Systems
Dynamics in the Fisheries of Northeast Arctic Cod

*Harald
Yndestad*

*Aalesund
College*

*Fogd
Grevesveg 9, 6009 Aalesund, Norway*

**Abstract**

Northeast Arctic cod is one
of the largest stocks of cod in the world. The cod biomass is influenced by temperature
cycles related to the earth nutation. A stationary dynamic cycle is estimated
in the recruitment and causes a dynamic instability between the biomass and the
quota of landings. The deterministic property of the cycle opens a new
perspective on forecasting biological economical resources in the Barents Sea.

**Keywords**

Northeast Arctic cod; Earth nutation; Stationary cycles; Forecasting;

**1 Introduction**

Northeast Arctic
cod is one of the largest stocks of cod in the world. During centuries this
stock of cod has been of utmost importance to the economic growth of western
Norway. People living by fishing have always known that the stock of cod has
dynamic properties. Some years the influx of cod is abundant and some years the
influx may be insufficient in relation to the demand. Better forecasting over a
time span of 5-10 years, will be crucial for better
planning of an economical and sustainable utilisation of the cod biomass.

In 1994 it was
found that the time series related to the quantities of North Atlantic cod has
a 6-7 year
cycle in the Fourier spectrum amplitude. The question then was: Is this a stationary cycle? If
so, there is a possibility of a more precise prediction of ecology resources
and hence a proper basis for fishery investments. Further research
revealed the
cycle in the fry abundance and even in the cod quotas. The same cycle was found
in the temperature of the Barents Sea and a relation was found between this
temperature cycle and the earth nutation, as the cause of causes (Yndestad,
1994c).

This paper focuses on some
methodical aspects of system dynamics of the Northeast Arctic stock of cod
related to a general system theory.

**2 System theory**

*A
system *is a set of social, biological, ecological, technological or material
partners co-operating for a common purpose. A doctrine of a general system
theory (Yndestad, 1996a) may be formulated by the dual views

System = System
ontology + System Epistemology

System ontology =
System Architecture + System Dynamics

System
Epistemology = System Ethic + System Learning

A general dynamic system S(t) may be expressed by a set of partners P(t) related by
the binding A(t).

S(t) = {A(t),{p_{1}(t),
p_{2}(t),,, p_{n}(t)}, n = f(t)} Î w

Where w is the common
purpose and n is the number of partners at the time t. In this case the purpose
is a sustainable biomass. The partners in the system are the earth system, the
temperature system in the Barents Sea, the ecological system in the Barents
Sea, the cod biomass system in the Barents Sea, the cod landings system and the
cod market system. The binding A(t) is the relation
between the partners. This theory implicates that systems are non-linear and
structural unstable by nature. When the state of one partner is changed, it
will affect all other partners. Systems are recursively made of subsystems

S_{i}(t) = {A_{i-1}(t),
S_{i-1}(t)} Î w

Where i is the abstraction level i = {0,1, 2,,}.
This implicates that systems are made of systems at lower abstraction levels.
In this case all systems from the market system to the planetary system have
subsystems. A consequence of this theory is that system dynamics on one level
will influence the state of all partners in the system. This property explains
why time series measured in systems has a Wiener spectre where the amplitude is
falling by 1/frequency. There is no theory for how to predict dynamics in such
a complex system. Even if we had online access to all the states in the system,
it would be extremely difficult to forecast 5 to 10 years ahead in time.

System epistemology is the
dual view of system ontology that represents a non-deterministic view of the
reality. Where *system ethics* is the
aim a system is striving fore. *System
learning* is the strategy, which a system chooses to reach the aim. Through
this doctrine, system is something more than a sum of interacting parts.
Systems are a dynamic process where the dynamics is a synthesis of the system
architecture, systems ethics and a free will of learning. This theory implies
that systems have knowledge to adapt to the dynamics of its partners. In this
case it means that the ecological systems in the Barents Sea have learned how
to take advantage of the stationary dynamics. The market system and the landing
system are adapting to the system dynamics by learning more about the
ecological system.

**The
cycle theory**

The theory in this paper is
based that system has a stationary cycle of 18.6 years related to the earth
nutation. According to the general system theory, a dynamic change in one
partner will influence the state of all others. In this case the earth nutation
represents a dominant energy compared to the other partners in the system. The
other partners are the temperature balances in the Barents Sea, the ecological
system in the Barents Sea, the cod biomass system, the landing system and the
market system. The earth nutation represents a small cyclic change on the earth
rotation, but since the other partners have much less energy, they will in the
long run learn how to adopt to the stationary dynamics
related to the earth nutation.

**3 System identification**

According to the general
system theory, systems have mutual influence on common partners.

**Earth
dynamics**

The earth dynamics is
influenced by the dynamics of the planetary system. The most important earth
dynamics p_{e}(t) has the relation

p_{e}(t) = f(w_{d}t, w_{y}t, w_{n}t, w_{p}t)

Where t is the time, w_{d} represents the
angle frequency of the earth axis rotation, w_{y} the earth
rotation around the sun, w_{n} the 18.6 years
earth axis nutation and w_{p} the earth
precession of 25800 years. Each of these cycles will influence the dynamics of
all the other partners in the system. The effect of the cycles w_{y} and w_{n} are well known.
In this paper we will discuss the relation to the earth nutation cycle w_{n}.

**Temperature
dynamics**

In the Barents Sea the warm
Golf stream from the south meets a cold stream from north. Russian scientists
have measured the temperature in the Barents Sea each month since 1900
(Bochkov, 1982). The mean temperature shows changes between 2.7 and 5.0 degrees
Celsius over the last hundred years. In this data series three cycles related
to the earth nutation has been estimated (Yndestad, 1996b). The estimated
cycles are

Where T is a sampling
interval of one year and n is the number of years since the reference year
1900. The mean temperature A_{0} = 3.9 and the cycle amplitudes A_{1}
= 0.4, A_{2} = 0.6, A_{3} = A_{4} = 0.4 degree Celsius.
The angle frequency w_{1} = 3*w_{n} = 55.8 years, w_{2} = w_{n}, = 18.6 years
and w_{3} = w_{n}/3 = 6.2 years.
The phase delay from the reference year 1900 is estimated to be j_{1} = 12/12, j_{2} = 9.6/12 and j_{3} = 336/12 years.
The random noise v(nT) has an unknown source. Since
the cycles are correlated to the 18.6 years earth nutation, the cycles are
stationary and deterministic. The signal/noise - relation is estimated to be
about 1.

**Biomass
dynamics**

The biomass of North Arctic
cod has changed between 750.000 tons and 4250.000 tons the last fifty years.
This biomass dynamics is modulated by the difference equation

Where the X(nT)-vector
represents biomass of each age class, A(nT) is a (n x m) matrix that represents
the recruitment, the growth rate and the mortality rate at each age class,
U(nT) is the biomass of landings and B(nT) is a (m x m) matrix that distributes
the landing on each year class. The recruitment and dynamics of cod is
dependent on the dynamics of the food chain in the Barents Sea. Since the food
chain in the Barents Sea is temperature dependent, it will adapt to a
stationary temperature cycle. This means there is a binding between the
temperature cycle and the dynamics of Northeast Arctic cod. These
relation is estimated to be (Yndestad, 1996c)

Where x_{8+(}nT) is
the spawn biomass, x_{n1}(nT) is number of one year recruited cod, p_{m}
is the mean production rate (number of one year cod/spawn biomass) and U_{RT}(nT)
represents the dynamic binding between the recruitment and the temperature in
the Barents Sea. The dynamic binding to the temperature cycle is estimated to
be

This indicates that the
recruitment of Northeast Arctic cod is exponentially related to the 18.6/3 =
6.2 years temperature cycle. This means that if the biomass is known, future
dynamics in cod recruitment can be estimated.

The dynamics of biomass growth
may be computed by the growth rate. The growth rate of Northeast Arctic cod has
the relation

Where x_{i}(nT)
is the biomass at the age i, nT is the time and a_{i} is the growth
rate at the age i. This growth rate is expected to be influenced by the sum of
three stationary temperature cycles plus a random disturbance from an unknown
source. This may be expressed as

Where v(nT)
is a temperature disturbance from an unknown source and a_{n}
represents the binding to this source. In this case the estimated mean growth
rate for each year class is a_{00} = 3.33, a_{10} = 2.10, a_{20}
= 1.95, a_{30} = 1.61, a_{40} = 1.55, a_{50} = 1.47, a_{60}
= 1.35, a_{70} = 1.26. The temperature dependent growth rate for each
year class is estimated to be a_{01} = 4.00, a_{11} = 2.25, a_{21}
= 2.33, a_{31} = 1.71, a_{41} = 1.56, a_{51} = 1.43, a_{61}
= 1.35. a_{71} = 1.26. This indicates that the 6.2 years related growth
cycle has approximately the same amplitude the mean growth rate.

**Landing
dynamics**

Biomass collapse Rate R

Figure 1 Landing rate from the year
1946 to 1997

The quota of Northeast
Arctic cod has changed between 212.000 tons and 1.340.000 tons the last 50
years. This quota is influenced by biomass level, the capacity of the landing
system, the dynamics in the market system and the political decision system.
Thus the marked system, the landing system and the cod biomass system is a part
of a value chain system where each subsystem will influence the dynamics of the
other. A landing rate is a policy index that indicates the regulation of
biomass landed for distribution to the market. This landing rate may be defined
as

Where x_{3+(}nT) is
the cod biomass and j_{d} is a quota
decision phase delay. For years there has been a decision phase delay j_{d}. Recruitment and
the growth are heavy influenced by the 6.2 years temperature cycle. Bio
dynamics and decision delay thus introduces a positive feedback or instability
in the biomass. The landing rate since 1946 is shown on figure 1. The figure
shows a typical pattern of an unstable regulation of the cod-landing quota.
Periodical cycles of landing rate, a biomass collapse in 1987 and a warning of
a new biomass collapse in 1997. This instability is a serious problem for the
biomass of Northeast Arctic cod and the fishing industry in Norway. The landing
rate controls the biomass of cod. When introducing the landing rate, in the
biomass model, we get the autonomous dynamic system

In this case we have
a-priori information on expected recruitment and growth related to
deterministic temperature cycles. A feed forward strategy and a feedback
strategy may then control the system dynamics of Northeast Arctic cod. The feed
forward strategy will take care of dynamics related to temperature dependent
growth and a feedback strategy will take care of dynamics in mortality.

**4 Discussion**

Time series of system
dynamics in nature usually have a Wiener spectre
(1/frequency) amplitude. In this case we have something different. The
stationary temperature cycle influence seems to influence the ecological and
the economical system of Northeast Arctic cod. A modulated third harmonic
temperature cycle of 6.2 year influences the recruitment and the stability of
quota regulation of landing. The 18.6 year cycle and the 3*18.6 = 55.8 year
cycle influence the maximum biomass level. If the temperature cycle theory is
confirmed by hydrographic mesuarment, it will a give a
new perspective to the understanding of the ecology dynamics in the Barents Sea
and the optimisation of fishery resources.

In 1978 Harald Yndestad
received a degree in cybernetics at University of Trondheim. For 10 years he
made research on complex IT systems at NDRE in Norway and has since 1982 been
ass. professor at Aalesund College. The last years he
has been engaged in research on general system theory and system dynamics of
Northeast Arctic cod.

**6 References**

Bochkov,
Y.A: (1982) Water temperature in the 0-200 m layer in the Kola-Meridian in the
Barents Sea, 1900-1981. Sb. Nauchn. Trud. PINRO,
Murmansk, 46: 113-122 (in Russian).

ICES:
(1995) Report of the arctic fisheries working group. ICES
Headquarters. Copenhagen. Denmark.

Loeng,
Ottersen, m.f: (1994) Statistical Modelling of Temperature Variability in the
Barents Sea. ICES C.M. 1994. Havforskningsinstituttet
i Bergen. Norway.

Yndestad, H: (1996a) A
General System Theory. Aalesund College. Norway.

Yndestad,
H: (1996b) Stationary Temperature Cycles in the Barents Sea. The
cause of causes. The 84'th international ICES
Annual Science Conference. Hydrography Committee.
Iceland.

Yndestad,
H: (1996c) Systems Dynamics of North Arctic Cod. The 84'th international ICES Annual Science
Conference. Hydrography Committee. Iceland.