6
Can Resource Pulses Improve Empower Acquisition of an
Ecosystem?
Seungjun Lee
ABSTRACT
The main question of this study was that if an ecosystem could be exposed to either constant external
resources or pulsed external resources of the same average intensity, which sources would maximize
empower acquisition of the system. To answer the question, this study tested how matching of pulsed
resources affects total empower acquisition of a system using numerical simulation models and an
alternative dynamic emergy accounting method. A producer-consumer model system was built and
simulated by varying the phases and frequencies of pulsed energy sources. It was hypothesized that
matching of frequency and phase among two or more pulsed energy sources increases the empower
acquisition of an ecosystem, compared with a system under constant energy sources. The simulation
results showed that in systems of two energy sources, matching phases and frequencies of the pulsed
energy sources involved in primary production is critical to improve total empower acquisition and
consumer energy storage.
INTRODUCTION
Resource pulsing has been acknowledged to be an important factor that influences ecosystem
structures and processes (Chesson, 2003; Ostfeld and Keesing, 2000; Yang et al., 2008). Some
ecologists defined resource pulses as episodes of resource availability characterized by low frequency,
large magnitude, and short duration (Yang et al., 2008). Those episodic resource pulses may affect
individual- or community-level behavior in an ecosystem. From a more general perspective, Odum et
al. (1995) suggested the prevalence of pulsing trends in external resources or internal variables in
comparison with the traditional steady-state paradigm for the succession of an ecosystem. While many
pulsing trends have been observed in nature (e.g., solar radiation, nutrient pulses, flood pulses,
population oscillations), the fundamental reasons for or benefits from the pulsing trends have not been
explained well.
Considering that many external resources are supplied to an ecosystem with variable temporal
pulsing patterns (e.g., daily insolation, seasonal fluctuation of food sources, episodic precipitation),
one may expect that the pulsed resource environment benefits ecosystems or individual organisms. For
example, an ecosystem may better maintain total energy flux under pulsed resource supplies.
Regarding the resource pulsing, there have been studies about the matching of pulse frequencies
between external resources and system components for system performance. For example, Campbell
(1984) found that system-level production is likely to be maximized under the matching of frequencies
between external resource and internal oscillation. In a similar context, Lodge et al. (1994) emphasized
the importance of synchrony of nutrient supply with plant uptake to minimize competition between
microbes and plants. The effects of matching of pulsing traits such as frequency and phase between
different external resources, however, have not been studied. When an energy transformation process
in a system occurs by using more than two pulsed external resources, matching of a trait such as phase
or frequency among the pulsed resources may benefit the system. If the average energy intensities are
39
the same between a constant external resource and a pulsed one, will the ecosystem draw more energy
flow when the two pulsed external resources are matched, compared to two constant external
resources? The effects of matching between pulsed resources seem to be equivocal because matching
of two or more fluctuating resources yields not only high production under the resources’ highest
points but also low production under the resources’ lowest points.
Emergy or Empower (emergy/time) is a measure of quality-corrected total energy input through a
system. Dynamic emergy accounting (Odum, 1996) is a useful tool for understanding time-dependent
emergy by simulating emergy, unit emergy value (UEV), and relevant currencies such as energy,
matter, and money. While emergy synthesis as a snapshot represents total emergy directly or indirectly
contributed to make a product at a certain time, dynamic emergy accounting shows how emergy or
UEV changes over time. Since Odum introduced the method for dynamic emergy accounting (Odum,
1996; Odum and Odum, 2000b), there have been only a few studies related to it (e.g., Campbell et al.,
1998; Tilley, 2010; Tilley and Brown, 2006; Vassallo et al., 2009).
This study aims to investigate if the matching of pulsed external energy sources improves
empower acquisition of an ecosystem using a numerical simulation model. First, I constructed an
alternative to Odum’s dynamic emergy accounting method for the simulation of emergy to make it
easier and more convenient to investigate the changes in emergy under frequency and phase
differences. Second, I built a simple producer-consumer model system and tested how the matching of
pulsed energy sources affects empower acquisition of the system by varying phase difference and
frequency combinations among the sources. I hypothesized that a system under pulsed external energy
sources draws more empower through the system under frequency and phase matching, compared with
a system under constant external energy sources, when the temporal average intensities between the
pulsed and constant energy sources are the same.
THEORY
Odum’s Dynamic Emergy Accounting Method
Odum and Odum (2000b) suggested a dynamic emergy accounting method by three conditional
equations derived from the diagrams in Figure 1 as follows:
dQ/dt > 0: dEmQ = EmJ – EmH
(1)
where EmJ = TrJ·J and EmH = TrQ·K2·Q·F
dQ/dt = 0: dEmQ = 0
(2)
dQ/dt < 0: dEmQ = TrQ ·dQ
(3)
where EmX is the emergy of X, TrX is the transformity (or UEV) of X.
Energy diagram
Emergy diagram
Figure 1 Odum’s energy and emergy Energy Systems Language (ESL) diagrams for the dynamic
accounting method (Odum and Odum, 2000b). 40
Inconsistencies in the Odums’ method
The depreciation pathway
When dQ/dt >0, or Q is accumulated, the emergy of the depreciation pathway K1·Q was
considered zero in Equation 1 because the heat sink is a necessary process to maintain Q but does not
carry emergy. Odum and Odum (2000b), however, stated regarding Equation 3 that “When the change
in emergy storage is negative, the loss in emergy is the loss of the energy times the transformity of the
storage, whether it is due to depreciation loss or whether it is due to the transfer of useful energy out.”
This statement is not compatible with Equation 1. That is, the depreciation pathway K1·Q was
regarded as carrying emergy in Equation 3 but not in Equation 1. Because heat sink is necessary to
maintain Q but does not carry emergy, emergy is not lost through the depreciation pathway.
When dQ/dt = 0, dEmQ = 0?
In the second conditional equation (Equation 2), Odum defined dEmQ = 0 when dQ/dt = 0. The
equation indicates that there is no emergy change in Q if energy in the state variable Q does not
change. Let’s imagine the energy storage Q is in a steady state with the same constant energy inflow
and outflow (dQ/dt = 0). Then, the inflow and outflow of emergy are calculated by energy and
transformity values of the inflow and outflow. The inflowing emergy is defined by TrJ·J, which is
constant. The outflowing emergy is defined by TrQ·K2·Q·F. If there is no consumption (K2 = 0) in
Fig. 1, the steady state is maintained only by the energy loss through the depreciation pathway (K1·Q)
and emergy is infinitely accumulated over time (dEmQ > 0). In this case, dEmQ = 0 only when the
energy inflow and outflow stop (J = 0 and K1·Q = 0). If K2 > 0, the state variable Q will gain or lose
emergy depending on the initial emergy inflow and outflow under the steady state of energy until
dEmQ becomes zero. Thus, in this case, dEmQ may not be zero initially but is likely to become zero
after a certain amount of time.
Heat sink in the diagram
The K1·Q pathway (Fig. 1) in many systems diagrams of ecosystems is generally regarded as
density-dependent death of organisms, of which some depreciates as heat and the rest is eventually
recycled for future production. Thus, the rest of the dead organic matter, which does not depreciate as
heat, carries emergy. For dynamic emergy accounting purpose, the K1·Q pathway should be divided
into a heat sink that depreciates and an outflow that carries emergy for the future use.
Are conditional equations necessary?
Recalling that emergy is the available energy of one kind previously used up directly and
indirectly to make something (Odum, 1996), emergy is flowing through the available energy pathways.
Thus, in a certain storage (state variable), emergy is accumulated with inflowing available energy and
lost with outflowing available energy. Outflowing pathways carry emergy only when they can be used
in the next processes as available energy. A heat sink (depreciation) pathway either from a storage or a
transformation process carries no emergy. So a differential equation for the emergy of a state variable
can be written by inflows and outflows of emergy without conditional equations.
An Alternative Method of Dynamic Emergy Accounting
Tilley (2010) suggested a unified equation for dynamic emergy accounting derived from Figure 1
without the energy source F, which will partially resolve the inconsistencies shown in the Odum’s
method as follows:
dEmQ = TrJ ·J – TrQ ·K2·Q
(4)
TrQ = EmQ/Q
(5)
Equation 4 (Tilley, 2010) can be formulated variably depending on the pathway configurations in a
diagram. Depending on how we delineate the boundary of a system, a certain storage would be an
41
intermediate one that is used for further production processes and the other can be a final one that is
not supposed to be processed any more. What Equation 4 eventually implies is that the change of
emergy in the state variable Q is the sum of all input emergy subtracted by all output emergy that is
used for further processes. In any case, a depreciation (heat sink) pathway does not carry emergy.
Tilley (2010), however, did not separate the density-dependent pathway K1·Q into a heat sink and an
available energy pathway. Equation 4 can be generalized by the following:
(6)
dEmQ   TrX  X   TrQ  QY
X
Y
(7)
TrQ  EmQ / Q
where X is the input available energy flux of each kind, QY is the output available energy flux of each
kind from Q.
METHODS
Model System
The effects of pulsed energy sources on empower acquisition of an ecosystem were studied using
a producer-consumer model. The model (Figure 2) represents a basic producer-consumer system
where the storages of producers (P) and consumers (Q) are reproduced by autocatalytic feedback
processes. P is reproduced by transforming energy sources 1 and 2, and Q is reproduced by consuming
P and using energy source 3. There are density-dependent deaths and heat sinks from P and Q. It was
assumed that energy source 1 is flow-limited. As an example of the flow-limited source, sunlight is
tapped by primary producers in an ecosystem and unused light is reflected by the system (Montague,
2007). The energy flow cannot be increased above the limit. Energy 2 and 3 were assumed to be
constant-force sources. The constant-force source is drawn by the system as much as the system
requires for its utilization. As the constant-force source generally exists in a very large amount
compared to the rate of use, the force or density of the energy source is constant regardless of the
energy flow drawn by the system. Production, however, is likely to increase with increasing force or
density level of the constant-force source because the probability that the producers or consumers in
the system find the energy source will increase. The flow of energy source 1 was defined as J1, and the
energy density of sources 2 and 3 were defined as E2 and E3, respectively.
Simulation of the Basic Model
Using the energy and emergy diagrams of the model (Figure 2), numerical simulation was
conducted (Tables 1 and 2). Production functions were written as multiplications of contributing
energy sources, state variables, and coefficients following Holling’s type I functional response
(Holling, 1959). Values for energy flows and storages were roughly estimated by assuming that the
state variables (P and Q) were in steady state, and the coefficient for each pathway was backcalculated from the pathway function and its flow value as suggested by Odum and Odum (2000a).
The steady-state model was simulated in 0.1 day intervals for 1000 days (10000 time steps) using the
software R (available at the R project for Statistical Computing, http://www.r-project.org). Energy,
emergy, the transformity of P and Q, and empower (emergy/time) values from the three energy sources
(EmJ1-R, EmJ2, EmJ3) were simulated.
Definition of the Constant and Pulsed Energy Sources
Figure 3 depicts the definitions of the frequencies of and the phase difference between two pulsed
energy sources. Frequency of a pulsed energy source was defined as the time for one cycle of repeating
oscillation of the source. Phase difference (Δθ) between two pulsed energy sources indicates the
temporal difference between the pulsing cycles of the two sources. In the producer-consumer model,
the pulse of energy source 1 represents the fluctuation of the energy flow rate over time, and the pulses
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Energy diagram
Emergy diagram
Figure 2. Energy and Emergy systems diagrams of the producer-consumer model.
of energy sources 2 and 3 indicate the fluctuations of energy densities over time. A pulsed energy
source was defined by a sine function, the temporal average intensity of which is equivalent to that of a
constant energy source as shown in the following equations:
(8)
Constant energy source: E s  c
Pulsed energy source: E  c sin  2 t     1
P



 a

(9)

where a is the frequency (days) of the pulsed energy source, t is the time (day), and θ (days) is the
phase of the function.
Tests of the Effects of Pulsed Energy Sources
The effects of the phase difference and frequency combinations among the three pulsed energy
sources on the empower acquisition of the system were examined by varying the equations for the
pulsed energy sources in the producer-consumer simulation model. For the test of phase difference, the
frequencies of the three energy sources were set at 5 days (5-5-5 for J1-E2-E3). Energy, emergy, the
transformity of P and Q, and empower from the three energy sources were simulated by varying Δθ
between the pulsed energy sources. For the test of frequency combinations, variable frequency
combinations among the three pulsed energy sources were applied in the simulation model with Δθ = 0.
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Table 1. Variables and equations for the producer-consumer model.
Variables and equations
Description
P
Storage of producers
Q
Storage of consumers
J1
Total flow rate of energy source 1
R
Flow rate of remaining energy source 1
E2
Density of energy source 2
E3
Density of energy source 3
J1-R = ES1in·P·R·E2
Inflow rate of captured energy source 1 by the system
J2 = ES2in·P·R·E2
Inflow rate of energy source 2 drawn by the system
J3 = ES3in·P·Q·E3
Inflow rate of energy source 3 drawn by the system
FPpro = Ppro·P·R·E2
Production rate of P
Energy consumption rate of P for autocatalytic feedback
FPfb = Pfb·P·R·E2
FPhs = Phs·P
Heat sink rate of P
FPdd = Pdd·P
Density-dependent death rate of P
FPcon = Pcon·P·Q·E3
Rate of P consumption for Q production
FQpro = Qpro·P·Q·E3
Production rate of Q
FQfb = Qfb·P·Q·E3
Energy consumption rate of Q for autocatalytic feedback
FQhs = Qhs·Q
Heat sink rate of Q
FQdd = Qdd·Q
Density-dependent death rate of Q
TrJ1
Transformity of energy source 1
TrJ2
Transformity of energy source 2
TrJ3
Transformity of energy source 3
TrP = EmP/P
Transformity of P
TrQ = EmQ/Q
Transformity of Q
EmJ1-R = ES1in·P·R·E2·TrJ1
Inflow rate of emergy from energy source 1
EmJ2 = ES2in·P·R·E2·TrJ2
Inflow rate of emergy from energy source 2
EmJ3 = ES3in·P·Q·E3·TrJ3
Inflow rate of emergy from energy source 3
EmPfb = Pfb·P·R·E2·TrP
Emergy flow rate of autocatalytic feedback of P
EmPdd = Pdd·P·TrP
Emergy flow rate of density-dependent death of P
EmPcon = Pcon·P·Q·E3·TrP
Emergy flow rate of P consumption for Q production
EmQfb = Qfb·P·Q·E3·TrQ
Emergy flow rate of autocatalytic feedback of Q
EmQdd = Qdd·Q·TrQ
Emergy flow rate of density-dependent death of Q
Emergy change of P per time
d(EmP)/dt = EmJ1-R +EmJ2
– EmPfb – EmPdd – EmPcon
d(EmQ)/dt = EmPcon + EmJ3 – EmQfb – EmQdd Emergy change of Q per time
Unit
J
J
J/day
J/day
J
J
J/day
J/day
J/day
J/day
J/day
J/day
J/day
J/day
J/day
J/day
J/day
J/day
sej/J
sej/J
sej/J
sej/J
sej/J
sej/day
sej/day
sej/day
sej/day
sej/day
sej/day
sej/day
sej/day
sej/day
sej/day
RESULTS
Simulation Results under the Constant Energy Sources
The producer-consumer model was simulated for 1000 days under the constant energy sources.
The time interval of simulation (dt = 0.1 day) was small enough so that the simulation results did not
change with a smaller dt. The state variables of P, Q, EmP, and EmQ leveled off approximately at 200
J, 600 J, 3.33×104 sej, and 2.17×106 sej after 327 days. Total empower through the system (EmJ1-R +
EmJ2 + EmJ3) at steady state was 1.15×105 sej/day. Replacement times of P and Q were 2 and 15 days,
respectively.
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Table 2 Steady-state values of the variables and coefficients for the producer-consumer model.
Variable or
Steady-state
Meaning
coefficient
Value
Unit
energy flowb
P
200a
J
Q
600a
J
J1
10000
J/day
R
5000
J/day
E2
100
J
E3
100
J
TrJ1
1
sej/J
TrJ2
100
sej/J
TrJ3
1000
sej/J
ES1in
5.00×10-5
J-2
5000
Rate of energy source 1 per unit P & energy sources 1 & 2
-6
ES2in
1.00×10
J-2
100
Rate of energy source 2 per unit P & energy sources 1 & 2
100
Rate of energy source 3 per unit P & Q & energy source 3
ES3in
8.33×10-6 J-2×day-1
Ppro
1.00×10-6
J-2
100
Reproduction proportion of P per unit energy sources 1 & 2
Pfb
2.00×10-7
J-2
20c
Consumption proportion of P per unit energy sources 1 & 2
Phs
0.05
day-1
10
Specific rate of heat sink from P
Pdd
0.1
day-1
20
Specific rate of density-dependent death of P
Pcon
4.17×10-6 J-2×day-1
50
Predation proportion of P by Q per unit Q & energy source 3
Qpro
3.33×10-6 J-2×day-1
40
Reproduction proportion of Q per unit P & energy source 3
Qfb
1.67×10-6 J-2×day-1
20d
Consumption proportion of Q per unit P & energy source 3
Qhs
1.67×10-2
day-1
10
Specific rate of heat sink from Q
Qdd
1.67×10-2
day-1
10
Specific rate of density-dependent death of Q
a Turnover rates of 2 and 15 days were assumed for P and Q, respectively.
b Steady-state energy flows were roughly estimated.
c It was assumed that 20% of production is used for feedback in P.
d It was assumed that 50% of production is used for feedback in Q.
Phase difference (∆θ)
Frequency of A
Frequency of B
2c
A
B
Pulsed resources
c
Constant resource
0
Figure 3. Frequency and phase difference in the pulsed energy sources.
Simulation Results under the Pulsed Energy Sources
The variables of P, Q, EmP, EmQ, and total empower oscillated under the pulsed energy sources.
When the frequencies of the three energy sources were identical (e.g., 2-2-2 days), P, EmP, and total
empower oscillated with one maximum and one minimum peak within an oscillation cycle like sine
functions. When the frequencies of the three energy sources were different (e.g., 2-15-15), however, P,
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EmP, and total empower oscillated with more than two maximum or minimum peaks within an
oscillation cycle. Although the oscillating patterns of the variables were dependent on the frequency
combination among the pulsed energy sources, each temporal average value was in a pulsing steady
state after day 500.
Effects of Phase Difference
Under the fixed frequency combination of 5-5-5 among the three pulsed energy sources, the
average energy storage of P was maximal at 250 J when the phase of E3 differed from those of the
other two energy sources by about 1.25 days (Figure 4-a). That is, the phase matching between J1 and
E2 maximized the energy storage of P when their phases differed from that of E3 by 1.25 days.
Figure 4. Average energy of P and Q and total empower under variable Δθ values among the three
pulsed energy sources. (a) P. (b) Q. (c) Total empower. (Pulsing frequencies of J1-E2-E3 were 5-5-5
in days. Each point is an average value from Day 500 to 1000. The phase of J1, E2, or E3 was varied
from those of the other two energy sources. The legend of J1, E2, and E3 indicates the phase of each
pulsed energy source was varied from those of the other two energy sources with a designated phase
difference (Δθ). The dashed lines indicate P, Q, and total empower under the constant energy sources.)
46
The energy storage of Q and total empower were maximal when the phase difference among the
pulsed energy sources was near zero (Figure 4-b). Q was maximal at 762 J when the phase difference
of E3 from the other energy sources was about 4 days (E3 in the legend). Q was 744 J when the phases
of the three energy sources were matched (Δθ = 0). Total empower was maximal at 1.44×105 sej/day
when the phase difference of E3 from the other energy sources was about 4.5 days. Total empower
was 1.43×105 sej/day at Δθ = 0.
The phase matching between J1 and E2 (phase deviation of E3 from J1 and E2) stabilized the
energy storage of Q and the total empower even under the phase deviation of E3 from J1 and E2 (Fig.
4-b,c). While the phase deviation of E3 from J1 and E2 (phase matching between J1 and E2) resulted
in the difference of Q up to 79 J (the difference between the maximum and the minimum) and the
difference of total empower up to 1.58×104 sej/day, the phase deviation of J1 or E2 from the other two
energy sources (J1 or E2 in the legend) caused the difference of Q up to 583 J or 592 J and the
difference of total empower up to 1.08×105 sej/day or 1.09×105 sej/day, respectively, depending on the
phase difference.
Effects of Frequency Combination
The storage of Q and total empower drawn by the system increased under the pulsed resources
when the frequencies of J1 and E2 were matched (Figure 5). The storage of Q was 600 J and total
empower was 1.15×105 sej/day when the energy sources were constant. The energy storage of Q and
total empower were greater than 600 J and 1.15×105 sej/day in all the tested frequency combinations
when the pulsing frequencies of J1 and E2 were identical. The energy storage of Q and total empower
under the pulsed energy sources were less than those under the constant energy sources when the
frequencies of J1 and E2 were different from each other (15-2-15, 2-15-15, 1-2-15). Frequency
matching between J1 and E2, however, did not guarantee high energy in P. The energy storage of P
was generally high when the frequencies of the pulsed resources were short (e.g., 1-1-1, 2-2-2, 2-2-15),
while low when the frequencies were long (e.g., 50-50-50, 100-100-100).
Figure 5. Average energy of P and Q and total empower under variable frequency combinations
among the three energy sources, where Δθ = 0. Each point is an average of all values from Day 500 to
1000. The three-number combination in each point indicates pulsing frequencies of J1-E2-E3 in days.
Numbers in the parentheses indicate total empower (×105 sej/day).
47
DISCUSSION
‘Maximum’ in the Maximum Empower Principle
The maximum empower principle (MEP) has been understood as an extremal goal function of
self-organization (Patten, 1995). Although it has been hypothesized that under the MEP a prevailing
system maximizes empower acquisition, it is difficult to find a theoretical ‘maximum’ point of
empower. Thus, in this study, the maximum empower was discussed in terms of the increase or
decrease of empower acquisition under the pulsed energy sources, compared with the constant energy
sources. The maximum empower and energy in the results of this study indicate the maximum values
of empower and energy under all the tested phase differences or frequency combinations.
Do Pulsed Energy Sources Improve the Empower Acquisition of an Ecosystem?
An alternative dynamic emergy accounting method was used in this study. Empower acquisition
of the producer-consumer model system under the pulsed energy sources was dependent on the
matching of phases among the sources. When the frequencies of the pulsed energy sources were
identical, the matching of the phases between the two energy sources involved in the production of
producers (P) significantly increased the energy storage of consumers (Q) and total empower of the
system. The highest energy storage of Q and total empower were obtained when the phase of E3
slightly deviated from the matched phase of J1 and E2. The slight phase deviation of about 10–20% of
the frequency (0.5–1.0 day phase difference out of 5 day frequency) for maximal energy storage of Q
and total empower appears to be caused by the phase of the remaining energy source R (For the
remaining source R, see Fig. 2 and Table 1. See Montague, 2007 for the equation and details of
remaining source R in a flow-limited source.). Although R shows the same pattern of a sine function as
J1, the phase of R slightly deviated from J1. This implies that there is a slight phase gap between the
external energy sources J1, E2, E3 and their internal supplies. Because the production of P is a
function of R and E2 and that of Q is a function of E3, the exact phase matching among J1, E2, and E3
may not maximize empower acquisition. A slight phase deviation among J1, E2, and E3 is thus likely
to maximize empower acquisition of the system by matching the internal timing of supplies of pulsed
resources.
Empower acquisition of the system under the pulsed energy sources was also dependent on the
matching of frequencies among the sources. The frequency matching between the two energy sources
involved in the production of P was an important factor for the increase of Q and total empower. By
matching the frequencies of the two energy sources (J1 and E2) and maximizing the production rate of
P, the production of Q can be maximized by utilizing P and the energy source E3.
Maximum Empower and Consumer Energy
In the producer-consumer model system, the energy storage in Q appeared to be correlated with
the total empower acquisition through the system (Figs 4 and 5). This may indicate that the energy
storage of consumers is an indicator of maximum empower through the system. Cai et al. (2004)
quoted Odum’s discussion that maximum total power acquisition requires maximum consumer
respiration. Because consumers play a critical role in a food web by controlling producers and energy
is transferred from producers to consumers, the energy storage of consumers is likely to represent how
much total useful energy is drawn for the survival of the system.
An Intuitive Mechanism
A simple mathematical fact provides an intuitive mechanism regarding the matching of phases or
frequencies among pulsed energy sources. One can imagine a case where phases and frequencies of
two pulsed energy sources are perfectly matched. The temporal function of each pulsed energy source
48
can be written as c[sin(t)+1], whose temporal average value is c. If the sine function is squared (i.e.,
perfect matching in both phase and frequency), the temporal average of the squared function would be
about 1.5c2, which is 50% more than c2. That is, if the production function is defined as the
multiplication of contributing variables and the average energy intensities are the same between the
pulsed and constant energy sources, the matching of the pulsed energy sources produces up to 50%
more available energy than that of the constant energy sources. However, since the amount of energy
flow through the system is determined by the system configuration, the simple mathematical fact may
not always guarantee that the matching of pulsed energy sources improves empower acquisition
through the system. In this study, the mechanism seemed to work for the producer-consumer system,
one of the most fundamental ecosystem models.
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