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The 18" International Conference of The System Dynamics Society
August 6-10, 2000. Bergen, Norway
The predestined fate
The Earth nutation as a forced oscillator on management of Northeast Arctic cod.
Harald Y ndestad
Aalesund University College, Box N-5104 Larsgaarden, 6021 Aalesund, Norway.
Tel: +47 70 16 12 00; fax: +47 70 16 13 00; e-mail: harald.yndestad@ hials.no
Abstract
The paper presents a system dynamics theory of the influence of the Earth's nutation on
management of Northeast arctic cod. According to this theory the Earth's axis dynamically
behaves as a forced oscillator on a non-linear sea system that modulates a set of harmonic and
sub harmonic low frequency temperature cycles in the sea system.
The paper reports a correlation between time harmonic cycles of the 18.6 year Earth nutation
and the temperature system and the biological system in the Barents Sea. The influence from
the Earth nutation is explained by a general systems theory where modulated temperature
cycles are forced oscillators on the biological system in the Barents Sea. The system dynamics
of the biological system are synchronized to the temperature cycle and amplified by a
biological stochastic resonance to the food systems. A stochastic resonance of 18.6/3=6.2 yr
between the management and the biomass dynamics introduces an unstable biomass.
Introduction
In the Barents Sea the inflow of warm North Arctic water meets a stream of cold Arctic water
from the north and cool mixed water circulates back to East Greenland. These streams may
vary in intensity and slightly in position and cause biological changes in the Barents Sea.
Since the first analysis by Helland-Hansen and Nansen (1909), changes in these streams have
been explained by climatic alterations in average wind and climatic variability (Loeng et al.,
1992; Dyke, 1996). There is, however, no clear answer as to how meteorological and
oceanographic conditions influence each other (Loeng et al., 1992).
Northeast Arctic cod is the largest stock of Gadus morhua cod in the world. The fishery of
this stock is located along the northern coast of Norway and in the Barents Sea. For centuries
this stock of cod has been the most important economic biomass for Norwegian fishermen
and of vital importance for settlement and economic growth in the western part of Norway.
4500000
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500000
0)
S
wy
[BStock |
PPP MOL KM Ar MM
PPS FL NS
> PP SY PO
LPP co o95
PPS * PP PS ST SY
lyr]
Figure 1. Time series of Northeast Arctic cod biomass stock from 1950 to 1998
The biomass of Northeast Arctic cod has always fluctuated and there have been several
theories on the causes of these fluctuations. Some years the influx of cod is abundant and
some years the influx may be insufficient in relation to the demand. People dependent on
fishing have always known the stock of cod has a short time and long time fluctuation. These
fluctuations have been explained by herring periods and cod periods, introduction of new
fishing equipment and more. Better forecasting in a time span of 5-10 years, will be crucial
for better planning of an economical and sustainable utilization of the cod biomass. When the
Norwegian marine research started at the beginning of this century, the main task was to
uncover how the nature influenced the stock of cod and the impact of fluctuation on people
living by fishing (Rollefsen, 1949).
Years of fluctuation in the biomass and the landings have been explained by limited food
resources, cannibalism, changes in landings from number of cod to tons of cod, to high quota
of landings (Nakken. el al., 1996) and assessment methods. Early scientific explanations of
cod fluctuations were changes in food, mortality, hydrographic relations, sea temperature and
positive feedback in recruitment (Rollefsen, 1949). More recently the fluctuations of
Northeast Arctic cod has been described in more detail (Nakken, 1994), but the fundamental
explanations are much the same.
I 1994 a modeled projection of the lifetime earning capacity of a Norwegian trawler was
carried out as an item of contractual research. An autocorrelation of the biomass indicated it
was not realistic to predict future biomass more than one year ahead. This motivated to look
for a more fundamental cause of marine fluctuation. By chance it was found that time series
of North Atlantic cod has a dominant 6-7 year cycle in the autocorrelation and the Fourier
amplitude spectrum. The same dominant cycle was found in cod recruitment and landings. A
next question was is this a stationary cycle? If this is a stationary cycle, this is a cause of
causes that has the information we may use to estimate future biomass and future quota of
landings. The income from the trawler than will be predestinated by the timing between the
biomass cycle and the trawler investment. The next step was to look for the source of the
stationary cycle.
The system dynamics doctrine from Newton is based on a ballistic view of reality. Energy is
flowing from one object to the next and in this flow delay will introduce dynamics. This
doctrine is realistic when the objects have stable relations. Free will and structural dynamics
changes relations between objects in nature. When the relations between objects is changed,
the dynamic property is changed and an uncertainty is introduced. This introduces an
fundamental limitation in forecasting by the Newton law of system dynamics.
Aristotle had a different doctrine of system dynamics in nature. He explained the dynamics
of objects by the four causes, namely the efficient cause, the material cause, the structure
cause, and the predestined fate. Predestined fate was decided by the “cause of causes”; the
positions of the Sun, the Moon and the stars. The Aristotle doctrine of the predestinated fate
represents a dual view of the reality. By this doctrine planetary dynamics will introduce
dynamics that sooner or later will influence all objects in nature.
In 1543 Copemicus associated the change of star positions with a changing direction of the
rotational axis of the Earth. Isaac Newton explained in “Principia” that the Earth is a spinning
object where the axis describes a circle about the North Pole. This motion is called
"precession" and proceeds, in about 25 800 yrs, along a cone with a half apex angle of 23.439
degrees and moves along the elliptic by 50.291 arc s yr‘. In 1754 Kant predicted that friction
with tidal forces on the Earth would cause a deceleration of the Earth's rotation. Euler
predicted in 1758 that the rotation of the Earth’s axis would slow the Earth's motion with
respect to an Earth-fixed reference frame (polar motion). Some years later in 1776 Laplace
made theoretical tidal modeling involving periodic hydrodynamics on a rotating sphere. In
the eighteenth century the English astronomer Bradley discovered that the Earth’s rotational
axis wobbled around the precession cone. This change was called the “nutation”. Better
instrumentation slowly modified the view of the movement of the Earth as a stable dynamic
process. Earth axis dynamics are now described by the four components: precession,
nutation, celestial pole offset, and polar motion. The nutation has an amplitude of 9.2 arc s
and a 18.6 yr cycle caused by the Moon. By new high-precision measurements more than
100 frequency components in the nutation have been discovered. The four dominant cycles
of the nutation are 18.6 yrs (precession period of the lunar orbit), 9.3 yrs (rotation period of
the Moon's perigee), 182.6 d (half a year) and 13.7 d (half a month). New geodetic
techniques now make it possible to detect Earth displacement influenced by the tide, the
Earth core and mantle, and from atmospheric disturbance.
This predestinated dynamics may have an important influence on the ecological change in
nature. In 1938 professor Petterson explained fluctuation of herring by a tidal 112 yr cycle
(Rollefsen, 1949). This is an sub harmonic cycle of the Earth nutation of 18.6 yrs. In
historical records of cod landings in Norway, Ottestad (1942) reported 11, 17.5, 23 and 57 yr
cycles of cod. These cycles are related harmonic and sub harmonic cycle of the Earth
nutation of 18.6 years (Y ndestad, 1999b). There is reported a correlation between the Earth
nutation and the temperature in the Barents Sea (Yndestad; 1996a; 1999a), a correlation
between the Earth nutation and the biomass of Northeast Arctic cod (Wyatt et al. 1992, 1994;
Yndestad, 1996b, 1999b) and a correlation to management of Northeast Arctic cod.
This paper focus on the system dynamics methods from two papers (Yndestad; 1999a,
1999b). The paper explains by general systems theory how Earth nutation influences
dynamics in the food chain from planetary dynamics to management of Northeast Arctic cod.
Materials and methods
The prepared temperature series is taken from the Kola section (Bochkov, 1982). The data are
measured along 33°30'E from 70°30'N to 72°30'N and have a sampling time of 1 month from
1900 to 1994. All history time series on Northeast Arctic Cod are based on the Report of
Arctic Fisheries (ICES, 1999). The biomass time series from 1999 to 2020 are forecasted by
the author and based on temperature dependent growth and a recruitment model (Y ndestad,
1999b).
General systems theory
General system theory is a means of understanding abstract organizations independent of time
and space. A system is a set of subsystems cooperating to a common purpose. This may be
expressed as
S(t) = {B(t),{Si(0, S2.(0..., Sal} } Ew (1)
where S(t) is the system, Sj(t) is a subsystem, B(t) is a dynamic binding between the
subsystems and w is the common system purpose. According to the general theory systems
are time varying, structurally unstable and mutually state dependent.
We have a planetary system: S(t)={B,(t),{S-(t),Sm(t),S.(t)}}} where S,(t) represents the Earth
system, S,,(t) the Moon system, S,(t) the Sun system, and B,(t) is the mutual dynamic
binding. In this case the planetary system represents a stable periodic system. The Earth
system has the subsystems S,(t)={B.(t),{Sp(t),Sq(t),Sw(t),S.(t),Sy(t)}} where S,(t) the Earth
axis system, S,(t) a warm Atlantic flow system, S,(t) a cold-water stream system, S,(t) an
unknown disturbance system and B,(t) the dynamic binding between the systems. The Barents
Sea system S,(t) has a set of food chain sub systems where S,,,(t) is the management system
and temperature system S;(t) has a set of sub systems in the sea.
System state dynamics
The state dynamics of the system element S(t) is described by the state space equation
a(t) =A(t) -x(t) 4B(t) -ult) 4C(t)-vit) a
y(t) =D + x(t) +w(t)
where x(t) represents the state vector at a system element S,(t), v(t) a disturbance vector from
an unknown source, w(t) estimate noise, A(t) is the dynamic growth matrix, B(t) is the
dynamic binding matrix to the external element and D the measurement matrix and u(t) is the
state vector from an extemal element S,(t) In this case u(t) is the planetary dynamics where
most of the energy is related to a set of stationary periodic cycles
-Sy- (3)
u(t) “Zo cos(Ww,t +9, )
where M is the munber of cycles, un represents cycle amplitude, Ww, = 21V/T, the angle
frequency and ¢; a phase delay. The most important cycles are the Earth seasonal frequency
Wy = 2n/1 (rad/yr), the Earth nutation w, = 2r/18.6 (rad/yr) and the precession w, = 21/26800
(rad/yr). The autocorrelation of the periodic u(t) has the property
2
R,, (1) =E[u(t) «u(t +7) —E[u(t)}] =F cos(ut)
where tis the time displacement. This means that the autocorrelation has a stationary cycle if
the time series has a stationary cycle.
Wiener spectrum
The energy E, from the unknown source may be estimated by Parseval’s theorem
too to 2
E, = flv) =t fr ca dw (§)
If the spectrum V (jw) is white noise, the spectral density is
, ore (6)
Syo( ja) $V"( jw) P=V,"
where V0’ is the noise variance. In this case the integrated energy will be infinite. Since this is
impossible, the temperature spectrum must be colored. Such process may be modulated by the
first order process
' (7)
v(t) =-a -v(t) +n(t)
where n(t) is the none-correlated white noise. The autocorrelation of this process (5) is
2 8
R,, (1) =Elv(t) -v(t +1) -Elvt)]?] _ sel m
where tis a time displacement. This indicates that the autocorrelation function of the time
series v(t) from an unknown source is expected to fall exponentially. The frequency transform
of the first order process (5) is a non-correlated spectrum
(9)
. V
V(jw) =—*
(0) atjw
This indicates that spectrum of the measured time series, is expected to fall by Vo/(a+jw) and
the power density spectrum will fall by \VGjuyP. If the system is a part of a more complex
system, the measured spectrum is expected to fall by Vo/(atjw)”.
Frequency response
A next question is how a system element S,(t) is influenced by external sources. The external
source is a forced oscillator u(t) and the non-correlated disturbance v(t). This system is linear
when we have the stationary relations A(t)=A, B(t)=B and C(t)=C. In this case the Fourier
transform of (6) is
_ _BU(ju) ,C-V(ju) _
80) aa) Gea)
Hy (jo) U (jo) +H, (jo) V (ja) 1
where H,(jw) and H,(jw) are the frequency transfer functions. Equation (10) indicates that a
stationary cycle U(jw) is forced on a system element S,(t) it will introduce a cycle response at
X (jw) where the amplitude and phase is changed by the transfer function H,(jw). When a non-
correlated spectrum V(jw) is forced on the transfer function H,(jw) it will introduce a non-
correlated spectrum at X (jw) where all amplitude and phase cycles are changed.
According to the general systems theory (1) systems have a mutual dynamic binding B(t)
between sub systems. We may than expect that all system are more or less non-linear by
nature. It is known from non-linear theory (Moon, 1987) that non-linear system will modulate
a set of harmonic and sub harmonic frequency cycles. A forced cycle U(juy) on a non-linear
transfer function H,(jw) will than introduce the frequency response
X,, (J) =H (jo)x( jah) SH, *X(jw—ney fm)
According equation (10) and (11) a non-linear system Him) will introduce a set of harmonic
and sub harmonic cycles from the forced stationary cycles U(jw) and from the non-correlated
spectrum V (jw). An inverse transform of (10) and (11) gives us the general property
(12)
x(t) => Hen -sinl™ ust + cam (t)] +v(t)
where w is a periodic cycle, Hin,m) the cycle amplitude, n the harmonic number, m the sub
harmonic number, mm/(t) the phase delay and v(t) a disturbance from an unknown source
having a non-correlated spectrum.
Small stationary cycles may be amplified by a stochastic resonance. If we have a general
system.
S(t) = {B(t),{Si(), So()}} = Ew (13)
The system elements in S(t) may have the frequency transfer functions H;(jw) and H2(jw) and
a mutual binding B(t). Mutual binding is a feedback situation. It may be shown that the total
transfer function of two feedback system has the property
SQ) = Hy Gj) /(1 +H, GW) HpGw)) (14)
This system will have a stochastic resonance and a maximum amplification when
P,(jus) Po(ju) =-1 (15)
The system is said to have stochastic resonance when the system partners in S(t) are stochastic
systems. This means that the all systems and sub systems in the food chain may have a
stochastic resonance related to stationary cycles in the system.
Management of Northeast Arctic cod
Current management of Northeast Arctic cod is based on the system dynamics and the control
strategy
X(t) =A(t) -x(t) +B(t) u(t) +C (t) -v(t)
y(t) =D -x(t —t) +w(t)
u(t) =F (t)- y(t)
(16)
where u(t) is the quota of landings vector and the landing rate F(t) is the control parameter.
The landings rate F(t) has been changed each year. Future estimate of the biomass is than only
predictable from one year to the next. The biomass shift from one year to the next is
computed from sation (16).
x(t,) =e» xt COG) u(t dt =A(t,) “Le —I u(t.)
x(t,) {I +A(t,)ITx(t,) +Tu(t,) an
x(t.) {I +A(t,)IPx(t,) <P -F (t,)[D -x(t, —) +wit,)]
where v(t)=0, a one year time interval T=t,-tp and I is an identity matrix. Equation (17)
describes how this control strategy influences the biomass dynamics. The control of the
biomass is based on choosing a proper quota of landing u(t) that moves the biomass to the
wanted state x(t:). There are some fundamental problems related to this control strategy. The
growth matrix A(t) has time variant stationary cycles of 6.2 yr, 18.6 yr and 55.8 yr due to the
Earth nutation influence on the Barents Sea temperature. Estimates of the growth matrix will
than change each year and biomass dynamics will introduce errors in the estimated data. The
estimate delay tof 2-3 yr will introduce a phase error in the estimate. A combination of the
phase error tand the stationary cycle of about 6 yr in the growth matrix A(t), will introduce
an instability in the biomass. This means that the current control strategy will introduce three
different types of instabilities.
Results
Sea temperature dynamics
es
BS be wel
‘Temperature (Deg. Celsius)
2.5
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Time (Year)
rigure 2: 1 emperature ume senes Irom the barents 5ea
Figure 2 shows the low pass filtered time series of the temperature from the Kola section in
the Barents Sea from 1900 to 1994 (Y ndestad; 1994). The figure shows fluctuations where the
i Periodogram
rUED
10"
10
temperature is changing +/- 0.5 degree Celsius.
Figure 3. Power density spectrum of the temperature time series from the Barents Sea.
Figure 3 shows the power density spectrum of temperature time series in Figure 2. The figure
confirms some fundamental properties from the general system theory. The power density
spectrum of the temperature is falling by k/(atw)’. This confirms the theory of energy
distributions from equation (9). Most power density is concentrated at the 1 yr seasonal angle
frequency Wt = 6.28 (rad/yr) forced from the Sun. The seasonal 1 yr cycle from the Sun
generates the harmonics 2u. and 3, and there is a trace of the sub harmonics w./2 and w,/4.
This is according to the modulation theory (10) and (11). At the lower end of the spectrum
there are some indications of the nutation harmonics frequency w./2 = 0.6 (rad/yr) or 9.3 yrs,
u/3 = 1.1 (rad/yr) or 6.2 yrs, and at w/4 = 1.3 (rad/yr) or at 4.6 yrs. This confirms the
modulation theory (11) and (12). The time series is dominated by the 1 yr seasonal cycle. The
correlation between time series and the Earth cycles of 18.6/3=6.2 yr, 18.6 yr and
3*18.6=55.8 yr is found to be about 0.5 (Y ndestad; 1999a). This indicates there is a relations
between harmonic cycles of the Earth nutation and the temperature series in the Barents Sea
and thus confirms the relation described is equation (11) and (12). If this theory is confirmed,
there is a stochastic resonance in the flow of water that amplifies the cycles from the Earth
nutation. This cycle is a deterministic process that will change the climate and ecological
system in the Barents Sea.
This confirms there is a binding B(t) in the system S(t) = {B(t),{S,(t), Si(t)}}} = w where
S,(t) is the planetary system and S,(t) is the temperature system in the Barents Sea.
Cod biomass dynamics
Recruite
4000000
3500000
3000000 f\__ x
2500000 4 al
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1000000
500000 4
0
Oo OD ok Oo YD
Po of” of
—* Estimate
=m ICES
© WO Ah A® Ve oD oh of
PD AD PAP AP AP AP
Figure 4. Number of recruitment of Northeast Arctic cod (ICES) and estimated recruitment
from 1946 to 1998.
Figure 4 shows the time series of the number of 3 year Northeast Arctic cod since 1946 to
1998. The ICES data shows the time series from the official ICES reports (ICES, 1999). A
spectrum analysis of this data indicates this time series is correlated to the Earth nutation
cycles of 18.6 years and 18.6/3=6.2 yrs (Y ndestad, 1999b). This correlation indicates a
relations between harmonic cycles of the Earth nutation and the recruitment and the biomass
growth of Northeast Arctic cod in the Barents Sea. This relation is connected to the food chain
in the Barents Sea and thus confirms the relation described is equation (11) and (12). This
fluctuation may be explained by a chain of reactions. First the Earth nutation is a forced
oscillator on the sea temperature system. Than the temperature system is a forced oscillator on
the food chain in the Barents Sea. This fluctuations is amplified by a stochastic resonance
between the cod biomass and the food chain in the Barents Sea. This confirms there is a
binding B(t) in the biomass system S(t) = {B(t),{Sp(t), Sc(t), S(t), Si(t)}} =€w where S,(t) is
the planetary system, S,(t) is the cod system, S,(t) is the food chain system and S,(t) is the
temperature system in the Barents Sea.
The deterministic property of the nutation cycle may be used to tune a dynamic biomass
model of growth and recruitment. This is shown in Figure 4 where the estimated time series
is computed from a biomass growth model and the Earth nutation cycles are parameters
(Yndestad, 1999b).
Management dynamics
Landings rate
one oa b cere 8 a N
BERR REFERKFEREERSRB SS GF
2 aeaae & aa 2 aa
[yr]
Figure 5. History of current landings rate from 1950 to 1998.
The biomass level is controlled by the quota of landings. The landings rate F(nT) is a control
parameter has the relation
_u(aT) (15)
F(nT
07) a>)
where u(nT) is the quota of landings at the year n and y(nT) is the ICES estimated total
biomass at the year n. The biomass is sustainable when the landings rate about F(nT) <0.3
(Y ndestad, 1999b). Figure 5 shows the landings rate from 1950 to 1998. The landings rate
F(nT) has a 6-7 year cycles. This fluctuation demonstrates an unstable situation. The
harmonic cycles from the Earth nutation do not stop at the biomass system as described in
equation (11). The management of Northeast Arctic cod has a relation to the biomass and this
relation will influence the cod management. In this case there is a stochastic resonance
between cod management and the 6.2 yr nutation cycle that make the system unstable. This
confirms there is a binding B(t) in the biomass system
S(t)={B(t), {Sp(t),Sma(t),So(t),Si(t),S:(t)} } = Ew where S,(t) is the planetary system, S;(t) is the
temperature system in the Barents Sea, S,(t) is the food chain system, S,(t) is the cod system
and S,a(t) is the management system.
Forecasting Northeast Arctic cod
4500000
4000000
3500000
3000000
g 2500000
2 2000000
1500000
1000000
500000
0
S ©
PP
GStock
SP SP cM? SP HF FP 5? GFW FP GF?
PP PD PD A Ph ap ap” oP
tyr]
Figure 6. History (1950-1998) and forecasted (1999-2020) total biomass, spawning stock
biomass and landings by feedback control.
When the source of changes in recruitment and growth is identified, we may introduce this
source in a dynamic biomass model. In this case the cause is a deterministic fluctuation. This
information is useful to forecast future biomass recourses. Management of Northeast Arctic
cod is not a deterministic process. But if the landings rate F(nT) = 0.25 the next 20 years,
we will have a better ability to forecast future biomass resources.
Figure 6 shows the time series of historical records of total biomass from 1950 to 1998 and a
forecasted biomass from 1999 to 2010. The forecasted biomass is based on a temperature
dependent growth model (Y ndestad; 1999b) and a constant landings rate F(nT) = 0.25. Figure
6 shows that the it takes about 20 years to build up the biomass and the deterministic 6.2 yr
fluctuations will increase when the biomass is growing.
Discussion
The Newton system dynamics doctrine is based on the law of energy flow and the law of
energy balance in nature. When relations between system elements are changing we have a
non-linear system where small changes in the relations may lead to large changes in the
system states. This limits the ability of long time forecasting dynamics in nature.
A Fourier transform of a time series represents a dual view of nature where we are looking for
frequency properties. In most cases a Fourier spectrum has information about the past and
little information about the future. A frequency transform of time series often has a Wiener
spectrum as shown in equation (9) and figure 3. This does not mean the future time series is
predicable. The phase of the spectrum is changed in the next time series. There are however
exceptions. If a stationary cycle is forced on a system element in nature, the system element
will respond by a stationary cycle as in equation (10). Planetary dynamics are stationary
cycles in nature. This paper indicates there may be some truth in the Aristotle doctrine on the
predestinated fate.
Correlation analysis indicates there is a binding B(t) between the system elements
S(t)={B(t), {Sp(t),Sma(t), S(t), Si(t),S:(t)} } = Ew where S,(t) is the planetary system, Si(t) is the
temperature system in the Barents Sea, S;(t) is the food chain system in the Barents Sea, S,(t)
is the cod biomass system and S,,,(t) is the cod management system. This relations are
explained by a chain of reactions. The Earth nutation is a stationary forced oscillator on the
sea system, the sea system is non-linear and introduces a set of harmonic and sub harmonic
temperature cycles amplified by stochastic resonance. The temperature cycles is a forced
oscillator on the food chain and introduces a stochastic resonance that amplifies the impact
from the temperature cycle. The biomass of Northeast Arctic cod is a forced oscillator on
biomass management. A stochastic resonance of 18.6/3=6.2 yr between the management and
the biomass dynamics introduces an unstable biomass.
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CC BY-NC-SA 4.0