, effort to find a better means of constructing the approximation is appropriate. Fitting the approximation at each time-step with simple distributional assumptions improves it considerably, but whereas the bias in the rate of predation mortality is then generally,10%, it was nevertheless.15% on certain days. Moreover, the biases had different phases for the different proto-moments, and this doubtless compounds the problem when these are combined (for example, in a variance estimate). The performance of the approximation is also somewhat improved by fitting the approximation to the effect of the function (numbers eaten), rather than fitting the log-normal function itself. Interestingly, the bias of the approximation varies in a highly systematic pattern (Figure 6a), suggesting that the bias results from some lawful process and hence may well lend itself to understanding and correction. The bilinear form approximation could be made quite accurately, provided the cumulative distribution of each proto-moment was first calculated (Figure 6b). However, this result suggests that the accuracy with which the bilinear form predicts predation is sensitive to distributional assumptions (compare Figures 6a and 6b) Further work to resolve this problem is clearly indicated. In the meantime, we must caution that if the bilinear form approximation is to be used, then its adequacy should be carefully checked against another approach, such as the reconstituted size distribution used here. Additionally, one might understandably wish to view the size distribution resulting from the protomoments, so the reconstructed size distribution method developed here is a useful adjunct to the proto-moment approach. Other improvements to the model In fitting the model, we only attempted to replicate input data distributions. Therefore, only point estimates were produced, and these have no indications of confidence interval. However, the simple recursive form of the proto-moment approach, particularly if a viable form of the bilinear form approximation for the rate of predation mortality could be found, should in principle lend itself to a statistical approach to the estimation of parameters. For example, Kalman filter or Bayesian approaches should be feasible for estimating at least some of the parameters. Conclusion The reconstituted size distribution model approach produces a realistic development through time for the length distributions, the density in numbers, and the biomass of young-of-the-year capelin and cod. Clearly, this description of the population dynamics of the two stocks depends on assumptions about growth and other parameters. The rate of predation mortality for the whole period was estimated as 0.03 and 0.11 d 21 for 2002 and 2003, respectively. These should be considered as maximum values, because the model operates on densities per m 2 and does not take into account changes in spatial overlap as the species develop and disperse over wider areas during late summer. The model estimates of the rate of predation mortality allow the accuracy of the bilinear matrix-based approach to be investigated and better understood, and provides a check on its accuracy. The proto-moment method in the particular case of young-of-the-year fish interactions might be suitable as a

11 Modelling growth, mortality, and predation interactions of cod juveniles and capelin larvae 1323 submodel in multibox models because of its parsimonious structure and because the size-structured immigration and emigration of animals between boxes could be described using the additive characteristics of proto-moments. In a broader sense, the results introduce a way of checking proto-moment-based multispecies modelling based on a bilinear approximation by using the reconstituted size distribution method in tandem; an important step in the development of the use of proto-moment approach for studying multispecies systems. This is the first attempt to expand a proto-moment-based population model to a multispecies situation, and the approach looks capable of estimating predation mortality accurately in a simple, yet challenging, example. Acknowledgements We thank Torstein Pedersen for useful comments on the manuscript, the illustrators at the Norwegian College of Fishery Science for their assistance with the artwork, and Gillian M. Pope for English proof-reading. We also acknowledge Panayiota Apostolaki, Éva Plagányi, and an anonymous reviewer for their valued comments on an earlier version of the manuscript. References Anon. 2002a. Toktrapport. Institute of Marine Research, Bergen, Norway, survey report no (in Norwegian). 9 pp. Anon. 2002b. Report of the International 0-Group Fish Survey in the Barents Sea and Adjacent Waters in August September. Institute of Marine Research, Bergen, Norway, survey report no pp. toktrapporter/2002 Anon. 2003a. TOKTRAPPORT Loddelarvetokt. Institute of Marine Research, Bergen, Norway, survey report no (in Norwegian). 9 pp. toktrapporter/2003 Anon. 2003b. Report from the joint Norwegian/Russian ecosystem survey in the Barents Sea, August October Institute of Marine Research, Bergen, Norway, survey report no pp Bjørke, B., and Sundby, S Distribution and abundance indices of postlarval and 0-group cod. In The Effect of Oceanographic Conditions on Distribution and Population Dynamics of Commercial Fish Stocks in the Barents Sea. Proceedings of the Third Soviet Norwegian Symposium, Murmansk, May 1986, pp Ed. by H. Loeng. Institute of Marine Research, Bergen, Norway. Deriso, R. B Harvesting strategies and parameter estimation of an age structured model. Canadian Journal of Fisheries and Aquatic Sciences, 37: Deriso, R. B., and Parma, A. M Dynamics of age and size for a stochastic population model. Canadian Journal of Fisheries and Aquatic Sciences, 45: Fournier, D. A., and Doonan, I. J A length-based stock assessment method utilizing a generalized delay-difference model. Canadian Journal of Fisheries and Aquatic Sciences, 44: Gjøsæter, H The population biology and exploitation of capelin (Mallotus villosus) in the Barents Sea. Sarsia, 83: Gjøsæter, H., and Bogstad, B Effects of the presence of herring (Clupea harengus) on the stock recruitment relationship of Barents Sea capelin (Mallotus villosus). Fisheries Research, 38: Godiksen, J. A., Hallfredsson, E. H., and Pedersen, T Effects of alternative prey on predation intensity from herring Clupea harengus and sandeel Ammodytes marinus on capelin Mallotus villosus larvae in the Barents Sea. Journal of Fish Biology, 69: Gundersen, A. C Distribution of capelin larvae during the first year of life in the Barents Sea, in the period In Proceedings of the Sixth IMR PINRO symposium, Bergen, June 1994, pp Ed. by A. Hylen. Institute of Marine Research, Bergen, Norway. Hallfredsson, E. H Fish predation on capelin larvae Mallotus villosus in the Barents Sea, in relation to recruitment. Doctor scientiarum thesis, The Norwegian College of Fisheries Science, University of Tromsø, Norway. 213 pp. Hamre, J Biodiversity and exploitation of the main fish stocks in the Norwegian Barents Sea ecosystem. Biodiversity and Conservation, 3: Helle, K., and Pennington, M The relation of the spatial distribution of early juvenile cod (Gadus morhua L.) in the Barents Sea to zooplankton density and water flux during the period ICES Journal of Marine Science, 56: Hilborn, R., and Walters, C. J Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty. Chapman and Hall, New York. 569 pp. Huse, G., and Toresen, R A comparative study of the feeding habits of herring (Clupea harengus, Clupeidae, L) and capelin (Mallotus villosus, Osmeridae, Müller) in the Barents Sea. Sarsia, 81: Huse, G., and Toresen, R Juvenile herring prey on Barents Sea capelin larvae. Sarsia, 85: ICES Report of the ad hoc Multispecies Assessment Working Group, Copenhagen, June ICES Document CM 1984/Assess: 20. ICES Report of the ad hoc Multispecies Assessment Working Group, Copenhagen, November ICES Document CM 1986/Assess: 9. Nilssen, E. M., Pedersen, T., Hopkins, C. C. E., Thyholdt, K., and Pope, J. G Recruitment variability and growth of North East Arctic cod: influence of physical environment, demography and predator prey energetics. ICES Marine Science Symposia, 198: Olsen, S Some results of the Norwegian capelin investigations Rapports et Procès-Verbaux des Réunions du Conseil International pour l Exploration de la Mer, 158: Ottersen, G., and Loeng, H Covariability in early growth and year-class strength of Barents Sea cod, haddock, and herring: the environmental link. ICES Journal of Marine Science, 57: Pope, J Golden ages or magic moments? Natural Resource Modeling, 16: Scharf, F. S., Juanes, F., and Rountree, R. A Predator size prey size relationships of marine fish predators: interspecific variation and effects of ontogeny and body size on trophic-niche breadth. Marine Ecology Progress Series, 208: Thorisson, K The food of larvae and pelagic juveniles of cod (Gadus morhua L.) in the coastal waters of Iceland. Rapports et Procès-Verbaux des Réunions du Conseil International pour l Exploration de la Mer, 191: doi: /icesjms/fsm133