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Journal of Marine Systems 71 (2008) 171 – 194 www.elsevier.com/locate/jmarsys
Towards bridging biogeochemical and fish-production models Wolfgang Fennel Institut für Ostseeforschung Warnemünde an der Universität Rostock D-18119, Warnemünde, Germany Received 10 April 2007; accepted 28 June 2007 Available online 18 July 2007
Abstract The paper presents a theoretical approach to formulate a model which comprises the full food web. The lower part of the food web is represented by a biogeochemical model which interacts explicitly with a fish-production model. The fish-production model component builds on existing theories but was substantially reformulated in order to facilitate the model coupling. The dynamics of the fish-production model is basically driven by the predator–prey interaction. We use the example of the Baltic Sea, which has a relatively simple foodweb structure. The fish biomass is dominated by three groups, sprat, herring and cod, which represent about 80% of fish biomass in the Baltic. The zooplanktivors sprat and herring are eaten by cod. In this paper we start the construction of the model as a simple box system, which can be considered as an isolated water column of 10 × 10 km2 times the water depth in the central Bornholm basin of the Baltic Sea. The stepwise building up of the model is illustrated by example simulations, which allow to assess the consistence of the theoretical approach and the choices of parameters. As last step we introduce a simple biogeochemical model and link it with the fish model. The resulting model system is strictly mass conserving without unspecified sources of food or so. We conduct experiments with the model system and show that it can reproduce features such as interannual variation in fish catches and trophic cascades. © 2007 Elsevier B.V. All rights reserved. Keywords: Biogeochemical models; Fish models; Predator–prey interaction; Reproduction; Mortality
1. Introduction In the research on marine ecosystems, there are two broad branches looking either at the lower or the upper parts of the food web, i.e., biogeochemistry and fish production. Although these branches are interlinked, they are more or less separately studied. This is also reflected in the development and application of models. Biogeochemical models see the fish only implicitly as mortality rates, while fish-production models see the lower food web basically through prescribed food, e.g. copepod biomass. To understand and manage marine ecosystems in the frame of an integrated ecosystem based approach, it is E-mail address:
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quite obvious that the two branches need to be connected. An example of an integrated study on the role of fish and fisheries in Baltic Sea nutrient dynamics was provided by Hjerne and Hansson (2002). Their work attempts to quantify the amount of phosphorous bound in fish and the removal of nutrients through fisheries. It was shown that removal by fisheries corresponds up to 18% of the anthropogenic load of the nutrients that reach the open sea. A first theoretical attempt to cover the full food web for modeling fisheries in the North Sea was undertaken by Andersen and Ursin (1977). An approach towards connecting NPZ models with a bioenergetics-based population model of Pacific herring was developed and analyzed in Megrey et al. (2007b) and Megrey et al.
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(2007a). The present paper intends to contribute towards the goal of bridging biogeochemical models and multispecies fish-production models. The model formulation should cover the whole food web with different size classes and time scales in a consistent manner, and the model-system must strictly conserve mass. In order to encounter this general goal, it is necessary to make the two classes of models compatible. This goal can only be reached by a stepwise approach. This study starts with considering the fish model aspects of the problem. The approach is inspired by the work on multi-species stock pro-duction models, Horbowy (1989) and Horbowy (2005), and employs elements of the theory developed for stage-resolving models of zooplankton, Fennel (2001). In order to quantify the dynamics of fish stocks, the model equations of fish-biomass should take the impact of predators on the survival of the prey into account, and relate predator growth to the prey biomass consumed. As a relatively simple natural system, with only a small number of important species, the Baltic Sea is chosen as example system. The main stocks are sprat, herring and cod, which represent about 80% of the fish biomass of the Baltic. The zooplanktivores sprat and herring are the main food of cod. The paper develops step by step the coupled biogeochemical and fish model, starting with the fish component and linking it later to the NPZD-model, where NPZD stands for Nutrients, Phytoplankton, Zooplankton and Detritus. The model structure is sketched in Fig. 1. The block arrows indicate the pathway from nutrients to fish while the thin arrows refer to recycling to nutrients, both directly and through the mineralization of the detritus pool.
The paper continues as follows: After the formulation of the dynamic equations of the fish in Sections 2– 4, we introduce in Sections 5 and 6, a relatively simple NPZD-model, which will be linked to the fish model. The resulting model system is then used for example simulations and to scrutinize whether the system is able to reproduce qualitatively features such as interannual variation in fish catches and trophic cascades. The paper concludes a summary and conclusions in Section 7. 2. Formulation of the theory 2.1. Biomass and individual growth Fish biomass is a key quantity for stock assessments. For a cohort of fish, the biomass, B, is the product of the mean individual mass, m, and the abundance, i.e. the number of individuals, N, per unit volume. The dynamics of B is obviously characterized by changes of both the number and the individual mass, d d d B ¼ m N þ N m: dt dt dt
ð1Þ
Ignoring reproduction for a moment, a given initial number of individuals, N0 = N(0), decreases at a death rate, μ, d N ¼ lN ; N ðtÞ ¼ N0 elt : dt
ð2Þ
For the evolution of the average individual mass in a developing cohort, the Bertalanffy formula is largely used, dm=dt ¼ Hm2=3 km;
ð3Þ
where H and k are empirical constants. The analytical solution is mðtÞ ¼
H H tk 3 1=3 þ m0 ; exp k k 3
with m0 being the initial mass. For t → ∞ the maximum mass follows as mmax = (H / k)3, i.e., the parameters H and k are related through the maximum mass. An equivalent formulation is, " Fig. 1. A sketch of a coupled model, which consists of a relatively simple biogeochemical model (NPZD) and a fish-production model for sprat, herring and cod.
mðtÞ ¼ mmax 1
m0 1 mmax
#3 1=3 ! tk : exp 3 ð4Þ
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Then we find the dynamical equation of the biomass by combining Eqs. (2) and (3) as, d B ¼ ml N þ Hm2=3 N km N dt ¼ lB þ Hm2=3 N kB:
ð5Þ
However, this equation of B is not closed, because the term, Nm2/3 , cannot be expressed by the biomass B. There are several choices of growth equations, which are often listed in the literature without ranking, e.g. Brown and Rothery (1993), and which can be described as a family of empirical models, as shown by Richards (1959), dm k m mmax 1m ¼ 1 ; dt 1m m with ν being a positive number less than one. This equation has an analytical closed form solution, mðtÞ ¼ mmax ð1 Am ejt Þ1m : 1
Here Aν is related to the initial and the maximum mass, m0 = m(0) and mmax = m(t → ∞). The case, ν = 0, is known as monomolecular growth, or in fishery science as Brody formula, see Beverton and Holt (1957), ddmt ¼ ) and j½mmax m, with m = m max (1 − A 0 e −κt A0 ¼ 1
m0 mmax .
For ν = 2/3, implying A2=3 ¼ 1
m0 mmax
1=3
and
κ =k /3, we retrieve Eq. (4). For ν = 2, the ‘logistic’ or ‘autocatalytic’ case, we have dm m mmax ¼j 1 ; m; with m ¼ dt mmax 1 þ A2 ejt mmax ð6Þ 1: and A2 ¼ m0 The three formulas have inflection points from which on the growth decelerates. The mostly used approach seems to be the Bertalanffy formula. Fitting such formulas to mass data, gives an excellent condensed information on data describing mass development of fish as function of time. However, the underlying dynamical Eq. (3) involves the above mentioned problem when relating dynamics of numbers of individuals, individual growth and total biomass. The Brody formula circumvents this problem, but depends on a maximum mass, implying the occurrence of an artificial maximum biomass, Bmax = Nmmax. An alternative way, which supports the formulation of size-class related prey–predator interaction, consists in the division of the mass of the individuals into
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mass intervals, with Xi−1 being the lower and Xi the upper limit of the mass-class ‘i’. The choices of the width of the mass intervals are motivated by the need for formulations of size dependent interaction of predator and prey. For each ‘size-class’ or ‘mass-class, ‘i’, we have, dmi ¼ gi mi : dt
ð7Þ
The effective growth rates, ‘gi’, represent gains through feeding minus loss through respiration, excretion etc., and are constant within the intervals. Only for the largest mass class we assume a variable rate to ensure that the effective individual growth tends to zero when a maximum individual mass is reached. Such an approach was successfully applied to stage-resolving modeling of copepods, Fennel (2001). The main reason for the transition from a continuous growth equation to a discrete set of mass interval is the potential to describe many overlapping cohorts at the same time. The accumulated knowledge from observations of the growth of average individuals of the main species is represented in a condensed manner by the parameters in the Bertalanffy formulas, see Horbowy (1989). We can use this information to estimate the piece-wise constant growth rates by mapping the Bertalanffy parameters of the corresponding mass intervals onto the constant rates. Since the solutions of both the Bertalanffy Eq. (3) and the piecewise constant growth Eq. (7) express mass as unique function of time, we can find gi as, gi ¼
1 Xi ln : ti ti1 Xi1
with Xi−1 ≤ mi ≤ Xi. The times ti are determined from Eq. (4) by, m(ti − 1) = Xi−1 and m(ti) = Xi. The Bertalanffy parameters, the choice of the mass intervals, and the resulting numerical values of the rates are listed for the three main species in Table 1. Since we deal with three species, we will denote the limits of the mass intervals by xXi and the rates by gxi, where the lower case ‘x’ stands for sprat, herring, or cod, x = (S, H, C). For the largest mass class, i.e., Xl ≤ m ≤ mmax, an exponential growth is no longer consistent because the growth must decrease when m approaches mmax, i.e., the ingested food is entirely balanced by metabolic processes when the maximum mass is reached. We choose a ‘logistic’ equations, see Eq. (6), written in the form, d mmax m m ¼ gl m; dt mmax m0
for
Xl V m V mmax ;
ð8Þ
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Table 1 List of model parameters Bertalanffy parameters Species
H/(gram− 1/3 year)
k/year
Cod (JH1989) Cod (JH2005) Herring Sprat
5.61 10.6 8.1 3.06
0.023 0.51 1.30 0.93
Maximum mass in gram Species
m0
xX1
xX2
xX3
xX4
xX5
xX6
xmmax
Cod Herring Sprat
2 2 1
5 5 5
30 10 10
60 30 15
200 60 20
800 150 –
1500 – –
104 250 35
Maximum growth rate in 1/day Species
gx1
gx2
gx3
gx4
gx5
gx6
gx7
Cod Herring Sprat
0.0082 0.0082 0.0036
0.0068 0.0068 0.0019
0.0038 0.0060 0.0009
0.0025 0.0026 0.0008
0.0021 0.0012 0.0005
0.0015 0.0005 –
0.0010 – –
The upper part lists the values of the Bertalanffy parameters, H and k which are adopted from Horbowy (1989), except for cod, where we use also data from Horbowy (2005), as indicated be JH1989 and JH2005. The middle part defines the mass intervals of the mass-classes and the lower part the piecewise constant rates, required in (7). The maximum masses, xXi are given in gram, the rates gxi have the units d− 1. Note that the lower case ‘x’ in xXi and gxi stands for the species, C, H and S, referring to cod, herring and sprat, respectively.
with gl being the effective growth rate of the largest mass class. The solution is, mðtÞ ¼ m ¼
1þ
mmax mmax m0
; 1 ejt
and j ¼ gl
mmax : mmax m0
Our approach requires a mechanism to promote the variables from one mass-class into the next one. With Eqs. (1) and (7), we find instead of Eq. (5) the set, d Bi ¼ transfer½ði 1ÞYi li Bi þ gi Bi dt transfer½iYði þ 1Þ: The symbolic expressions, e.g. transfer[(i − 1) → i], indicate a promotion of the state variables in the lower ‘mass-classes’ (i − 1), to higher next higher one (i). The ‘transfer’ can be expressed by τiBi, where the transfer rate, τ, is controlled by the average individual mass, mi = Bi / Ni. 2.2. Construction of the model equations In order to simulate fish biomass and abundance we have to define quantitatively the system under consideration. Although the ultimate goal will be a spatial explicit three-dimensional model, we start the process of model construction with a box-model, i.e., we consider the biomass and number of individuals in a
box-like water column, covering an area of 102 km2 in the central Bornholm Basin in the Baltic Sea. Assuming a water depth of H = 100 m, the volume of the box is 10 km × 10 km × 0.1 km = 10 km3 or 1010 m3. The surface of the box corresponds to about 20% of the area of central Bornholm Basin. In the further evolution of the model, this box can serve as one or several grid cells in a three-dimensional model. If we ignore for a moment reproduction, the development of the biomass in a specified volume is driven by growth and losses through mortality and transfer from one mass class to the next one. We can construct a set of evolution equations for the abundances per massclass, say for example i = 1 to i = 7, as d B1 ¼ ðg1 l1 Þ s1 B1 ; dt d B2 ¼ s1 B1 þ ðg2 l2 ÞB2 s2 B2 ; dt d B3 ¼ s2 B2 þ ðg3 l3 ÞB3 s3 B3 ; dt d B7 ¼ s6 B6 þ ðg7 l7 ÞB7 ; N dt
ð9Þ
where τi are the rates at which the biomass is leaving a mass-class, when approaching the corresponding mass class limit, and μi are the mortalities, which will be specified later. To define the transfer functions, we use
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filter-functions f (mi, Xi), which enables the transfer within a mass interval xXi − △ b m b xXi, f ðmi ; xXi Þ ¼ hðY Þ
ðY þ DÞ ðhðY Þ hðY þ DÞÞ; D
with Y = mi − xXi. Here the θ-function was used, which is one for positive, and zero for negative arguments, (θ(y) = 1 for y N 0 and θ(y) = 0 for y b 0). The shape of the filter function is sketched in Fig. 2. The transfer function for biomass can then be written as si ¼ nf ðmi ; xXi Þ;
ð10Þ
where ξ sets the transfer speed. Below we will choose ξ = 10 / d. Since the dynamics of the abundance is driven by reproduction, which is ignored for the moment, mortality and transfer, but not by growth, the equation set follows as, d N1 dt d N2 dt d N3 dt d N7 dt
¼ l1 N1
s1 B1 ; x X1
s1 s2 B 1 l2 N 2 B2 ; xX1 x X2 s2 s3 ¼ B 2 l3 N 3 B3 ; xX2 x X3 s6 ¼ B 6 l7 N 7 N xX6 ¼
ð11Þ
In order to describe the corresponding transfer for the number of individuals, in Eq. (11), we scale the transfer of biomass by the maximum mass in the corresponding mass interval. We note that a more rigorous justification of these sets of equations can be achieved by involving the dynamical equation of the population density, e.g. chapter 4 in Fennel and Neumann (2004).
Integrating the Eqs. (9) and (11), with arbitrary initial numbers of individuals of the initial mass of m0 = 2 g for cod and herring, and m0 = 1 g for sprat, with the growth rates listed in Table 1 and for zero mortality, reproduces the mass development shown in Figs. 3,4 and 5. However, now the individual mass of an animal is replaced by the average mass defined by the ratio biomass over number of individuals, B /N. In other words, we can reproduce the growth of an average individual by looking at the development of biomass and numbers of a population. 3. Fish-production modeling with predator–prey interaction The natural next step is the explicit consideration of predator–prey interaction, i.e., the piece-wise constant growth rates of the predator, cod, have to be expressed by functions describing the feeding process. In order to be specific, we start considering herring as prey and cod as predator. If once the theory is formulated for two groups, it is straightforward to extend the equations by including sprat. There are some general rules which provide guidance to formulate the relationships:
˙ ˙ ˙
The prey item must be smaller than the predator, to allow the predator to eat the prey, the feeding rate of the predator must be limited by the available prey, any consumption of prey biomass implies a reduction of the number of prey individuals.
We assume that the second mass-class of cod, Bcod 2 , can feed on the first mass-class of herring, Bher , while the 1 third class of cod, Bcod , feeds on the first and second class 3 her of herring, Bher 1 and B2 , and so forth. Thus, the mass cod class Bi feeds on all Bher k under the condition that k b i. To introduce food limitation, we choose an Ivlev approach, GðBÞ ¼ 1 expðIt BÞ;
Fig. 2. Sketch of the filter function with a linear transition interval △.
175
ð12Þ
where B is the biomass concentration of the prey and Iν the so-called ‘Ivlev’ constant. We start as a first guess with the choice Iν ∼ 10− 9 km3 g− 1. The consumed food is utilized to both growths and maintenance of the vital functions of predators, i.e., the model must quantitatively track the losses, which can be recycled and enter the model food web. The growth through feeding is then controlled by the limiting prey and some maximum feeding rates, gimax , minus loss rates, li, which refer to the amount of food used for metabolic processes others as growth. The general structure of the feeding function is then like ðgimax li ÞGðBÞ:
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Fig. 3. Development of an average individual cod, as given by a Bertalanffy formula, (5), for the two sets of parameters, see Table 1, and a piecewise constant rate approach, (7).
We derive the maximum grazing rates for the predator from the effective rates gCi, listed in Table 1, by assuming that roughly half of the ingested food is used for metabolic processes and the other half for growth. Only for the highest mass class the loss increases and can approach the feeding rate. For the loss term, L7, in the equation of Bcod 7 , we choose, L7 ¼ l7 þ l7
m CX6 ; Cmmax CX6
ð13Þ
which implies L7 ¼ l7 ¼ 0:5gCmax for m near the lower 7 limit of the mass interval and L7 YgCmax near the maxi7 mum mass. The metabolic losses li refer to recycling processes, such as respiration and excretion. A detailed resolution of these processes is not needed as long as the coupling to the lower food web is not yet taken into account. The parameter values are listed in Table 2. The
Fig. 5. As Fig. 3, but for sprat.
¼ gCi þ li . Our relationship between the rates is, gCmax 7 choice of a bulk loss rate is comparable with the sum of all contributions used in more detailed bioenergetic models, see e.g. Megrey et al. (2007b). Obviously, weight functions are needed to ensure that the predator will not consume more than the specific maximum rate allows. For example, the fourth mass class her her of cod feeds on Bher 1 ; B2 and B3 implying a growth max her her proportional to gC4 ðGðB1 Þ þ GðBher 2 Þ þ GðB3 ÞÞ. If her her her GðB1 Þ þ GðB2 Þ þ GðB3 Þ exceeds one, the growth is higher than the maximum rate. This can happen if several mass classes of the prey exist at the same time. To avoid this complication, we introduce weight functions of the P her type Bher = B i k k , which ensure that only a limited amount of the food items will be consumed. In order to formulate the interactions between predator and prey explicitly, we differentiate between predator–prey and prey–predator interaction, P and Π, respectively. The former can be expressed for cod and herring as, k1 X her Bher i GðBi Þ
Pk ðherÞ ¼ gCmax Bcod k k
i¼1
; ð2 V k V 7Þ;
kP 1 i¼1
ð14Þ
Bher i
the latter is written as her Pi ðherÞ ¼ GðBher i ÞBi
7 X gCmax Bcod k k : iP ¼1 k¼iþ1 Bher k
ð15Þ
k¼1
Fig. 4. As Fig. 3, but for herring.
These formulas express that the predators catch smaller prey. Since for the smallest mass class both prey and predator are of the same size, we assume that Bcod i consumes zooplankton and grows at certain rate gC1 , which will be specified later.
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177
Table 2 Maximum feeding rates, gmax Ci and loss rates, li, for the model—cod Mass-class
1
2
3
4
5
6
7
gmax Ci
0.0164 0.0082
0.0137 0.0068
0.0077 0.0038
0.0049 0.0025
0.0051 0.0026
0.0038 0.0019
0.0024 0.0012
li
Note that these rates are related to the effective growth rates by gCi = gmax Ci − li.
Sprat can be included in a straightforward manner. Since many prey items of both herring and sprat may be on the same location at the same time, we have to ensure that the maximum feeding of cod is limited to a physiologic maximum. In order to formulate the predator–prey and prey–predator interactions terms for two groups of prey we introduce sum-terms of the prey biomass as, r1 ¼ BSpra þ Bher 1 ; 1 r3 ¼
3 X
r2 ¼
ðBSpra þ Bher k Þ; k
k¼2 X ðBSpra þ Bher k Þ; k
r4 ¼
k¼1
r5 ¼
5 X
k¼1 4 X
ðBSpra þ Bher k Þ; k
k¼1
ðBSpra k
þ
Bher k Þ;
r6 ¼
k¼1
5 X
BSpra k
k¼1
þ
6 X
Bher k :
k¼1
ð16Þ Then we can generalize the expression (14), Pk ð H; S Þ ¼
gCmax Bcod k k k1 X her ðBher i GðBi Þ i¼1
ð21Þ
The six equations for the biomass of herring, (i = 1... 6), are, d her her her B ¼ sHi1 Bher i1 þ ðgHi li ÞBi sHi Bi Pi ðHÞ: dt i ð22Þ Since the consumption of prey biomass by cod reduces also the number of prey individuals, the equations for the abundance follow as, d her Bher Bher Pi ðherÞ Ni ¼ sHi1 i1 li Niher sHi1 i ; dt HXi1 HXi mher i ð23Þ
mher i ¼
þ ðBSpra GðBSpra ÞÞ i i ; ð17Þ
with 2 ≤ k ≤ 7 and H and S refer to herring and sprat. Moreover, the prey–predator terms for sprat and herring become Pi ðSÞ ¼
d cod Bcod Bcod Ni ¼ sCi1 i1 li Nicod sCi i : dt CXi1 CXi
where the average individual mass is given by,
rk1
GðBSpra ÞBSpra i i
. The equation set for the abundance of with mi ¼ li =gCmax i cod is not affected by the predator prey interaction and follows from Eq. (11) as,
7 X gCmax Bcod k k : r k1 k¼iþ1
Bher i : Niher
For the sprat state variables we need two sets of five equations, (i = 1... 5), for biomass, d Spra Spra ¼ sSi1 BSpra sSi BSpra Pi ðSÞ; B i i1 þ ðgSi li ÞBi dt i ð24Þ and for the abundance,
ð18Þ
d Spra BSpra BSpra Pi ðSÞ Ni ¼ sSi1 i1 li NiSpra sSi i Spra ; dt SXi1 SXi mi ð25Þ
and with Pi ðHÞ ¼
her GðBher i ÞBi
7 X gCmax Bcod k k : rk1 k¼iþ1
ð19Þ
Then we have for cod seven biomass and abundance equations, (i = 1... 7), d cod cod B ¼ sCi1 Bcod sCi Bcod i1 þ ð1 mi ÞPi ðH; SÞ li Bi i ; dt i
ð20Þ
¼ mSpra i
BSpra i NiSpra
:
3.1. Example simulation In order to illustrate the interaction of cod with herring and sprat by means of an example simulation, we integrate Eqs. (20)–(25) with initial values of cod, herring and sprat. We select, somewhat arbitrary, nonzero
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values for cod mass class five, and mass class two for herring and sprat. All the other mass classes have zero abundance at the start. We choose for t = 0 in particular N5cod = 104 ind. and B5cod = CX4N5cod with CX4 = 200 g, N1her = 105 ind. and B1her = HX1N2her, with HX1 = 5 g, and N1spra = 3106 ind. and B1spra = SX1N2spra with SX1 = 5 g. For clarity, mortalities are set to zero. The model is integrated over six years and the results are sketched in Figs. 6–9. We start with the discussion of the mean individual masses, see Fig. 6. In the beginning, cod and the two prey species grow as expected from the Bertalanffy formula. However, the dynamical changes are limited to a period of about four years. In that time span the predator consumes all prey and then the growth stops. This becomes obvious from Fig. 6, left lower panel, where the total biomass for each species is shown. The predator uses up the prey biomass. This scenario is visualized in Figs. 7–9. Owing to the zero mortality, the abundance of cod is constant, while the biomass increases as long as prey is available, Fig. 9, top and bottom, respectively. Basically, cod feeds on
both prey stocks, except for a short phase after somewhat more than one and a half year, when the herring has advanced into a mass class, which is too big for the current mass class of cod. According to the choices of the predator–prey interaction term, this mass class of prey cannot be eaten by the cod. During this period of time the herring abundance remains constant, see Fig. 8. However, as soon as the predator enters the next mass class, the consumption of herring continues. Sprat is consumed all the time, Fig. 7. During the first one or two years the biomass of prey increases, because the gain of mass is faster than the decrease in abundance. Without reproduction the model runs into final stages without further development when the prey biomass was completely used up. Other choices of initial conditions, (not shown), with a smaller initial mass class of the predator and larger mass classes of prey, can produce stages where the growth of the predator mass stops while the prey grow up until it reaches eventually its maximum mass. We note that with the current settings the constant grazing rates of prey imply that sprat and
Fig. 6. An example of the dynamics of predator–prey interaction: Cod starts with masses-class ‘five’ acting on herring and sprat, which start with of mass class ‘two’. Shown is the growth of the mean individual mass of cod, (upper left panel), herring (upper right panel), sprat (lower left panel). The horizontal lines indicate the maximum levels reached by each mass class. The total biomass of the three species is plotted in the lower right panel, (cod — solid, herring — dashed, and sprat — dotted).
W. Fennel / Journal of Marine Systems 71 (2008) 171–194
179
Fig. 7. The same case as Fig. 6, but shown are the abundance (top) and biomass development (bottom) of cod. Note that the vertical indentations indicate the transfers, when concentration and abundance of a mass class drop to zero while the state variables of the next mass class emerge.
herring graze on an unlimited zooplankton pool, which is not explicitly represented in the equations. 4. Modeling reproduction and fishing mortality In the previous section the interactions of predator and prey was illustrated by integrating a simple initial value problem. Since reproduction was not included, the model runs into final stages. In the present section we include this important element of the dynamics and take also fishing mortality into account. 4.1. An off-spring approach Modeling reproduction can be done directly by allowing adult fish to lay very many eggs, with a very low survival rate. This issue is usually addressed in recruitment studies, which look at the problem of match– mismatch of fish larvae and food availability, implying a spatially resolved model. In the current study we simplify the problem by looking at the off-spring at the end of the larval phase. This approach circumvents the explicit resolution of the eggs' survival problem and avoids dealing with very rapid changes of high egg abundances. The timing of the off-spring generation is, however, assumed to be the same as the time span of the spawning, i.e., a few months. During this span the female adults can lose 10–20% of their biomass, which re-appears as offspring biomass. The reproduction rates can then be
derived from the fish biomass by subtracting the mass loss from the biomass of the spawning stages and adding this off-spring mass to the state variables of the smallest mass class. The equivalent abundance is found by dividing the off-spring biomass by the initial weight, m0, of the smallest mass class. We assume implicitly that the surviving larvae have accumulated biomass by grazing on zooplankton. This is a preliminary ad hoc assumption, because we cannot resolve the critical phase of development of larval stages with a bulk NPZD model that sees zooplankton only in terms of an aggregated state variable. For the reproductive period we choose time windows, defined by days of the year, for the three groups. For cod the time window spans from day 60 to day 160. For herring and sprat we take the time interval, from day 60 to day 120, TwindowC ¼ hðt 60 dÞ hðt 150 dÞ; and TwindowH ¼ TwindowS ¼ hðt 60 dÞ hðt 120 dÞ: Then the off-spring rates, osi, are chosen as OSS4 ¼ rS4 TwindowS ; OSS5 ¼ rS5 TwindowS ; OSH5 ¼ rH5 TwindowH ; OSH6 ¼ rH6 TwindowH ; OSC6 ¼ rC6 TwindowC ; OSC7 ¼ rC7 TwindowC ;
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Fig. 8. As Fig. 7, but shown are the abundance (top) and biomass development (bottom) of herring.
where the reproduction rates rxi are listed in Table 3. This approach resembles those in Horbowy (2005), where a certain amount of biomass is added at certain time steps to mimic the recruits, and Megrey et al. (2007b), where each year new recruits are added.
4.2. Fishing mortality To model fishing mortality, which will be denoted by FXi, we choose instantaneous rates, which are multiplied by the biomass and abundance. This is different from the
Fig. 9. As Fig. 7, but shown are the abundance (top) and biomass development (bottom) of sprat.
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181
Table 3 Reproduction rates for cod, herring and sprat rS4
rS5 −4
3 · 10
−1
d
rH5 −4
5 · 10
−1
d
rH6 −5
5 · 10
−1
d
technical term ‘fishing mortality’ often used in fishery sciences, that expresses the percentage amount of the stock removed by fishery within one year or so. We require the abundances of the fish to be higher than certain threshold values, because for too few individuals the chance to catch some fish is very small, ind: cod FC6 ¼ fC6 h N6 20 3 ; km ind: cod FC7 ¼ fC7 h N7 10 3 ; km ind: her FH5 ¼ fH5 h N5 1000 3 ; km ind: her FH6 ¼ fH6 h N6 500 3 ; km ind: Spra FS4 ¼ fS4 h N4 1000 3 ; km ind: Spra FS5 ¼ fS5 h N5 500 3 : km Examples for choices of the rates fS4 to C4 are listed in Table 4. Reproduction and fishing mortality enter the cod equations as d cod cod cod cod B ¼ OSC6 Bcod 6 þ OSC7 B7 þ ðgC1 l1 ÞB1 sC1 B1 ; dt 1 d cod cod cod B ¼ sC1 Bcod 1 l2 B2 sC2 B2 þ ð1 m2 ÞP2 ðH; SÞ; N dt 2 d cod cod cod B ¼ sC5 Bcod 5 ðOSC6 þ FC6 þ l6 ÞB6 sC6 B6 dt 6 þð1 m6 ÞP6 ðH; SÞ; d cod cod B7 ¼ sC6 Bcod 6 ðOSC7 þ FC7 þ l7 ÞB7 þ ð1 m7 ÞP7 ðH; SÞ; dt
ð26Þ and, d cod N dt 1 d cod N dt 2 d cod N dt 6 d cod N dt 7
1 Bcod cod cod 1 ðOSC6 Bcod ; 6 þ OSC7 B7 Þ l1 N1 sC1 m0 CX1 cod cod B B ¼ sC1 1 l1 N2cod sC2 2 ; N CX1 CX2 Bcod Bcod cod 5 ¼ sC5 ðl6 þ FC6 ÞN6 sC6 6 ; CX2 CX6 Bcod cod 6 ¼ sC6 ðl7 þ FC7 ÞN7 : CX6 ¼
ð27Þ Since the offspring and fishing mortality terms do not affect the predator prey interactions. The modifications
rC6 −5
5 · 10
−1
d
−5
10
rC7 −1
d
5 · 10− 5 d− 1
of the herring and sprat equations are straightforward, by adding the corresponding terms accordingly. 4.3. Example simulations To illustrate the effects of reproduction and fishing mortality on the model species, we look at example simulations. We choose an example with a relatively high fishing mortality acting on the largest stage of herring and on the two largest mass classes of cod, see ‘case 1’ in Table 4 for the numerical values. As in the previous section, we set all mortality rates μi to zero. The spectrum of the initial distribution of prey and predator was chosen as a more or less evenly distributed abundance of the prey stages, while cod does initially only occur in the second largest mass class, see Fig. 10. Since the small prey fish develops rapidly, this choice of initial values for cod allows to follow the propagation of cod after the reproduction from off-spring stages towards the uppermost mass class. Because the development of herring and sprat is quite similar, we show only the results for herring. After a short period, all herring stages are present, as shown in Fig. 11. In the upper and lower panel of Fig. 11, the development of the first three and the last three mass classes is plotted. Looking at one of the mass classes we see sharp increases and drops, which indicate the changes of abundance in the mass class if new individuals enter or leave the stage. For higher mass classes stages can coexist. This is a striking difference to the case of no reproduction, where the abundance and biomass of all species propagate in single cohorts towards the final stages, see Fig. 6. The smaller stages propagate quickly through the stages, while it takes years for the development of the adult stages, mass class ‘5’ and ‘6’, see Fig. 11 lower panel. A general increase in abundance can be seen for all mass classes, implying also a general increase of the biomass (not shown). This is due to the missing food limitation of the growth rate of prey. The model cod starts with the lower end of mass class ‘6’, and produces off-spring while it grows towards the next mass class. The propagation of the off-spring through the mass classes is visualized in Fig. 12, which shows in the lower panel that development in the mass class ‘6’ required almost one decade to reach the uppermost class. Later, for the years 16 to 18, the
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Table 4 Four example cases of daily fishing mortality rates for cod, herring and sprat fS 4 Case 1 Case 2
0 0
fS 5 0 2.7 · 10− 4 d− 1
fH 5 0 0
animals propagate much faster, with the expected growth rate, through the stage six, corresponding to the growth indicated in Fig. 3. The reason for this difference in the speed of development can easily understood by the role of the food limiting Ivlev function (12). In the starting phase, compare Fig. 10, the prey biomass is of the order of 5 to 15 tons/km3, implying a value of G(B) ≈ 10− 3, while with increasing prey concentrations, say of the order of up to 3000 tons/km3, (not shown), we have G (B) ≈ 0.95. Thus, the different development times are due to different prey biomass concentrations. During the time span of the model years 12 to 16 and the model years 18 to 20, a substantial amount of adult cod is removed by fishing mortality, implying an interruption of reproduction during the model years 14 to 16. The corresponding annual catches are shown in Fig. 13. The interannual variations in the catches seen in the model are entirely caused by the speed at which the stock recovers by the development of the off-spring from small to adult mass classes. The speed of this process is also controlled by the food availability, i.e., prey concentrations. After twenty model years, the total fish biomass has increased, as shown in terms of catches in Fig. 13. 5. Coupling a simple biogeochemical model and the fish model The next natural step is linking the fish model to a model component of the lower part of the food web, as indicated in Fig. 1. This implies limited food for the zooplanktivors in a consistent way. Such a coupled model includes pathways from fish to nutrients and detritus explicitly. All loss terms in the fish model have now to be specified because there is now a feedback to the NPZD model controlling the recycling processes. 5.1. A simple NPZD model Biogeochemical models of the ocean and, in particular of the Baltic Sea, have reached a remarkable degree of realism. However, for simplicity, we start here with a simplified model, which describes the dynamics of bulk variables for phytoplankton and zooplankton, P and Z, as well as nutrient, N, and detritus D. The nutrient, N, is
fH 6
fC6
−3 −1
10 d 2.7 · 10− 4 d− 1
fC7 −4
−1
6.8 · 10 d 6.8 · 10− 4 d− 1
8 · 10− 3 d− 1 8 · 10− 3 d− 1
chosen as nitrate, but with fixed conversion rates to phosphorus and carbon, i.e. we consider strict Redfield ratios of elemental composition, for simplicity. The dynamics of the four state variables is governed by, dN ¼ uðN ; T ÞP þ lPN P þ lDN D þ lZN Z; dt
ð28Þ
dP ¼ uðN ; T ÞP lPN P gðP; T ÞZ lPD P; dt
ð29Þ
dZ ¼ gðP; T ÞZ lZN Z lZD Z; dt
ð30Þ
dD ¼ lZD Z þ lPD P lDN D; dt
ð31Þ
where u(N, T) is the uptake, and g(P, T) the grazing rate. The rates lXY describe loss and transformation processes, which transfer a variable, X, to another variable, Y. For example, lPN is the respiration rate of phytoplankton (loss of P to N), and lND is mineralization rate of detritus (loss of D to N). The uptake rate involves nutrient limitation and depends on temperature (Epply factor) as, eaðT T0 Þ N 2 ðhðt 100 dÞ a2 þ N 2 hðt 320 dÞÞDðtÞ;
uðN ; T Þ ¼ hðT T0 Þu0
with the half saturation constant α2 = 40 mmolC2m− 6, a threshold temperature T0 = 2.5 °C, the maximum uptake rate at T = T0, u0 = 0.5 d− 1, and the Epply constant, α = 0.063(°C)− 1. The threshold temperature T0 reflects a crude approach of the formation of a upper mixed layer if T exceeds T0, and Δ(t) is the normalized length of the day, as shown in Fig. 14 (bottom). For simplicity all state variables are formally expressed in carbon units, by multiplication of the nitrogen based values by the Redfield ratio rRedfield = 6.625.
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183
Fig. 10. Initial distribution of biomass (right) and abundance (left) of cod, herring and sprat.
Fig. 11. The abundance of all mass classes of herring during 20 year model simulation. The upper panel shows the mass classes 1 to 3, the lower panel the stages 4 to 6.
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Fig. 12. The abundance of all mass classes of cod during 20 year model simulation. The upper panel shows the mass classes 1 to 3, the lower panel the stages 4 to 7.
For the zooplankton grazing we have chosen gðP; T Þ ¼ beaT hðP P0 Þð1 expðP2 IP ÞÞ; −1
with β = 0.16 d being the maximum grazing rate at T = 0, P0 is a background or seed concentration of phytoplankton, P0 = 0.1 mmolC/m3 and IP = 0.0036 mmolC− 2 m6 is the ‘Ivlev’-constant for zooplankton grazing on phytoplankton. The respirational losses of phytoplankton and zooplankton are simply related to the nutrient uptake and ingested food, lPN = 0.1u(N, T) and lZN = 0.3g(P, T ), respectively. The losses of P and Z to detritus D through mortality are chosen as lPD = 0.02/d and lZD = 0.03/d. The mineralization of detritus to nutrients is also assumed to depend on temperature lDN = 0.01exp(αT )/d. All rates have the dimension d− 1. The model is initialized with a winter value of nitrate, 6 mmolNm− 3, which corresponds to about 40 mmolCm− 3, while the start values of phytoplankton and zooplankton are set to a small background value, P(0) = Z(0) = 0.6625 mmolCm− 3. The parameters are listed in Table 5. 5.2. The model linkage — seen from the NPZD model The fish model and the NPZD model are linked through mainly three channels: the consumption of zooplankton by the sprat, herring and the early stages of
cod, GF, and the recycling of fish biomass to nutrients LFN and to detritus LFD. dN ¼ lðN ÞP þ lPN P þ lDN D þ lZN Z þ LFN ; dt
ð32Þ
dP ¼ lðN ÞP lPN P gðPÞZ lPD P; dt
ð33Þ
dZ ¼ gðPÞZ lZN Z lZD Z GF ; dt
ð34Þ
dD ¼ lZD Z lPD P lDN D þ LFD : dt
ð35Þ
The term GF in zooplankton Eq. (34) represents the removal of zooplankton biomass through all fish classes consuming zooplankton at certain rates, giX ðZÞ, x =(sprat, herring, cod), multiplied by the corresponding biomass BXi , GF ¼ fcon þ
ð
6 X i¼1
2 X i¼1
giher ðZÞBher i þ
5 X
giSpra ðZÞBSpra i
i¼1
Þ
gicod ðZÞBcod ; i
where fcon is a conversion factor which expresses the ratio of biomass in g/km3, to carbon units in mmolC/m3 for the NPZD model, e.g. 1 g/km3 = N fcon mmolC/m3 with
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185
Fig. 13. Simulated time series of the annual catches of cod and herring for a 20 years period.
fcon =10− 8. All consumption rates, gix ðZÞ, are food limited according to an Ivlev formula. giX ðZÞ ¼ gXmax ð1 expðIZ ZÞÞeaT : i
ð36Þ
We choose the parameter IZ as IZ = 1 mmolC− 1m3. All mass classes of herring and sprat, and the two smallest classes of cod, consume zooplankton. Since we assume
that a substantial part of the zooplankton death rate is due to feeding of sprat and herring, we reduce the zooplankton mortality from lZD = 0.03/d in the truncated model by one third to lZD = 0.02/d in the fully coupled model. For all fish mass classes, we can assume that a part of the consumption flows into respirational losses, described by rates LSi N, LHi N and LCi N for sprat, herring and cod, respectively. The same amount is transferred into detritus by excretion at rates LSi D, LHi D and LCi D. A certain share of
Fig. 14. Annual cycle of nutrients, detritus (top), phytoplankton and zooplankton (middle), and temperature and normalized length of the day (bottom). Note that all state variables are converted to carbon units.
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Table 5 List of parameters for the full coupled model NPZD model Process
Notation
Numeric value
Unit
Uptake constant Half saturation Temperature threshold Respiration of phytopl. Mortality of phytopl. Grazing constant Ivlev constant for Z Truncation mortality Coupled model mort. Initial value nitrate Initial value phytopl. Initial value zoopl.
μ0 α2 T0 lPN lPD β IP lZD lZD N(0) P(0) Z(0)
0.5 40 2.5 0.1μ(N,T ) 0.02 0.16 0.0036 0.03 0.02 6 0.6625 0.6625
d− 1 mmolC2m− 6 °C d− 1 d− 1 d− 1 mmolC− 2m6 d− 1 d− 1 mmolNm− 3 mmolCm− 3 mmolCm− 3
Rates for the fish model Mass class i
1
2
3
4
5
6
7
μC* i /d− 1 μH* i /d− 1 μ*Si /d− 1 μCi /d− 1 μHi /d− 1 μSi /d− 1 −1 gcod i /d −1 gCmax /d 1 max − 1 gHi /d −1 gSmax /d i
– – – 0.001 0.05 0.05 0.0082 – 0.0053 0.0053 – –
0.05 0.05 0.05 – – – 0.0066 0.0066 0.0029 0.0029 0.15 0.05
0.05 0.05 0.05 – – – – 0.0077 0.0090 0.0014 0.15 0.05
0.05 0.05 0.05 – – – – 0.0049 0.0039 0.0012 0.15 0.05
0.05 0.05 – – – 1012 km3 BS5 g – 0.0041 0.0018 0.0008 0.2 0.05
0.05 – – 10− 5 12 3 10 km BH 6 g – – 0.0030 0.0006 – 0.2 0.05
– – – 1010 km3 BC7 g – – – 0.0019 – – 0.2 0.05
vi
Lbasic C i
gCcod ðZÞ i
the respirational losses is required for basic metabolism, which is also active when no food consumption occurs. For sprat and herring we use
here νi refers to the share of consumed food needed for respiration, i.e.,
LSi N ¼ ðgisprat ðZÞ þ gSmax Þ=16; i
mi ¼
and
li max gC i
:
LHi N ¼ ðgiher ðZÞ þ gHmax Þ=8: i The smallest mass class feeds on zooplankton, while the second mass class eats both zooplankton and small prey fish. For the larger mass classes of cod we take only consumption of sprat and herring into account. This has implications for the loss rates of cod, i.e., we have LC1 N ¼
0:25gicod ðZÞ;
ð37Þ
The model coupling to the NPZD model with respect to nutrients amounts to the addition of the term L FN, (Loss of Fish to Nutrients), in Eq. (32), that reads LFN ¼ fcon þ
LC2 N Bcod 2
¼
ð0:25g2cod ðZÞ
þ
cod eaT Lbasic C2 ÞB2
þ m2 P2 ðH; SÞ;
ð38Þ cod LCi N Bcod ¼ eaT Lbasic i C2 B2 þ mi Pi ðH; SÞ;
for
i ¼ 3 to 7:
ð39Þ
ð
5 X i¼1
7 X i¼1
LSi N BSpra þ i
Þ
6 X
LHi N Bher i
i¼1
LCi N Bcod : i
Apart from biomass losses through fishing mortality, the total biomass must strictly be conserved. Mortality of fish constitutes a further pathway to the model detritus. This implies for the NPZD model that an input
W. Fennel / Journal of Marine Systems 71 (2008) 171–194
term LFD, (Loss of Fish to Detritus), is required in the equation of detritus, Eq. (35), LFD ¼ fcon þ
ð
5 X
LSi D BSpra þ i
i¼1
7 X i¼1
LCi D Bcod i
Þ
6 X i¼1
þ fcon ðlS1 BSpra 1 ð40Þ
To describe the loss of fish biomass to detritus through excretion, we assume a similar relation as for he respirational losses, LCi D ¼ LCi N ;
LHi D ¼ LHi N ;
LSi D ¼ LSi N :
show all the different contributions. The cod biomass equations are d cod cod B ¼ OSC6 Bcod 6 þ OSC7 B7 dt 1 cod þðg1cod LC1 N LC1 D lC1 ÞBcod 1 s C1 B 1 ;
LHi D Bher i
cod starv Spra þ lH1 Bher 1 þ lC1 B1 þ ðlS5 þ lS5 ÞB5 starv her þ lH5 Bher 5 þ ðlH6 þ lH6 ÞB6 starv cod þðlC7 þ lC7 ÞB7 Þ:
187
ð41Þ
The mortality rates are specified in the next subsection. While LFN acts only on the fish biomass, the loss of biomass due to mortality included in LFD, affects also the number of fish. 5.3. The model linkage — seen from the fish model The general structure of the fish model, as governed by the equation sets (20)–(23), remains largely unchanged. However, we have to adjust a few expressions in the equations to implement the link to the NPZD model. First the grazing rates of fish on zooplankton are now food limited by a limiting function of the type (36), while in the uncoupled system the food for prey fish was not limited. With respect to the predator prey interaction we modify the Ivlev constant in Eq. (12) by enhancing it by two orders of magnitude, i.e., Iν is changed to ICH and ICS, which are now of the order of ICH ∼ ICS ∼ 10− 7 km3 g− 1 . This is motivated by the results of the example simulation in Subsection 4.3, where the development of cod was much to slow for a reasonable amount of prey, but then the growth was as expected after the prey field have grown to very high concentration of biomass. In the case of a finite system with limited resources a reasonably fast growth can only be reached through an increased Ivlev constant. The growth rates of the fish biomass variables is also assumed to depend on temperature by an Eppley-factor, exp(aT), i.e., we replace the G(B)'s in Eqs. (17)–(19) by G(B) exp(aT). We write the full equation set explicitly to
d cod cod cod B ¼ sC1 Bcod 1 þ ðg2 ðLC2 þ LC2 D ÞB2 dt 2 sC2 Bcod 2 þ P2 ðH; SÞ; d cod cod cod B ¼ sC2 B2 ðLC3 N þ LC3 D ÞBcod 3 sC3 B3 þ P3 ðH; SÞ; dt 3 d cod cod cod B ¼ sC3 Bcod 3 ðLC4 N þ LC4 D ÞB4 sC4 B4 þ P4 ðH; SÞ; dt 4 d cod cod cod B ¼ sC4 Bcod 4 ðLC5 N þ LC5 D ÞB5 sC5 B5 þ P5 ðH; SÞ; dt 5 d cod cod B ¼ sC5 Bcod 5 ðLC6 N þ LC6 D þ OSC6 þ FC6 þ lC6 ÞB6 dt 6 sC6 Bcod 6 þ P6 ðH; SÞ; d cod cod B ¼ sC6 B6 ðLC7 N þ LC7 D þ OSC7 þ FC7 þ lC7 dt 7 cod þlstarv ð42Þ C7 ÞB7 þ P7 ðH; SÞ:
We have included natural mortality rates, μC1, μC6, μC7, for the first and the two largest mass classes. Moreover for mass class ‘7’ we have also introduced a starvation mortality. For small fish, such as sprat, the starvation mortality can be involved when the individuals mass falls below the lower limits of the corresponding mass interval. For adult cod such an approach is not reasonable, because the interval along the mass axis would be too wide and it would take years before starvation mortality could be activated. Therefore, we choose an approach where starvation mortality is involved if the effective growth rate becomes negative, i.e., loss exceeds the consumption of food. lstarv C7 ¼ 0:005hðLC7 N þ LC7 D P7 ðH; SÞÞ=d:
ð43Þ
An analogous approach is applied to herring for the uppermost mass class. For the largest mass class of all three species, we assume that the natural mortality is proportional to the biomass, i.e., 1010 cod 1012 her B7 ; lH6 ¼ B6 ; and d d 1012 Spra B5 : lS5 ¼ d
lC7 ¼
ð44Þ
This approach implicates a ‘carrying capacity’ of each group.
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The full equation set for the abundance of cod is explicitly, d cod N dt 1 d cod N dt 2 d cod N dt 3 d cod N dt 4 d cod N dt 5 d cod N dt 6 d cod N dt 7
1 Bcod cod cod 1 ðosC6 Bcod ; 6 þ osC7 B7 Þ lC1 N1 sC1 CX1 m0 Bcod Bcod ¼ sC1 1 lCT2 N2cod sC2 2 ; CX1 CX2 Bcod Bcod cod 2 T ¼ sC2 lC3 N3 sC3 3 ; CX2 CX3 Bcod Bcod cod 3 T ¼ sC3 lC4 N4 sC4 4 ; CX3 CX4 Bcod Bcod ¼ sC5 4 lCT5 N5cod sC5 5 ; CX4 CX5 Bcod Bcod 5 ¼ sC5 ðlC6 þ lCT6 þ FC6 ÞN6cod sC6 6 ; CX5 CX6 Bcod starv cod 6 ¼ sC6 ðlC7 þ lC7 þ FC7 ÞN7 : CX6 ¼
The equations for the abundance are, d her 1 her her N ¼ ðosH5 Bher 5 þ osH6 B6 Þ lH1 N1 dt 1 m0 Bher C 1 ðherÞ sH1 1 ; HX1 mher 1 her d her B Bher P2 ðherÞ N2 ¼ sH1 1 lHT2 N2her sH2 2 ; dt HX1 HX2 mher 2 her her d her B B P3 ðherÞ N ¼ sH2 2 lHT3 N3her sH3 3 ; HX2 HX3 dt 3 mher 3 her her d her B B P4 ðherÞ N ¼ sH3 3 lHT4 N4her sH4 4 ; dt 4 HX3 HX4 mher 4 her her d her B B P5 ðherÞ N ¼ sH4 4 ðlHT5 þ FH5 ÞN5her sH5 5 HX4 HX5 dt 5 mher 5 d her Bher P ðherÞ 6 starv her 5 N ¼ sH5 ðFH6 þ lH6 þ lH6 ÞN6 : HX5 dt 6 mher 6
ð48Þ
ð45Þ For the middle mass classes, i = 2 to 6, we have introduced further starvation mortalities, μ⁎i , which act only on the abundances and mimics the death of a certain amount of ill-conditioned animals. These mortality rates are ‘switched-on’ as soon as the average individual mass falls below the lower limit of the corresponding mass class, i.e., lCTi ¼ 0:05d 1 Hðmi 0:8CXi1 Þ:
ð46Þ
Since, contrary to the biomass, the number of individuals is not constraint by the law of mass conservation, this approach is consistent and can be useful. An analogous definition is applied for herring, i = 2 to 5, and sprat, i = 2 to 4. We write also all equations for the prey groups herring and sprat explicitly. For the herring biomass it follows, d her her her B ¼ osH5 Bher 5 þ osH6 B6 þ ðg1 LH1 N dt 1 her LH1 D lH1 ÞBher 1 sH1 B1 P1 ðherÞ; d her her her her B ¼ sH1 Bher 1 þ ðg2 LH2 N LH2 D ÞB2 sH2 B2 dt 2 P2 ðherÞ; d her her her her B ¼ sH2 Bher 2 þ ðg3 LH3 N LH3 D ÞB3 sH3 B3 dt 3 P3 ðherÞ; d her her her B ¼ sH3 Bher 3 þ ðg4 LH4 N LH4 D lH4 ÞB4 dt 4 sH4 Bher 4 P4 ðherÞ; d her her B ¼ sH4 B4 þ ðg5her LH5 N LH5 D ÞBher 5 dt 5 her ðosH5 þ FH5 þ lH5 ÞB5 sH5 Bher 5 P5 ðherÞ; d her her her her B ¼ sH5 B5 þ ðg6 LH6 N LH6 D ÞB6 ðosH6 dt 6 her þFH6 þ lH6 þ lstarv H6 ÞB6 P6 ðherÞ: ð47Þ
For sprat biomass the full equation set reads, d Spra ¼ osS4 BSpra þ osS5 BSpra þ ðgS1 LS1 N LS1 D lS1 ÞBSpra B 5 4 1 dt 1 Spra sS1 B1 P1 ðSÞ d Spra ¼ sS1 BSpra þ ðgS2 LS2 N LS2 D ÞBSpra sS2 BSpra P2 ðSÞ; B 1 2 2 dt 2 d Spra Spra Spra Spra ¼ sS2 B2 þ ðgS3 LS3 N LS3 D ÞB3 sS3 B3 P3 ðSÞ; B dt 3 d Spra ¼ sS3 BSpra þ ðgS4 LS4 N LS4 D ÞBSpra ðosS4 þ FS4 ÞBSpra B 3 4 4 dt 4 sS4 BSpra P ðSÞ; 4 4 d Spra Spra þ ðgS5 LS5 N LS5 D lS5 lstarv B5 ¼ sS4 BSpra S5 ÞB5 4 dt ðosS5 þ FS5 ÞBSpra sS5 BSpra P5 ðSÞ: 5 5
ð49Þ
The equations for the sprat abundance follow as, d Spra 1 BSpra Spra Spra 1 ðosS4 BSpra þ os B Þ l N s N1 ¼ S5 S S 1 5 4 1 1 SX1 dt m0 P1 ðSÞ Spra ; m1 d Spra BSpra BSpra P2 ðSÞ N2 ¼ sS1 1 lST2 N2Spra sS2 2 Spra ; dt SX1 SX2 m2 Spra d Spra BSpra B P3 ðSÞ Spra 2 3 T N ¼ sS2 lS3 N3 sS3 Spra ; dt 3 SX2 SX3 m3
d Spra BSpra BSpra P4 ðSÞ N4 ¼ sS3 3 ðFS4 þ lST4 ÞN4Spra sS4 4 Spra ; dt SX3 SX4 m4 Spra d Spra BSpra B Spra sS5 5 N5 ¼ sS4 4 ðFS5 þ lS5 þ lstarv S5 ÞN5 SX4 SX5 dt P5 ðSÞ Spra : ð50Þ m5
A compilation of the involved parameters is given in Table 5.
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6. Model experiments After the development of the model we will perform some numerical experiments to study the model performance and try to simulate qualitatively effects consistent with the general knowledge of the system. We conduct two runs over forty years with the initial distributions of the fish variables as shown in Fig. 15. In the first run the fishing mortality is set to zero, while in the second run a fishing mortality is imposed on the adult mass classes of all three fish groups. Apart from the question, how the fish model reacts on the food limitations for the zooplanktivors, we can also ask, how the biogeochemical model of the lower food web is affected by the interactive link to the fish model. The background mortality of the model zooplankton was reduced by about 30% in the interacting model, to lZD = 0.02 instead of lZD = 0.03 in the truncated NPZD model. In the interacting model the zooplanktivors sprat and herring are feeding on the zooplankton and impose an additional predation mortality on the bulk
189
zooplankton. Since the control of the NPZD model by physical parameters, i.e. light and temperature, are kept stationary in this model version and since the system is closed, i.e. no imports or exports, except fishing, it is clear that all visible variations in the state variables are imposed by signals propagating down from the fish model. 6.1. Zero fishing mortality First we run the model without any fishing mortality. The rationale is that a stable state of the coupled model system with no anthropogenic influence must, in principle, exist. The spectra of the initial distributions are shown in Fig. 15. Some results of a 40 years simulation are shown in Fig. 16 by means of the total abundance for sprat, herring, and cod, (upper panel). The development of the total biomass of the three fish groups and the zooplankton is shown in the lower panel of Fig. 16. The zooplankton biomass was converted from carbon units per m3 to biomass per km3.
Fig. 15. The initial distribution of the biomass and abundance of the different mass classes of cod (top), herring (middle), and sprat(bottom) as used in the model.
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Fig. 16. Time development of the total abundance of cod, herring and sprat (top), and of the biomass of zooplankton, cod, herring and sprat (bottom) for 40 years simulation (No fishing mortality).
While the adjustment of the total prey biomass, (sprat and herring), occurs in the first few years, cod needs about 15–20 model years to adjust from the initial conditions to a quasi-steady state. During the second time slice, 20–40 model years, the variations in the model variables are small and reflect basically the annual cycle. Only for cod we observe an interannual oscillation in the total biomass with a period of about three years that is not seen in the total numbers of cod. This oscillation
can be attributed to the large mass interval of adult cod and the associated long development time within the interval, compare Fig. 3. Before the biomass dependent mortality becomes large enough to reduce the total mass of living fish, a growth phase of several model years is required. After a phase of enhanced mortality, some time is needed until the biomass increases again. A comparison of the results of the truncated NPZD model, Fig. 14, with the results of the coupled model for
Fig. 17. The undisturbed, (no fishing mortality), development of the annual cycle of the NPZD-model component with interaction with the fish model.
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Fig. 18. Time development of the total abundance of cod, herring and sprat (top), and of the biomass of zooplankton, cod, herring and sprat (bottom) for the 40 years simulation with ‘high fishing mortality’.
the time slice from 20 to 40 years, Fig. 17, shows that both models produce very similar yearly cycles of the state variables. However, in the coupled model is somewhat less material in the nutrient, detritus and plankton state variables. The missing potential biomass was transferred to the fish variables during the spin-up process and the mutual adjustment of both model components.
6.2. High fishing mortality In our second experiment we apply a relatively high fishing mortality of the largest mass classes of herring and sprat and the two largest mass class of cod, see ‘case 2’ in Table 4. Fig. 18 shows the time development of the total abundance of cod, herring and sprat, (upper panel),
Fig. 19. Time series of the annual catches of cod, herring and sprat for 40 years as seen in the coupled model.
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Fig. 20. As Fig. 17 but with fishing mortality.
and of the biomass of zooplankton, cod, herring and sprat, (lower panel), for 40 years. The development of the zooplankton and the fish biomass in Fig. 18 indicates an adjustment time of about 10 to 12 years before a stable regime with a clear annual cycle is reached. However, due to fishing mortality, the biomass of the system is decreasing. The catches of cod, herring an sprat, which are controlled by the choice of the fishing mortality rates are shown in Fig. 19. The catches of cod show oscillations with a period of 5 to 6 years. For a comparison of the truncated NPZD model, Fig. 14, with the coupled model with fishing mortality we show the state variables of the biogeochemical model for the time slice of 20 to 40 years in Fig. 20. Since fishing mortality implies export of biomass through fisheries, the models system is not closed. Hence we expect a slow reduction of the potential biomass in all state variables, including the lower part of the food web. In the NPZD model component, it appears that in relation to phytoplankton, the zooplankton biomass decrease stronger, Fig. 20. The spring bloom peaks broaden and the signature of the yearly cycle might indicate some eutrophication. This can be explained by the decrease of cod, implying a higher grazing pressure of sprat and herring on zooplankton, and can be considered as a tropic cascade. 7. Conclusions The paper describes work towards bridging biogeochemical and fish models. The basic ideas and a first
implementation in terms of a spatial simple box model are outlined. The model applies to the Baltic Sea, which was chosen because of its relatively simple food web. To construct the model, the fish populations were structured into a set of mass classes, where the choice of mass intervals was motivated by the need for consistent size relations between prey and predator. The formulation of the model equations is based on the quantitative description of predator–prey and prey–predator interactions. The model takes explicitly the reduction of prey biomass and abundance in relation to the growth of predators into account. This model concept does not require considerations of specific cohorts, but can treat several different overlapping cohorts. The fish model was then linked to a simple version of a NPZD model. A challenge of this approach arises from the necessity to deal with rather different time scales. The NPZD model works on seasonal to yearly time scales. The primary production starts the development of the food web and is controlled by the annual variations of the physical forcing. Zooplankton and detritus operate on the yearly time scales, while the fish population model involves life cycles of 10 to 20 years. Thus, it takes several years for a nutrient signal to propagate through the food web within the mass classes of prey and predators. The described approach provides a first attempt to interlink models and assesses the plausibility of such a theory. Further work needs to be done to consolidate the underlying assumptions and the choices of parameters. The most critical mechanisms to formulate in the model are reproduction and mortality of the adult stages.
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Reproduction involves the rather complex recruitment process from egg production through the fragile larval stages and includes the match–mismatch problem. This process was in the current model version abridged by an off-spring approach. With regard to mortality, we have chosen a density dependent approach for the adult, where the mortality rate of the adult depends on their biomass. This is a way to avoid exaggerated accumulations of biomass in the adult stages, in particular for zero fishing mortality. The model system must be able to generate a stable natural state without removal of biomass through fisheries. The biomass dependent mortality of the adult stages amounts to introducing a ‘carrying capacity’ of the considered system. The growth rates are derived from data by mapping fitted Bertalanffy growth curves onto piece-wise constant rates. The share of consumed food used for basic metabolic and other losses and for growth was roughly assumed to vary from to 50/50 for cod and herring and 25/75 for sprat. This is a crude approach that needs to be refined by detailed information on metabolism, specific dynamic action, egestion and excretion, as for example discussed in Megrey et al. (2007b). However, since the purpose of this paper was to develop a theoretical approach, it seems to be sufficient that our bulk assumptions are in the same range. Some parameters like mortality are more difficult to constrain and are chosen as plausible values. We have conducted and described several example— simulations, which showed that the model is in principle consistent. Experiments with high fishing mortality showed an interannual variability of cod catches, which resemble qualitatively time variations in data derived by multi-species stock assessment methods for the Baltic, see Anonymous (2005). Although such variations are often attributed to bottom up effects, it is interesting that the variations seen in the model are entirely due to the interactions of prey and predators in conjunction with the different time scale of their life cycles. To check whether the order of magnitudes of the modeled catches correspond to observed data, we can use, for example, the total catches of 45 tons km− 3 for the model year 30, see Fig. 19. Multiplying this by the volume of the central Baltic, 13 · 103 km3, gives an overall annual catch of 585 · 103 tons. This value is somewhat less than of the catch data of about 700 · 103 tons, for the 1970ties, Thurow (1997), when the observed winter level of nitrate in the surface waters was about 3 mmolN m− 3, Nehring (1995). The nitrate value corresponds to the winter level for the year 30 in Fig. 20. Note that the value of 20 mmolC m− 3 is equal to 3 mmolN m− 3·rRedfield with
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rRedf ield = 6.625. For the model year 15 the nitrate level corresponds to 4.5 mmolN m− 3, (not shown), which was observed in the 1980ties, Nehring (1995). The catches in the model as shown in Fig. 19 are 75 tons km− 3, implying for the central Baltic 75 tons km− 3 · 13 · 103 km3 = 975 · 103 tons, which is surprisingly close to the catch data, Thurow (1997). This is encouraging although the relative distribution of the catches of the different species was not yet reproduced by the model. Apart from consolidating the model equations and the parameters choices, future work will include the spatial dimension, the distribution of fish stocks in response to environmental variations, as well as a more detailed description of egg laying and match–mismatch consideration in time and space. In a three dimensional perspective it is obvious that different stocks, which grow differently fast at different places, will intermingle due to currents or active migration. Such a system can consistently be described with our approach. The discussion of integrated model systems with the potential to describe food web processes can be helpful to understand and quantify the consequences of truncated food webs in models. The usually applied truncation of food web processes in biogeochemical models implies parameterization of unresolved processes in terms of resolved ones. How appropriate such constructed parameterizations are can only be assessed by experiments with models of higher process resolution. 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, vol. 7. Contributions from the Danish Institute for Fishery and Marine Research, Charlottenlund. Anonymous, 2005. Report of the study group on multispecies assessment in the baltic (SGMAB). Report ICESCM 2005/H:06, ICES. Beverton, R.J.H., Holt, S.J., 1957. On the Dynamics of Exploited Fish Population. Fish. Invest, London. Brown, D., Rothery, P., 1993. Models in Biology: Mathematics, Statistics and Computing. Wiley, Chichester. Fennel, W., 2001. Modeling of copepods with links to circulation models. Journal of Plankton Research 23, 1217–1232. Fennel, W., Neumann, T., 2004. Introduction to the Modelling of Marine Ecosystems, 1st Edition. Elsevier Oceanographic Series, vol. 72. Elsevier, Amsterdam. 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 stock-production model. Journal of Applied Ichthyology 21, 198–204.
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Megrey, B.A., Rose, K.A., Ito ichi, S., Hay, D.E., Werner, F.E., Yamanaka, Y., Aita, M.N., 2007a. North Pacific basin-scale differences in lower and higher trophic level marine ecosystem responses to climate impacts using a nutrient–phytoplankton–zooplankton model coupled to a fish bioenergetics model. Ecological Modelling 202, 190–210. Megrey, B.A., Rose, K.A., Klumb, R.A., Hay, D.E., Werner, F.E., Eslinger, D.L., Smith, S.L., 2007b. A bioenergetics-based population model of Pacific herring (Clupea harengus pallasi) coupled to a lower trophic level nutrient–phytoplankton–zoo-
plankton model: description, calibration, and sensitivity analysis. Ecological Modelling 202, 144–164. Nehring, D., 1995. Nährsalze. In: Rheinheimer, G. (Ed.), Meereskunde der Ostsee. Springer-Verlag, Berlin, pp. 97–103. Richards, F.J., 1959. A flexible growth function for empirical use. Journal of Experimental Botany 10, 290–300. Thurow, F., 1997. Estimation of the total fish biomass in the Baltic Sea during the 20th century. ICES Journal of Marine Science 54, 444–461.
