Journal of Marine Systems 81 (2010) 184–195

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Journal of Marine Systems j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j m a r s y s

A nutrient to fish model for the example of the Baltic Sea Wolfgang Fennel Leibniz Institut für Ostseeforschung, Warnemünde an der Universität Restock, D-18119 Warnemünde, Germany

a r t i c l e

i n f o

Article history: Received 8 October 2008 Received in revised form 25 August 2009 Accepted 23 September 2009 Available online 28 December 2009 Keywords: Fish models Reproduction Fishing Mortality Eutrophication

a b s t r a c t The paper describes an application of a NPZDF-model, which couples interactively a biogeochemical and a fish production model. The Baltic Sea is chosen as an example system, where the fish community is dominated by two prey species (sprat and herring) and one predator (cod). The linkage of the model components is established through feeding of prey fish on zooplankton and recycling of fish biomass to nutrients and detritus. The fish dynamics is driven by size dependent predator–prey interactions, reproduction and mortality. A challenge is the quantitative description of the multitude of processes involved. The model formulation is constrained by strict mass conservation. Examples of experimental runs over 40 years are discussed: A baseline simulation gives an approximate hindcast scenario of stock variations and catches over the time period of the years from 1963 to 2003 with increasing and declining cod catches. As known from observations, the stock size of sprat shows a reverse trend, while the herring stock appears to be relatively stable. The modeled results display several observed features, e.g. stock sizes and magnitude of changes, but the response times and phases of the variations are not well reproduced. Moreover, scenarios are simulated to indicate how moderate adjustments of fishing rates of cod over different time periods would change the stock and catches. © 2009 Elsevier B.V. All rights reserved.

1. Introduction To understand and manage marine ecosystems in the frame of an integrated ecosystem based approach requires consideration of the full marine food web. A first modeling contribution towards the goal of covering the full food web in the North Sea was undertaken by Andersen and Ursin (1977). However this approach was probably too ambitious for that time. Work on substantially simplified multispecies stock production models for the Baltic, was done by Horbowy (1989, 2005). An attempt to quantify the amount of phosphorous bound in fish and the removal of nutrients through fisheries in Baltic Sea was provided by Hjerne and Hansson (2002). The interannual variation of the catches of cod, sprat and herring are well documented from the year 1963 to now (Fig. 1). Cod and sprat variations appear to be closely related, high cod catches correlate with low sprat catches and visa verse. The catches of herring are less variable, (e.g. Köster et al., 2003). The reason for the interannual variations could be bottom up (e.g. there could be poor reproduction of cod due to hyperoxic or anoxic condition in the halocline where the cod eggs stay), or top down (e.g. due to high fishery pressure). To understand and quantify the underlying dynamics, models are needed that deal with large parts of the food web.

E-mail address: [email protected]. 0924-7963/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmarsys.2009.12.007

Recently a nutrient to fish model was proposed, which links interactively a NPZD-model and a fish model for the Baltic Sea, where the fish biomass is represented by sprat, herring and cod, (Fennel, 2008). The model includes the complete life cycles of the fish and takes the interactions of predator and prey explicitly into account. The zooplanktivors sprat and herring are basically the prey of cod. The resulting product is a NPZDF-model that we call the ‘Warnemünde Food web Model’ (WFM). A consistent integration of rather different time scales in a NPZDF-model is a theoretical challenge. Primary production, which starts the flux of matter through the food-web, is controlled by the seasonal variations of the physical forcing; zooplankton operates on the yearly time scale, while fish involves life cycles of 10 to 20 years. The construction and analysis of such full food web models is also of great importance for theoretical understanding of possibilities and limitations of truncated models. Principally, both fish models with a truncated lower food web, or NPZD-models with a truncated upper food web, require some parameterizations to mimic unresolved processes in terms of resolved ones. How appropriate such parameterizations are, can only be assessed by experiments with full models, (Fennel, 2009). This paper aims at a hindcast simulation of the observed stock dynamics and catches in the Baltic Sea for the period from 1963 to 2003. With relatively simple assumptions of the external forcing functions, we make an attempt to reproduce the main features of the stock dynamics. The model is based on the paper Fennel (2008), which describes the basic ideas of the model development and demonstrates the general consistence and plausibility of the

W. Fennel / Journal of Marine Systems 81 (2010) 184–195

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Fig. 1. Catches of cod, sprat and herring in the Baltic Sea from 1963 to 2003. (Data source: ICES).

approach. Moreover, two model integration for a period of 40 years were presented; one run without fishing mortality and a second one with a constant fishing rate. The model was forced by a recurring annual cycle of water temperature and length of the day. External inputs of nutrients were ignored. Without fishing mortality the model system reached a steady state, while for nonzero fishing mortality the corresponding mass export leads to a slow decrease of all state variables from nutrients to fish. In the present paper we consider external forcing of the model system by varying nutrient loads, changing reproduction rates of cod, as a proxy variable for oxygen conditions in the halocline of the central Baltic Sea, and changing fishing mortality of cod. Moreover, the bioenergetics of the fish growth has been refined by adopting seasonally changing energy content of the fish as proposed in Megrey et al. (2007). We will show also examples how a hypothetical fishery management action in the model could improve sustainable fishing. Previous attempts to quantify the development of future catches of the main species of the Baltic Sea in response to drastic management actions, (Harvey et al., 2003; Hansson et al., 2007), were based on Ecopath with Ecosim software tools. The approach discussed in this paper presents an independent way to model the system. The paper is organized as follows: After the introduction we provide in Section 2 a brief outline of the model and describe example simulations in Section 3. The paper concludes with discussions and conclusions in Section 4. 2. Model description The full NPZDF-model (WFM) was already described elsewhere, (Fennel, 2008). Therefore we give here only a brief description of the main structure and the basic features. We indicate some refinements of the description of bioenergetic process and of predator prey interaction terms. A compilation of the process rates and choices of the involved parameters is given in the Appendix A. The current version is a box model and it will take some time to arrive at a spatially resolved model by embedding the fish component in a threedimensional biogeochemical and circulation model. To justify such a simple approach we note that the primary production is limited to an upper layer of thickness of about 20 m. However, the phytoplankton,

P, uses up the nutrients, N, in particular nitrogen, in the water column above the halocline (the upper 50 to 60 m). This is well observed in the Baltic because the spring bloom is usually associated with convection events which entrain nutrients into the upper layer, e.g. Nausch et al. (2008). Detritus, D, refers to sinking dead organic material, which will be recycled to nutrients through mineralization or transferred to a detritus pool in the sediment, Dsed. We take the accumulation of sinking material in the sediments into account, but ignore the biogeochemistry near the sea bed. Zooplankton, Z, can vertically migrate to find food in the water column and fish can move and search for food. Thus, in a first approximation we may ignore the spatial distribution of fish. For the calculation of the state variables of the fish model we consider a box of a volume of 1 km3. To obtain the order of magnitude of key variables, such as stock biomass or catches, for the whole system, we multiply the corresponding state variables by the volume between sea surface and the depth of 60 m of the Baltic Proper, which amounts to Vprod ∼ 8 · 103 km3, (see (Seifert and Kayser, 1995) and www.io-warnemuende.de/iowtopo). The primary production at the beginning of the food chain is limited by nutrients from this part of the water body. 2.1. The NPZD-model A simple NPZD-model component, which is able to reproduce the typical annual cycling of nutrients through the bulk-phytoplankton, bulk-zooplankton, and bulk-detritus can be formulated in terms of the following set of equations, dN import = −uðNÞP + lPN P + lDN D + lZN Z + N + LFN ; dt

ð1Þ

dP = uðNÞP−lPN P−gðPÞZ−lPD P; dt

ð2Þ

dZ = gðPÞZ−lZN Z−lZD Z−GF ; dt

ð3Þ

dD = lZD Z + lPD P−lDN D−lDDsed D + LFD ; dt

ð4Þ

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W. Fennel / Journal of Marine Systems 81 (2010) 184–195

dDsed = lDDsed D: dt

ð5Þ

Here the rates lXY (where X and Y stand for N, P, Z, D, and Dsed), describe the transfer from the state variable X to the state variable Y. Nutrient uptake and grazing of zooplankton are described by limiting function u(N) and g(P). The term Nimport describes the external input of nutrients; the accumulated material at the sea bed is described by Dsed. The connection with the fish model is indicated by the fluxes of zooplankton biomass to fish, GF in Eq. (3), and the fluxes of material from fish to nutrients and detritus, LFN and LFD, in Eqs. (1) and (4), respectively. The term GF represents the sum of the expressions that describe the consumption of zooplankton by all size-classes of prey fish and the smallest mass class of cod. The terms LFN and LFD comprise all losses of fish biomass due to respiration, excretion, and mortality, except predation and fishing mortality. We use a strict Redfield stoichiometry and a fixed conversion ratio between biomass and carbon. 2.2. The fish model The fish model comprises planktivorous sprat and herring, and piscivorous cod. It describes completely closed life cycles and includes predator–prey interaction. To characterize the life cycle of fish, we define certain mass levels xXi− 1 and xXi, where x stands for sprat, (x = S, i = 1, to 5), herring (x = H, i = 1 to 6) or cod (x = C, i = 1 to 7). In the model, the mass of sprat ranges from 1 g to 35 g, of herring from 2 g to 250 g, and of cod from 2 g to 10 kg. The choice of mass intervals was motivated by the need for consistent size relations between prey and predator. During the development, the average fish, characterized by its mass (biomass over numbers of fish), propagates through the mass intervals. The equations of the fish model describe the evolution of the state variables biomass and numbers of fish in every mass class and involve several rates corresponding to different processes. To illustrate the structure of the model, we list the dynamic equations for cod and herring, Eqs. (6) to (9). There are five types of driving processes: grazing on zooplankton, feeding on prey fish, mortality, losses of material through metabolism, and reproduction. Grazing on zooplankton drives the growth of sprat, herring and of the two smallest mass classes of cod. The grazing terms, e.g. gCod or gSprat , are food limited through a Ivlev1 1 type function, g1xi ∼ (1− exp(IzZ)). The predator–prey interaction takes the size-classes of prey and predator into account. For example, the interaction of cod and herring is described by, max Cod

Pk ðHerÞ = gCk Bk

H Her Her ∑k−1 i = 1 pk;i Bi GðBi Þ H Her ∑k−1 i = 1 pk;i Bi

QðTÞ;

is the maximum of food that can consumed by cod of mass where gCmax k class ‘k’, and G(BiHer) ∼ (1− exp(IC BiHer)) is an Ivlev-function for the cod that control the food limitation for small prey concentration. Preferences of food are expressed through the term pH k,i, which weight the relative importance of the interaction of cod mass class ‘k’ and herring mass class ‘i’. This parameter ensures that the largest predator does not eat the smallest prey as long a larger prey is available. The term Q(t) was introduced to mimic the seasonal energy density of the fish in a similar manner as in Megrey et al. (2007). The bioenergetic function, which is given in the Appendix A, was motivated by the experimental findings of Arrhenius (1998a,b) and Arrhenius and Hansen (1996). From the viewpoint of the prey, the prey–predator interaction is described by, Πi ðHerÞ =

gCmax pHk;i BCod k Her Her k GðBi ÞBi ∑ i−1 H Her k = i + 1 ∑k = 1 pk;i Bk 7

μC7, and starvation mortality, e.g. μ Cstarv . Loss through respiration and 7 excretion is covered by rates LCiN and LCiD, which describe the transfer of fish biomass to respirational and excretory products. Part of the losses is in proportion to the consumption of foods, and other parts refer to basic metabolism, which must be maintained even if no food is available. Reproduction is approximated by an off-spring approach, where the mass loss of matured fish due to egg production is channeled as off-spring into the smallest mass class. The off-spring rates, for example osC7, are nonzero during the reproduction phase and are constrained by the mass loss of 20% the biomass of the matured fish during the reproductive season. Thus, less than 20% of the adult biomass can be converted into new recruits. This is consistent with the findings of Köster et al. (2003), (Fig. 3 in that paper), where the number of recruits and spawning biomass were estimated by a Multi Species Virtual Population Analysis, (MSVPA). A compilation of process descriptions and parameter choices is listed in the Appendix A. The model equations consist of differential equations for biomass concentration and abundance of each mass class. For cod biomass the equations are, d Cod Cod Cod Cod Cod Cod = osC6 B6 + osC7 B7 + ðg1 −LC1 N −LC1 D −μC1 ÞB1 −τC1 B1 ; B dt 1 ::::::: :::::: d Cod Cod Cod Cod = τCi−1 Bi−1 −ðLCi N + LCi D ÞBi −τCi Bi + Pi ; B dt i ::::::: ::::::: d Cod Cod starv Cod = τC6 B6 −ðLC7 N + LC7 D + osC7 + FC7 + μC7 + μC7 ÞB7 + P7 : B dt 7 ð6Þ

The rates τxi promote the variables from one stage to the next one if a mean individual (biomass over abundance) mass reaches the upper limit of the corresponding mass interval. For the abundance we have, d Cod 1 BCod Cod Cod Cod N1 = ðosC6 B6 + osC7 B7 Þ−μC1 N1 −τC1 1 ; dt m0 CX1 :::::::::: ::::::::: d Cod BCod BCod Cod Ni = τCi−1 i−1 −μ TC Ni −τCi i ; i dt CXi−1 CXi :::::::::: :::::::::::: d Cod BCod starv Cod N7 = τC6 6 −ðμ C7 + μ C7 + FC7 ÞN7 : dt CX6 and BCod The two largest mass classes, BCod 6 7 , which correspond to the age of three years and older, are fished. For the prey species the equations for biomass and abundance have a similar structure. We note the example of herring, d Her Her Her Her Her B = osH5 B5 + osH6 B6 + ðg1 −LH1 N −LH1 D −μH1 ÞB1 dt 1 −τH1 BHer 1 −Π1 ðHerÞ; ::::::::: ::::: d Her Her Her Her B = τHi−1 Bi−1 + ðgi −LHi N −LHi D ÞBi dt i Her

Q ðtÞ:

This expression represents the predation mortality of prey. Further mortality terms corresponds to fishing, e.g FC7, natural mortality, e.g

ð7Þ

−τHi Bi ::::::::: :::::::

−Πi ðHerÞ;

d Her Her Her Her B = τH5 B5 + ðg6 −LH6 N −LH6 D ÞB6 dt 6 −ðosH6 + FH6 + μH6 + μHstarv ÞBHer 6 −Π6 ðHerÞ; 6

ð8Þ

W. Fennel / Journal of Marine Systems 81 (2010) 184–195

and, d Her 1 BHer Π ðHerÞ Her Her Her N = ðosH5 B5 + osH6 B6 Þ−μH1 N1 −τH1 1 − 1 Her ; dt 1 m0 HX1 m1 ::::::: :::::::::::: Her

Her

d Her B B Π ðHerÞ Her N = τHi−1 i−1 −μ THi Ni −τHi i − i Her ; dt i HXi−1 HXi mi ::::::: :::::::::: d Her BHer Π ðHerÞ starv Her N6 = τH5 5 −ðFH6 + μ H6 + μ H6 ÞN6 − 6 Her : dt HX5 m6 ð9Þ Since the removal of prey-biomass through predation mortality also implies a reduction of the number of prey, the prey–predator interaction terms, Πi(Her), appear both in Eqs. (8) and (9). 3. Example simulations After the brief model description we perform simulations over 40 years that correspond to the time period from the years 1963 to 2003. The simulations start with a set of initial values for the state variables and are forced by time varying reproduction conditions, external nutrient loads and fishing mortality. 3.1. Design of model simulations At the outset, we study a hindcast or reference scenario, S0, with a realistic, relatively high fishing mortality of cod. In a series of runs we consider then different cases of changing fishing mortality of cod, while the other forcing functions are the same as in the reference scenario. The riverine nitrogen loads into the Baltic Proper increased from about 100 ktons/y in 1960 to a twofold level of about 200 ktons/y at the end of the 1980's. In the 1990's the level is found in the middle between the start value and the maximum, (about 150 ktons/y), (Humborg, private communication), (Neumann and Schernewski, 2008). The maximum nitrogen flux in the Baltic Proper of about 200 ktons/y is equivalent to 0.03 mmol C m3 d. The variation of the flux used in this model is shown in the middle panel of Fig. 2. Further, we vary the reproduction rate of cod to reflect in a crude way the effects of salt water inflows on the reproduction conditions for cod. We assume in particular an improvement of the reproduction of cod after the major salt water inflow in 1973, i.e. ten years after the beginning of our simulation period. This is implemented by a multiplier acting on the off-spring rates, OSC6 and OSC7, as sketched in the top level of Fig. 2, (top). The fishing on cod is assumed to increase during the first decade and being constant thereafter. The initial increase is suggested by the general economic development after the war and the increase in catches, which indicates higher fishing efforts. After the initial phase, the level of fishing mortality is relatively high and corresponds to the order of magnitude reported by Köster et al. (2003), Fig. 2 bottom ― panel. The shown average annual fishing mortality, FC , was estimated from ――― Cod

――― Cod

FC N6 + FC7 N7 ― FC = 6 ――― ――― N6Cod + N7Cod ;

Cod ≈ 2. In Köster et al. (2003) it was shown that with a ratio of NCod 6 /N7 a short interruption of high fishing mortality over a period of two years was enforced in the Baltic in 1992 and 1993. We will have a brief look at the stock abundance of cod for the corresponding scenario, S2 (the index refers to the time period of reduced fishing of cod in years). Moreover, we will study the response of the model to longer periods

187

of reduced fishing mortality of cod for five and seven years, scenarios S5 and S7. The scenarios are listed in Table 1. 3.2. Hindcast scenario ‘S0’ A selection of the results of the first run is shown in Figs. 3–7. The development of the total numbers of model sprat, herring and cod is shown in the upper panel of Fig. 3; time series of the corresponding total biomass of sprat, herring, cod, and zooplankton are depicted in the lower panel. Note that the quantities are numbers or biomass per 1 km3. The figures show both annual and interannual variations. The total numbers of sprat, herring, and cod show a strong annual cycle, which is determined by reproduction. The annual variation of the prey-biomass is mainly driven by the seasonality of feeding. The seasonality of the mass fluxes between the lower and upper trophic levels can also be seen in Fig. 7, which shows the flow of material through the interface between the NPZD- and fish model. High fluxes towards the fish are found in the middle of the year. The zooplankton biomass shows the strongest yearly oscillations, (Fig. 3, bottom). The abundance of cod increases in the first few years, remaining high for four years, and declines steadily after then. The stock size of sprat shows a reverse trend. After a decreasing abundance in the first 15 years follows a steady increase. The modeled herring stock is relatively stable over the whole simulation period. Only at the end of the 40 year period a slight decrease in abundance can be detected. This decrease is more clearly visible in the biomass, (Fig. 3, bottom). The general structure of the interannual variations of stock abundance is in a good agreement with estimations using Multi Species Virtual Population Analysis (MSVPA), which are discussed in Köster et al. (2003). In their Fig. 2, the cod abundance varies between 5 · 108 and 3 · 109, the sprat abundance varies between 5 · 1010 and 4 · 1011, and herring stays close to 5 · 1010. Multiplication of the modeled stock abundances by the productive volume of the Baltic Proper, 8 · 103 km3, we get quite similar intervals, ranging from 8 · 107 to 1.3 · 109 for cod (5 · 108 to 3 · 109 MSVPA), from 8 · 1010 to 3 · 1011 for sprat (5 · 1010 to 4 · 1011 MSVPA), and about 4 · 1010 for herring (5 · 1010 MSVPA). However, in the model the biomass of both sprat and herring is increasing from the year 20, implying an increasing mass of the average individuals, while a decline of weight-at-age was observed. Only in the last 4 years of the simulation the herring biomass decreases slightly stronger than its abundance, which amounts to a decrease of the weight-at-age. The slight decline of herring biomass is related to the decrease of zooplankton in the last ten years of the simulation. During this period, the sprat is released from the strong predation by cod and grows faster than herring. In the model, the sprat with its shorter life cycle wins the competition for zooplankton and continues to grow faster than herring. The trends of the fish stocks are obviously controlled by fishing mortality. The decline of the stock abundance of cod is shown in Fig. 4, where the succession of the different mass classes is also indicated. The simulated total catches of sprat, herring, and cod of the Baltic Proper for 40 years are shown in Fig. 5. Obviously, some features of the variations of the modeled catches correspond qualitatively well to the reported data, Fig. 1. We find realistic orders of magnitude and an inverse variation of the cod and sprat catches, while the herring catches vary less strongly, as known from observations. On the other hand, the phases of the modeled catch variations differ from the observed behavior. For example, in the model simulation the phase of high cod and low sprat catches occurs earlier and is of shorter duration. The eutrophication scenario is reproduced by the model as known from observations, e.g. Nausch et al. (2008). The modeled winter values of the nutrients respond in a realistic manner to increasing loads during the first 30 years, see Fig. 6. The two-way interaction of the upper and lower part of the food web implies mass fluxes between the upper and lower food web. This

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W. Fennel / Journal of Marine Systems 81 (2010) 184–195

Fig. 2. Multiplier to mimic time variations of cod reproduction (top), history of riverine nutrient loads expressed as the increment of carbon concentration, (middle), and annual average fishing rates of cod (bottom) for the hindcast scenario ‘S0’.

is shown in Fig. 7 in terms of GF, which describes the flux of matter from zooplankton to fish through grazing of fish on zooplankton, and fluxes from fish to detritus and nutrients, LFD and LFN. The mass flux to fish through feeding on zooplankton represents the strongest annual signal, while the contribution of respiration and excretion are smaller. This imbalance is enhanced through the export of fish biomass by fishing, because the fish removed from the system cannot contribute to the recycling flows. The loss of matter is approximately compensated be the external nutrient input. 3.3. Time varying fishing mortality of cod In the following we consider the effects of varying fishing mortality of cod as listed in Table 1 and shown in Fig. 8. During the period from 1992 to 1993, the fishing mortality of cod in the Baltic Sea was substantially reduced due to rigid enforcement of management measures. Afterwards, the fishing mortality increased again to the former high level, (ICES, 1998). For this scenario, (top of Fig. 8), we show only the stock abundance of cod (Fig. 9), which demonstrates that the model response to a reduction of fishing mortality for a period of two years has only a minor effect. This suggests that a longer reduction period is required for a significant recovery of the stock. Hence, we simulate two cases of a reduced fishing mortality; one over the period of five years (scenario S5) and one over seven years (scenario S7), as indicated in the middle and bottom panel of Fig. 8.

Table 1 List of the model scenarios with different fishing of cod. Scenario

Period of reduced fishing after year 20

Remarks

S0 S2

– 2 years

S5 S7

5 years 7 years

Reference/hindcast run (Fig. 2, bottom) Enforced reduction, (Fig. 8, top) e.g. Köster et al. (2003), Hypothetical fishing reduction (Fig. 8, middle) Hypothetical fishing reduction (Fig. 8, bottom)

Next we analyze the response of the stock abundance in the model to lower fishing mortality of cod for five years (scenario ‘S5’). A motivation of the ‘management action in the model’ could be the need to respond to a low reproduction after the year 18, see Fig. 2 (top), with a moderation of the fishing pressure. The effect on the stock abundance of cod is shown in Fig. 10. The stock recovers fairly rapidly and remains on a reasonable high level for the duration of the period of a reduced fishing rate. Once the high fishing rate is restored, however, the stock declines again. After about 5 to 6 years the stock abundance assumes the same level as at the beginning of the reduced fishing period. The modeled catches of cod show an significant impact of the reduction over a time scale of about 10 years, Fig. 11. In the last 10 years of the simulation, the total catches are slightly smaller than in scenario ‘S0’. The response of the fish model in terms of the total number and total biomass of the species is drawn in Fig. 12. Again there is a clear inverse relationship of the stock abundance of sprat and cod. The sprat stock increases in the last 8 years when the cod stock declines. The herring stock shows only little variations. While the sprat abundance exceeds the numbers of herring, the biomass of herring is larger than that of sprat. The decline of cod biomass is stopped for about a decade; the sprat biomass is smaller than in scenario ‘S0’, because of the higher predation. In our third example, scenario ‘S7’, we prolong the fishery management measure to seven years, i.e., we reduce the fishing mortality of cod for the years 20 to 27, as sketched in the bottom panel of Fig. 8. The response of the fish model in terms of the total number and total biomass of sprat herring and cod is similar as the scenario S5, (Fig. 12), with an inverse behavior of cod and sprat and only small variations in the herring stock. When the cod stock recovers the sprat biomass decreases due to enhanced predation. The impacts of the management action on the stock abundance and catches of cod are shown in Figs. 13 and 14. The stock recovers fairly strong and reaches its maximum even one year after the end of the reduction period. About 10 years after restoring the high fishing rate, the stock declines to the level of the year 20, just before the reduction. The results of the scenario S7 demonstrate that the stock increase during the period of reduced fishing mortality is somewhat faster than the decline of the stock abundance after restoring the high fishing rate. Generally, the

W. Fennel / Journal of Marine Systems 81 (2010) 184–195

189

Fig. 3. Upper panel: The variation of total abundance of cod, sprat and herring per km3. Lower panel: The variations of the total biomass of cod, sprat, herring, and zooplankton per km3, for the hindcast scenario ‘S0’.

stock recovery is fairly stable for a period of about 15 years and the catches of cod are significantly higher and more stable than in the other scenarios. Sprat stock and catches are smaller than in the previously considered scenarios.

these relationships with our model data. We define the spawning stock biomass by the annual mean of total mass of all matured fish, i.e., mass class 4 and 5 for sprat, mass class 5 and 6 for herring, and mass classes 6 and 7 for cod. For example, the spawning stock biomass of herring for one year follows from,

3.4. Stock recruitment relationships

d

Her

In fishery sciences the relation between spawning stock biomass and recruitment played a central role. Since our model estimates dynamically the recruits from the spawning biomass, we may analyze

Fig. 4. Development of abundances of the seven mass classes of cod (per 1 km3), for the hindcast scenario ‘S0’. Note that the corresponding variations of the total number of cod (per 1 km3) is shown in Fig. 3.

bB

NSSB =

365 1 Her Her ∫ dtðB5 ðtÞ + B6 ðtÞÞ; ðd365 −d0 Þ d 0

where d0 is the first and d365 the last day of a year. The number of recruits are determined from the mean numbers of individuals in the

Fig. 5. Development of the total catches of cod, herring and sprat in the Baltic Proper during the 40 year run of the hindcast scenario ‘S0’.

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Fig. 6. The time development of the NPZD-part of the model; top: nutrients and detritus; bottom: phytoplankton and zooplankton, for the hindcast scenario ‘S0’.

first mass class averaged over the reproduction interval, (d60 to d120 for sprat and herring and d60 to d150 for cod, see Appendix A). For example, for herring we calculate the average number of recruits from, d

Her

bN1

N =

120 1 Her ∫ dt N1 ðtÞ: ðd120 −d60 Þ d 60

The transfer of biomass from the matured mass classes to the recruits is controlled by the reproduction rates, e.g., OSC7 and OSH6, in

the last equations in the sets (6) and (8). Dividing these mass fluxes by the initial masses, e.g. mC0 and mH 0 in the first equations of the sets (7) and (9), gives the corresponding numbers of individuals. We use the findings from two runs, S0 and S7, see Table 1, to plot the mean number of recruits per year versus the spawning biomass per year for cod, sprat and herring, Fig. 15. The black crosses (+ for S0) and green stars (⋆ for S7) coincide for the first 20 years, but show different paths in response to the changed fishing mortality of cod. The results show that, the stock-recruitment relations of all three species are significantly affected by the change of the fishing mortality

Fig. 7. The time development of material fluxes between the lower and upper part of the food web through grazing of fish on zooplankton, GF (solid), and fluxes from fish to detritus and nutrients, LFD (dashed) and LFN (dash–dots), for the hindcast scenario ‘S0’.

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Fig. 8. Management measures in terms of reduced fishing mortality of cod for the years 20 to 22, scenario ‘S2’ (top), for the years 20 to 25 scenario ‘S5’ (middle), and for the years 20 to 27, scenario ‘S7’ (bottom).

For a food web model ranging from nutrients to predatory fish, a consistent description of the interactions up and down the food web is important. We applied a NPZDF-model with fish populations structured into a set of mass classes. Our model is based on a two-way interaction,

with a consistent description of the mass fluxes between the state variables of the lower and upper food web. Our approach uses a formulation of the dynamics of biomass (B) and numbers of animals (N), which amounts to equations of the type dBdt =geffB −μB and dN/dt = −μN, with B =mN. Here geff stands for an effective growth rate, μ is a mortality rate, and m is the individual mass. For several overlapping cohorts, the mass, m, represents the average individual mass. The cohort approach considers a growth equation for the individual fish, dm/dt =geffm, combined with the development of the numbers of fish dN/dt = −μN, e.g. Megrey et al. (2007). Obviously, for a single cohort our model is equivalent to the cohort approach because of the relation dB/dt =mdN/dt +Ndm/dt. Populations can be structured in age, length or mass classes. Since all metabolistic processes and interactions

Fig. 9. Development of abundances of the seven mass classes of cod (per 1 km3) for the scenario ‘S2’. The two year period is indicated by the vertical lines.

Fig. 10. Development of abundances of the seven mass classes of cod (per 1 km3) for the scenario ‘S5’. The five-year period is indicated by the vertical lines.

of cod. We find an approximately linear relation between spawning stock biomass and recruitment, but with some variations. There is no significant indication of density dependence. In several cases the stock-recruitment relationships are ambiguous, since for certain biomass values we find distinct recruitment levels. This implies that the history of stock developments may also play a role. 4. Discussion and conclusions

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Fig. 11. As Fig. 5, but for the scenario ‘S5’.

between state variables can be described by mass fluxes between the variables, an approach with the population structured in mass classes appears to be convenient. Structuring populations in age or length classes implies that changing age or length need to be converted into equivalent variations of mass. The cohort approach is in particular useful and convenient for tracking individuals or groups of individuals (super-individuals) in space and time. But, if we consider longer time periods, e.g. decades, we have to deal with many overlapping cohorts of both prey and predators, which interact among each other and with the lower food web. For those cases it might be complicated and tedious to keep track of all mass fluxes between all the state variables. Contrary to the cohort approach, we use equations for biomass and numbers of every mass class and employ the development of a mean individual mass (biomass/numbers) as controlling quantity. This

Fig. 13. Development of abundances of the seven mass classes of cod (per 1 km3) for the scenario ‘S7’. The seven-year period is indicated by the vertical lines.

approach uses basically similar bioenergetic growth equations as in the Wisconsin model described in Megrey et al. (2007). However, the growth rates are applied onto the biomass within a series of consecutive mass intervals. Since at every time step the corresponding numbers of fish in the mass intervals are known, this approach is equivalent to applying the growth equation to a single average individual. The model comprises completely closed life cycles of sprat, herring and cod and their interactions by prey–predator and predator–prey interactions. The approach enables us to include several different overlapping cohorts. This provides the opportunity to include the spatial dimension by embedding the model into a three-dimensional circulation model to simulate the distribution of fish stocks in response to environmental variations.

Fig. 12. As Fig. 3, but for the scenario ‘S5’.

W. Fennel / Journal of Marine Systems 81 (2010) 184–195

Fig. 14. As Fig. 5, but for the scenario ‘S7’.

Many state variables and process descriptions were introduced in the fish model and several variables and parameters needed in the process equations are poorly known. But, rather than just adding state variables, we have involved a structure in terms of biomass and abundance resolved into mass classes. The choice of biomass and abundance as dynamical state variables implies that the model variables correspond directly to observable quantities. In the model equations, the growth processes are constrained by observational findings. Reported growth data fitted to Bertalanffy growth curves, e.g. Horbowy (2005), can be mapped onto the growth parameters of a Wisconsin type model for every mass interval. The reproduction rates used in the model are consistent with the relationships between numbers of new recruits and spawning biomass derived by a MSVPA, Köster et al. (2003). Nevertheless, consolidation of process descriptions will remain to be a central issue.

193

We considered simulations of the period 1963 to 2003. The model was forced by increasing nutrient loads, an initially growing and then high fishing mortality of cod, and a crude approximation of the cod reproduction in response to water renewal by major salt water inflows. The model could qualitatively reproduce the interannual variations of the observed catches, where catches of cod and sprat show an inverse behavior. At the end of the simulation period, the cod stock declined while the sprat stock increased. The herring stock shows only little variations. At the end of the 40-year simulation the herring biomass declines faster than the abundance, indicating a lower weight-at-age. Thus, the general time history and order of magnitudes of stocks and catches are well reproduced, but the phases and the response times differ from the observations. It should be noted, that our model starts with initial conditions and is then integrated over 40 years with certain forcing functions, while the MSVPA rely on observational data, which carry implicitly a substantial volume of information of the system. After the baseline scenario S0, we studied three example scenarios with a reduced fishing mortality of cod for 2, 5 and 7 years, (scenarios S2, S5, and S7). The scenario S2, i.e. two years of reduced fishing, is a realistic case, which was applied in the Baltic in 1992 to 1993. The model showed that a reduced fishing over 2 years has only a minor and short term effect. More significant impacts were found for the five-year and seven-year scenarios, which showed a lasting improvement of the cod stock and catches in the model. Thus, the model indicates that management measures relevant for the cod in the Baltic require an enforcement over a period of several years. Although this result can qualitatively be expected for a long living fish like cod, it might be worth to highlight that it can be established in a quantitative manner by the model. Since the model was forced by external factors reducing the reproduction conditions of cod, the results show clearly that high fishing mortality is the main reason for the declining cod stock in the model. This is evident from the scenarios with longer periods of reduced fishing mortality. Despite the model results appear to be promising, it should be noted that the model needs further elaboration. The observed increase

Fig. 15. Stock-recruitment relationships of herring (top), sprat (middle), and cod (bottom) for the scenario ‘S0'and ‘S7’. The black lines connect the data points (+) for ‘S’0, while the green dotted lines connect the data point (⋆) for S7. The first twenty data points coincide because the change starts after 20 years.

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of nutrient concentrations in wintertime is reproduced, but the biogeochemistry is simplified in the current version of the coupled model. The very simple NPZD-component can be replaced by a more complex biogeochemical model, which includes interactions with sediments and benthic–pelagic coupling, (Neumann et al., 2002; Fennel and Neumann, 2003). In the current version, the model is basically restricted to a box. However, expanding it towards the spatial arrangement of grid cells of a 3d-model system is a straightforward procedure. The box can already be considered as a grid cell of a 3d-model with coarse resolution. Such a model could resolve the migration of fish during their life cycles. Regarding the choice of parameters that describe the growth and interaction processes, some readjustment might be appropriate. However, even in highly resolved 3d-model systems many important local processes, such as foraging, encounter rates, feeding success, and avoidance, will remain unresolved sub-grid processes and require parameterizations. Acknowledgement The author thanks Eero Aero for inspiring discussions and Kenneth Rose for helpful comments. Two anonymous referees provided critical, but useful comments. Appendix A We summarize the dynamic process descriptions and choices of parameters used in the model. Most of them were described in Fennel (2008), but there are also some refinements. The predator–prey and prey–predator interactions terms for two groups of prey, herring and sprat, are defined as, max cod

Pk ðH; SÞ = gCk Bk

Her Her Spra ∑k−1 GðBSpra ÞÞ i =1 pk;i ðBi GðBi Þ + pk;i Bi i Q ðtÞ; σk−1

with 2 ≤ k ≤ 7, Πi ðSÞ =

Her Spra GðBi ÞBi

Her

Πi ðHÞ = GðBi

∑

σk−1

k=i + 1

gCmax pk;i Bcod k k

7

her

ÞBi

gCmax pk;i Bcod k k

7

∑

σk−1

k=i + 1

Spra

QðtÞ; and

3

Spra

σ3 = ∑ ðp4;i Bi i=1 5

Spra

σ5 = ∑ p6;i Bi i=1

6

Her

+ ∑ p6;i Bi i=1

4

Þ;

Spra

+ p3;i Bi

Spra

+ p5;i Bi

σ4 = ∑ ðp5;i Bi i=1

;

3

4

5

6

7

30 10 10 0.0066 0.0066 0.0103 0.0029 2 · 10− 11 2 · 10− 11 – – – – – – – –

60 30 15 – 0.0077 0.0090 0.0014 – – 10− 7 10− 7 – – – – – –

200 60 20 – 0.0049 0.0039 0.0012 – – 10− 7 10− 7 310− 4 – – – – –

800 150 35 – 0.0041 0.0018 0.0008 – – 10− 7 10− 7 310− 4 510− 5 – 2.710− 4 – –

1500 250 – – 0.0030 0.0006 – – – 10− 7 10− 7 – 510− 5 10− 5 – 2.710− 4 6.810− 4

104 – – – 0.0012 – – – – 5 · 10− 8 5 · 10− 8 – – 510− 5 – – 610− 3

was chosen to mimic a similar term in the model of Megrey et al. (2007). An experimental motivation for this approach was given in Arrhenius (1998a,b) and Arrhenius and Hansen (1996).    d −t ðt−d320 Þ QðtÞ = 1:3 0:1 + Θðd100 −tÞ 100 + Θðt−d320 Þ 0:9− 145 145  0:9 : + ðΘðt−d100 Þ−Θðt−d320 ÞÞðt−d100 Þ d220

Here Θ(x) is the step function, (Θ(x) = 1 for x N 0 and Θ(x) = 0 for x b 0), and dx refers to the day of the year. The Feeding of spat and herring on zooplankton is food limited trough G(Z) = eaT(1 − exp (−IZZ)), with the Ivlev constant IZ = 1 m3/mm C. Thus, the feeding max terms are gS,H G(Z)Q(t), while the loss terms are LHi,SiN = i = gHi,Si S,H max aT 0.0625(gi + e gHi,Si )/d and LHi,SiD = 0.0625eaTgS,H i /d. The loss rates for cod, which include both the share of food used for vital processes and basic metabolism, are,

0:03eaT gCmax k

LC7 = 0:19P7 +

Her

i=1

Her

2

5 5 5 0.0082 – 0.0123 0.0053 10− 11 10− 11 – – – – – – – –

LCk = 0:4Pk +

σ2 = ∑ ðp3;i Bi

+ p4;i Bi

1

CXi/g HXi/g SXi/g gcod /d− 1 i gCmax /d− 1 i −1 gHmax /d i −1 /d gSmax i ICHi/g− 2 km6 ICSi/g− 2 km6 ICHi/g− 1 km3 ICSi/g− 1 km3 rSi/d− 1 rHi/d− 1 rCi/d− 1 fSi/d− 1 fHi/d− 1 fCi/d− 1

for k = 2 to 4: for k = 5 to 6; and

0:01eaT gCmax : 7

QðtÞ:

i=2

Her

+ p2;1 B1 ;

Mass class i

LCk = 0:3Pi + 0:03eaT gCmax k

In the preference terms, pk,i corresponds the first index, ‘k’, to the mass class cod and the second index ‘i’ to the mass class of prey. The σ's represent the following sums,

σ1 = p2;1 B1

Table 2 List of parameters for the fish model: mass intervals, maximum growth rates and not used for metabolism, off-spring, and fishing rates.

5

H S Twindow = Twindow = Θðt−d60 Þ−Θðt−d120 Þ;

S

osSi = rSi Twindow ; for i = 4 and i = 5;

Þ;

H

Spra

σ6 = ∑ p7;i Bi i=1

C

Twindow = Θðt−d60 Þ−Θðt−d150 Þ; and

Then the off-spring rates can be written as,

Þ;

Her

The reproduction (off-spring rates) is restricted to certain time windows, which can be expressed in terms of Θ-functions as,

6

Her

+ ∑ p7;i Bi i=1

:

aT S,H 2 The Ivlev-function reads G(BS,H i ) =e (1− exp(−ICHi,CSi(Bi ) )), for aT S,H i = 1 and i = 2, G(BS,H i ) =e (1− exp(−ICHi,CSiBi )), while for i N 2 the Ivlev constants, ICHi and ICSi, are listed in Table 2. The choice of quadratic argument for the first two mass class has proven to give more stable result. The effect of temperature is included in a simplified manner in terms of an Eppley factor, exp(aT), with a = 0.063/°C. The preferencefactors, which are chosen to be the same for herring and sprat, are all unit, except p5,1 = 0.2; p6,1 = 0.1; p6,2 = 0.6; p7,1 = 0.05; p7,2 = 0.3; p7,3 = 0.6. The term Q(t) reflects the annual cycle of food quality and

osHi = rHi Twindow ; for i = 5 and i = 6; and C

osCi = rCi Twindow ; for i = 6 and i = 7: For the fishing rates we assume that the fishery is closed during the reproduction period, implying, C FC i = fCi ð1−Twindow Þ; for i = 6 and i = 7; H Þ; for i = 5 and i = 6; FHi = fHi ð1−Twindow

and S

FSi = fSi ð1−Twindow Þ for i = 5:

W. Fennel / Journal of Marine Systems 81 (2010) 184–195

The nutrient limited uptake rate u and the food limited grazing terms are,

Table 3 Rates and parameters of the NPZD-model. Notation

Numeric value

Unit

Process Uptake constant Half saturation Temperature threshold Respiration of phytopl. Mortality of phytopl. Grazing constant Ivlev constant for Z Mortality of zoopl. Sedimentation rate

u0 α2 T0 lPN lPD β IP lZD lDDsed

0.5 40 2.5 0.1u(N, T) 0.02 0.16 0.0036 0.01(0.03) 0.002

d− 1 mmol C2 m− 6 °C d− 1 d− 1 d− 1 mmol C− 2 m6 d− 1 d− 1

Initial values Nitrate Phytoplankton Zooplankton Detritus Sediment-detritus

N(0) P(0) Z(0) D(0) Dsed(0)

1(6.625) 0.6625 0.3 1.5(9.94) 0

mmol N m− 3(mmol C m− 3) mmol C m− 3 mmol C m− 3 mmol N m− 3(mmol C m− 3) mmol C m− 3

Further mortality terms describe natural mortality and starvation. Note that mxi = Bx1/Nxi is the average individual mass of the species x and the mass class i,

μ C1 = ð0:01 + 0:1Θðg1Cod −LC1 N −LC1 D ÞÞ = d; μ Ck = ð10−5 + 0:002ΘðPk −LCk N ÞÞ = d for k = 2 to k = 6: μ ⁎Ck = 0:05Θð0:8CXk −mCk Þ = d for k = 2 to k = 6; μ C7 = 10−10 BCod 7 = g = d; = 0:05ΘðP7 −LC7 N Þ = d: μ starv C7

μ H1 = ð0:01 + 0:01Θðg1Her −LH1 N −LH1 D ÞÞ = d μ ⁎Hi = 0:05Θð0:8HXi −mHi Þ = d; for i = 2 to i = 5; −12 = g = d; μ H6 = BHer 5 10 starv

μ H6

Her

= 0:01Θðg6 −LH6 N Þ = d

Spra

μ S1 = ð0:01 + 0:1Θðg1 μ ⁎Si

−LS1 N −LS1 D ÞÞ = d;

= 0:05Θð0:8SXi −mSi Þ = d; for i = 2 to i = 4;

μ S5 = 0:005Θðg starv μ S5

=

Spra

Spra −12 B5 10

195

−LS5 N Þ = d; = g = d;

Regarding the coupling of the upper and lower parts of the food web, we note that the model ‘currency’ is nitrogen, but with strict Redfield ratio to carbon, (C:N = 106:16), while the conversion between carbon and biomass is assumed to be 1 mmol C m− 3 = 100 tons km− 3 (Table 3).

uðN; TÞ = ΘðT−T0 Þu0 aT

eaðT−T0 Þ N2 ðΘðt−d100 Þ−Θðt−d320 ÞÞΔðtÞ; α2 + N2 2

gðP; TÞ = βe ΘðP−P0 Þð1−expð−P IP ÞÞ; The natural mortality rate of zooplankton is set to lZD = 0.01/d. If the predation on zooplankton is small and the variable Z exceeds a threshold of 5 mmol C m3, then lZD is enhanced to a higher values (0.03/d). The maximum nitrogen import flux, Nimport, is chosen as 4.510− 3 mmol N m− 3 d− 1, or expressed in carbon units, Nimport = 0.03 mmol C m3 d. Time variations are taken into account through multiplies as shown in Fig. 2 (middle). References Andersen, K.P., Ursin, E., 1977. A multispecies extension to the Beverton and Holt theory of fishing, with accounts of phosphorus circulation and primary production. Report 7, Contributions from the Danish Institute for Fishery and Marine Research, Charlottenlund. Arrhenius, F., 1998a. Growth and seasonal changes in energy content of young Baltic Sea herring (Clupea harengus L.). ICES Journal of Marine Science 53, 792–801. Arrhenius, F., 1998b. Variable length of daily feeding period in bioenergetics modelling: a test with 0-group Baltic herring. Journal of Fish Biology 52, 855–860. Arrhenius, F., Hansen, S., 1996. Food intake and seasonal changes in energy content of young Baltic Sea sprat (Sprattus sprattus L.). ICES Journal of Marine Science 55, 319–324. Fennel, W., 2008. Towards bridging biogeochemical and fish production models. Journal of Marine Systems 71, 171–194. Fennel, W., 2009. Parameterizations of truncated food web models from the perspective of an end-to-end model approach. Journal of Marine Systems 76, 171–185. Fennel, W., Neumann, T., 2003. Variability of copepods as seen in a coupled physical biological model of the Baltic Sea. ICES Marine Science Symposia 219. Hansson, S., Hjerne, O., Harvey, C.J., Kitchell, J.F., Cox, S.P., Essington, T.E., 2007. Managing baltic sea fisheries under contrasting production and predation regimes: ecosystem model analyses. Ambio 60, 259–265. Harvey, C.J., Essington, T.E., S.P.C., Hansson, S., Kitchell, J.F., 2003. An ecosystem model of food web and fisheries interactions in the Baltic Sea. ICES Journal of Marine Science 36, 939–950. Hjerne, O., Hansson, S., 2002. The role of fish and fisheries in Baltic Sea nutrient dynamics. Limnology and Oceanography 47, 1023–1032. Horbowy, J., 1989. A multispecies model of fish stocks in the Baltic Sea. Dana 7, 23–43. Horbowy, J., 2005. The dynamics of Baltic fish stocks based on a multispecies stockproduction model. Journal of Applied Ichthyology 21, 198–204. ICES, 1998. Report of the Baltic Fisheries Assessment Working Group. ICES CM 1998/ ACFM: 18. Köster, F.W., Möllmann, C., Neuenfeldt, S., Vinther, M., St.John, M.A., Tomkiewicz, J., Voss, R., Schnack, D., 2003. Fish stock development in the central baltic sea (1974– 1999) in relation to varability in the environment. ICES Marine Science Symposia 219, 294–306. Megrey, B.A., Rose, K.A., Klumb, R.A., Hay, D.E., Werner, F.E., Eslinger, D.L., Smith, S.L., 2007. A bioenergetics-based population model of Pacific herring (clupea harengus pallasi) coupled to a lower trophic level nutrient-phytoplankton–zooplankton model: description, calibration, and sensitivity analysis. Ecological Modelling 202, 144–164. Nausch, G., Nehring, D., Nagel, K., 2008. Nutrient concentrations,trends and their relation to eutrophication. State and evolution of the Baltic Sea. John Wiley and Sons, Inc, Hoboken New Jersey, pp. 337–366. Neumann, T., Schernewski, G., 2008. Eutrophication in the Baltic Sea and shifts in the nitrogen fixation analyzed with a 3d ecosystem model. Journal of Marine Systems 74 (1–2), 592–602. Neumann, T., Fennel, W., Kremp, C., 2002. Experimental simulations with an ecosystem model of the Baltic Sea: a nutrient load reduction experiment. Global Biogeochemical Cycles 16, 7-1–7-19. Seifert, T., Kayser, B., 1995. A High Resolution Spherical Grid Topography of the Baltic Sea.