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Resource pulses can increase power acquisition of an ecosystem
Ecological Modelling 271 (2014) 21–31
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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Resource pulses can increase power acquisition of an ecosystem Seungjun Lee ∗ Department of Environmental Engineering Sciences, Center for Environmental Policy, Phelps Lab., University of Florida, Gainesville, FL 32611, USA
a r t i c l e
i n f o
Article history: Available online 11 January 2013 Keywords: Dynamic emergy accounting Maximum empower principle Pulsing paradigm Resource matching Resource pulse Simulation model
a b s t r a c t Pulsing is prevalent in nature. As resource pulse has been recognized as one of the major factors influencing ecosystem structures and processes, it is important to investigate why nature pulses and what benefits an ecosystem obtains from pulsed resources. The main question of this study was that if a system could be exposed to either constant external resources or pulsed external resources of the same temporal average intensity, which resources would maximize power acquisition of a 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 a refined dynamic emergy accounting method. A producer–consumer model system was built and simulated by varying 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 a system, 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 increase total empower acquisition and consumer energy storage. The primary mechanism was that the matching of pulsed resources in phase and frequency promotes energy acquisition of primary producers that is further efficiently transferred for the production of consumers. Energy acquisition of consumers was strongly correlated with total empower acquisition of the system presumably because the consumers are in the high energy hierarchical position controlling the producers thus contributing to the total empower acquisition through the system. © 2012 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Hypotheses on the self-organization of ecosystems Goal functions for the self-organization of ecosystems have long been discussed among ecologists particularly who have studied energetics or system-level properties. Lotka (1922) stressed that a prevailing system tends to increase total available energy flux through the system during natural selection. Later, the hypothesis of the maximum power principle (MPP) proposed by Lotka (1922) has been partially supported and complemented by Odum. Odum proposed the maximum empower principle (MEP) that hypothesizes natural selection goes to a system that maximizes empower, the quality-corrected energy flux (Brown et al., 2004). He suggested empower instead of power as the maximized flow because the quality difference among various energy types (Odum, 1988) has been a problem in defining energy terms, especially when many energy sources of different qualities drive a process (Patterson, 1996).
Although the MPP or MEP has been hypothesized as a goal function for the self-organization of an ecosystem, only a few empirical or modeling studies on the hypothesis have been reported (e.g., Cai et al., 2006; DeLong, 2008). The scarcity of the studies on the MPP or MEP may be attributed to the difficulty in identifying and quantifying available energy flux through complex energy networks (Cai et al., 2004). Meanwhile, Ulanowicz (1997) proposed ascendency as a hypothesis on the development of ecosystems. Ascendency, an index encompassing both qualitative and quantitative aspects of system development, is defined as the multiplication of average mutual information and total system throughput. Although the ascendency includes mutual information as a factor of system development, it appears that both the MPP and ascendency hypotheses agree that a critical component of self-organization in a prevailing system is the total energy flux through the system.
1.2. Pulsing paradigm and resource pulse
∗ Tel.: +1 352 392 2426; fax: +1 352 392 3624. E-mail address: slee@ufl.edu 0304-3800/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2012.11.028
Resource pulse 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
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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 pulse, flood pulse, population oscillation), 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 instance, an ecosystem may better obtain 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 well. 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 the same between a constant external resource and a pulsed one, would the ecosystem draw more energy flux under the matching of the two pulsed external resources than that of the 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. 1.3. Dynamic emergy accounting 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), however, there have been only a few studies related to it (e.g., Tilley, 2010; Tilley and Brown, 2006; Vassallo et al., 2009). As Tilley (2010) discussed, the dynamic emergy accounting method introduced by Odum (1996) conflicts with the concept of emergy so it is necessary to refine the method to accurately simulate the trajectories of emergy or UEV.
frequency combination among the sources. I hypothesized that a system under pulsed external energy sources draws more empower through the system under the 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. 2. Theory 2.1. 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 Fig. 1 as follows: dQ > 0 : dEmQ = EmJ − EmH dT where EmJ = TrJ ·J and EmH = TrQ ·K2·Q·F
(1)
dQ = 0 : dEmQ = 0 dT
(2)
dQ < 0 : dEmQ = TrQ · dQ (3) dT where EmX is the emergy of X, TrX is the transformity (or UEV) of X. 2.2. Inconsistencies in the Odum’s method 2.2.1. The depreciation pathway When dQ/dT > 0, or Q is accumulated, the emergy of the depreciation pathway K1·Q was considered zero in Eq. (1) because heat sink is a necessary process to maintain Q but does not carry emergy. Odum and Odum (2000b), however, stated regarding Eq. (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 Eq. (1). That is, the depreciation pathway K1·Q was regarded as carrying emergy in Eq. (3) but not in Eq. (1). Because heat sink is necessary to maintain Q but does not carry emergy, emergy is not lost through the depreciation pathway.
Energy diagram F
Energy Source
J
Q
K2 Q F
K1 Q
Emergy diagram
F
1.4. Study plan This study aims to investigate how the matching of pulsed external energy sources influences empower (emergy/time) acquisition of an ecosystem using a numerical simulation model. First, I refined Odum’s dynamic emergy accounting method for the simulation of emergy. 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
EmJ
EmQ
EmH
Zero Fig. 1. Odum’s energy and emergy systems diagrams for the dynamic accounting method (Odum and Odum, 2000b).
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2.2.2. When dQ/dT = 0, dEmQ = 0? In the second conditional equation (Eq. (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 us 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. 2.2.3. 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. 2.2.4. 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 of emergy for a state variable can be written by inflows and outflows of emergy without conditional equations. 2.3. A new method in dynamic emergy accounting Tilley (2010) suggested a unified equation for dynamic emergy accounting derived from Fig. 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
TrQ =
EmQ Q
(4)
(5)
Eq. (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 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 Eq. (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
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K1·Q into a heat sink and an available energy. Eq. (4) can be generalized by the following: dEmQ = TrQ =
X
TrX · X −
Y
TrQ · QY
EmQ Q
(6)
(7)
where X is the input available energy flux of each kind, QY is the output available energy flux of each kind from Q. 3. Methods 3.1. Model system The effects of pulsed energy sources on empower acquisition of an ecosystem were studied using a producer–consumer model. The model (Fig. 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 flowlimited. 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. Energies 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 very large amount, 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 density of energy sources 2 and 3 were defined as E2 and E3, respectively. 3.2. Simulation of the basic model Using the energy and emergy systems diagrams (Fig. 2), a numerical simulation model was built (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. Values for energy flows and storages were roughly estimated by assuming the steady-state of the state variables (P and Q), and the coefficient for each pathway function was back-calculated from the pathway function and its flow value as suggested by Odum and Odum (2000a). Energy values and their transformities (Table 2) were selected by considering the hierarchy among the energy sources (i.e., J1, E2, E3, TrJ1 , TrJ2 , TrJ3 ). It was assumed that the consumption rate from P to Q is half of the production rate of P. The initial steady-state values of P and Q were set at 200 J and 600 J by considering the reasonable energy flow rates through pathways and the replacement times of P and Q. The steady-state model was simulated in 0.1 day intervals for 1000 days (10,000 steps) using R (available at the R project for Statistical Computing, http://www.r-project.org). Energy, emergy, transformity of P and Q, and empower (emergy/time) values from the three energy sources (EmJ1−R , EmJ2 , EmJ3 ) were simulated. 3.3. Definition of the constant and pulsed energy sources Fig. 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 ()
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Energy diagram Energy Source 2 E2
Energy Source 3 E3
J3
J2 FPfb Energy Source 1
J1
FQfb FPcon
P FPpro
Q FQpro
FQdd
FPdd R
FQhs
FPhs
Emergy diagram EmJ3
EmPfb
EmJ1-R + EmJ2
P
EmQfb Q
EmPcon EmPcon + EmJ3 EmPdd
Zero
Zero
EmQdd
Fig. 2. Energy and emergy systems diagrams of the producer–consumer model.
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 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 the following equations: Constant energy source : ES = c
(8)
Table 1 Variables and equations for the producer–consumer model. Variables and equations
Description
Unit
P Q J1 R E2 E3 J1 − R = ES1in·P·R·E2 J2 = ES2in·P·R·E2 J3 = ES3in·P·Q·E3 FPpro = Ppro·P·R·E2 FPfb = Pfb·P·R·E2 FPhs = Phs·P FPdd = Pdd·P FPcon = Pcon·P·Q·E3 FQpro = Qpro·P·Q·E3 FQfb = Qfb·P·Q·E3 FQhs = Qhs·Q FQdd = Qdd·Q TrJ1 TrJ2 TrJ3 TrP = EmP /P TrQ = EmQ /Q EmJ1−R = ES1in·P·R·E2·TrJ1 EmJ2 = ES2in·P·R·E2·TrJ2 EmJ3 = ES3in·P·Q·E3·TrJ3 EmPfb = Pfb·P·R·E2·TrP EmPdd = Pdd·P·TrP EmPcon = Pcon·P·Q·E3·TrP EmQfb = Qfb·P·Q·E3·TrQ EmQdd = Qdd·Q·TrQ d(EmP )/dT = EmJ1−R + EmJ2 − EmPfb − EmPdd − EmPcon d(EmQ )/dT = EmPcon + EmJ3 − EmQfb − EmQdd
Storage of producers Storage of consumers Total flow rate of energy source 1 Flow rate of remaining energy source 1 Density of energy source 2 Density of energy source 3 Inflow rate of captured energy source 1 by the system Inflow rate of energy source 2 drawn by the system Inflow rate of energy source 3 drawn by the system Production rate of P Energy consumption rate of P for autocatalytic feedback Heat sink rate of P Density-dependent death rate of P Rate of P consumption for Q production Production rate of Q Energy consumption rate of Q for autocatalytic feedback Heat sink rate of Q Density-dependent death rate of Q Transformity of energy source 1 Transformity of energy source 2 Transformity of energy source 3 Transformity of P Transformity of Q Inflow rate of emergy from energy source 1 Inflow rate of emergy from energy source 2 Inflow rate of emergy from energy source 3 Emergy flow rate of autocatalytic feedback of P Emergy flow rate of density-dependent death of P Emergy flow rate of P consumption for Q production Emergy flow rate of autocatalytic feedback of Q Emergy flow rate of density-dependent death of Q Emergy change of P per time Emergy change of Q per time
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
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Table 2 Steady-state values of the variables and coefficients for the producer–consumer model. Variable or coefficient
Value a
200 600a 10,000 5000 100 100 1 100 1000 5.00 × 10−5 1.00 × 10−6 8.33 × 10−6 1.00 × 10−6 2.00 × 10−7 0.05 0.1 4.17 × 10−6 3.33 × 10−6 1.67 × 10−6 1.67 × 10−2 1.67 × 10−2
P Q J1 R E2 E3 TrJ1 TrJ2 TrJ3 ES1in ES2in ES3in Ppro Pfb Phs Pdd Pcon Qpro Qfb Qhs Qdd a b c d
Unit
Steady-state energy flowb
Meaning
J J J/day J/day J J sej/J sej/J sej/J J−2 J−2 J−2 × day−1 J−2 J−2 day−1 day−1 J−2 × day−1 J−2 × day−1 J−2 × day−1 day−1 day−1
– – – – – – – – – 5000 100 100 100 20c 10 20 50 40 20d 10 10
– – – – – – – – – Rate of energy source 1 per unit P and energy sources 1 and 2 Rate of energy source 2 per unit P and energy sources 1 and 2 Rate of energy source 3 per unit P and Q and energy source 3 Reproduction proportion of P per unit energy sources 1 and 2 Consumption proportion of P per unit energy sources 1 and 2 Specific rate of heat sink from P Specific rate of density-dependent death of P Predation proportion of P by Q per unit Q and energy source 3 Reproduction proportion of Q per unit P and energy source 3 Consumption proportion of Q per unit P and energy source 3 Specific rate of heat sink from Q Specific rate of density-dependent death of Q
Replacement times of 2 and 15 days were assumed for P and Q, respectively. Steady-state energy flows were roughly estimated. It was assumed that 20% of production is used for feedback in P. It was assumed that 50% of production is used for feedback in Q.
Pulsed energy source : EP = c
sin
2 a
(t − ) + 1
(9)
where a is the frequency (days) of the pulsed energy source, t is the time (day), and (days) is the phase of the function. 3.4. Tests of the effects of pulsed energy sources and analyses of mechanisms The effects of phase difference and frequency combination 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, transformity of P and Q, and empower from the three energy sources were simulated by varying between the pulsed energy sources. For the test of frequency combination, variable frequency combinations among the three pulsed energy sources were applied in the simulation model with = 0. Average flows and storages of energy and emergy in the pathways and state variables were calculated to analyze how the matching of pulsed energy sources in phase and frequency changes the acquisition of empower and energy in the system. Average
replacement times of P and Q were calculated to analyze the energy replacement and transfer rates within the system under the matching of the pulsed energy sources. Replacement time under the steady state of P or Q was calculated by the following equation: Replacement time(day) =
energy storage (J) production rate (J/day)
3.5. Tests of the effects of pulsed energy sources with an additional energy source As more than two pulsed energy sources can be used in the production of consumers (Q) and the matching of them may change the energy and emergy flow patterns in the system, the effects of
Energy Source 3 E3
J3 Frequency of A Phase difference (
2c
A
B
Frequency of B
Energy (J)
Constant resource
Energy Source 4 E4
J4 FQfb = Qfb P Q E3 E4
Pulsed resources
)
(10)
FPcon = Pcon P Q E3 E4 FQpro = Qpro P Q E3 E4
c
0 Time (Day)
Fig. 3. Frequency and phase difference in the pulsed energy sources.
Fig. 4. Part of the energy systems diagrams of the producer–consumer model with the addition of energy source 4.
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frequency combination on the empower acquisition of the system were examined by adding additional constant-force energy source 4, inflowing along with energy source 3 for the production of Q (Fig. 4). The density of energy source 4 was defined as E4, whose value was 10 J and transformity was 10,000 sej/J. The steady-state flow rate of the energy source 4 was 10 J/day. I tested how the matching of frequencies among energy sources involved in the same or different transformation processes (i.e., production of P or Q) affects the empower acquisition of the system by varying the frequency combination among the four energy sources. 4. Results 4.1. Simulation results under the constant energy sources
4.2. Simulation results under the pulsed energy sources 4.2.1. Temporal patterns of the variables The variables of P, Q, EmP , EmQ , and total empower oscillated under the pulsed energy sources. Fig. 5 shows the simulation results under the frequency (days) combinations of 2–2–2 and 2–15–15 among the three pulsed energy sources (i.e., 2–2–2 means the pulsing frequencies of J1, E2, and E3 are 2, 2, and 2 days, respectively. Likewise, 2–15–15 means the frequencies of J1, E2, and E3 are 2, 15, and 15 days, respectively.). When the frequencies of the three energy sources were identical (i.e., 2–2–2), P, EmP , and total empower oscillated with one maximum and one minimum peaks within an oscillation cycle like sine functions. When the frequencies of the three energy sources were different (i.e., 2–15–15), however,
270
800
240
700
210
600
Q (J)
P (J)
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 not to change the simulation results 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.
180 150
400
120
300 520
540
560
580
600
4.5
3.0
4.0
2.5
EmQ ( 106 sej)
EmP ( 104 sej)
500
3.5 3.0 2.5
500
520
540
560
580
600
500
520
540
560
580
600
500
520
540
560
580
600
2.0 1.5 1.0
500
520
540
560
580
600
3.0
Total empower ( 105 sej/day)
Total empower ( 105 sej/day)
500
2.5 2.0 1.5 1.0 0.5 0.0 500
520
540
560
Time (day)
580
600
3.0 2.5 2.0 1.5 1.0 0.5 0.0
Time (day)
Fig. 5. Examples of simulation results under the pulsed energy sources (Black lines: frequency combination of 2–2–2. Gray lines: frequency combination of 2–15–15. Only simulation results from Day 500 to 600 are shown for better representation of the patterns. Pulsing patterns after Day 600 are the same as those from Day 500 to 600.).
S. Lee / Ecological Modelling 271 (2014) 21–31
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Fig. 7. Average energy of P and Q and total empower under variable frequency combinations among the three energy sources, where = 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).).
The phase matching between J1 and E2 (phase deviation of E3 from J1 and E2) relatively stabilized the energy storage of Q and total empower even under the phase deviation of E3 from J1 and E2 (Fig. 6b and 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. Fig. 6. Average energy of P and Q and total empower under variable 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 (). The dashed lines indicate P, Q, and total empower under the constant energy sources.).
P, 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 steady state (pulsing steady state) after day 500. 4.2.2. 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 (Fig. 6a). 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. The energy storage of Q and total empower were maximal when the phase difference among the pulsed energy sources was near zero (Fig. 6b and c). Q was maximal at 762 J when the phase difference of E3 from the other energy sources was about 4 days. Q was 744 J when the phases of the three energy sources were matched ( = 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 = 0.
4.2.3. 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 (Fig. 7). Simulation was conducted under variable frequency combinations that were selected in consideration of the replacement times of P and Q. I speculated that maximal empower acquisition may occur when the pulsing frequencies of resources are close to the replacement times of P and Q (2 and 15 days). From this point, I simulated the model with variable frequency combinations within 100 days. 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 (i.e., 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). 4.3. Analyses of mechanisms To analyze energy and emergy flow patterns under the pulsed energy sources, average flow rates and storages were calculated from the simulation results (Fig. 8). Under the pulsing frequency combination of 15–15–15 ( = 0), average inflow rates of energy sources 1, 2, and 3 increased from 5000 J, 100 J, and 100 J to 5614 J, 112 J, and 129 J, respectively. More energy accumulated in P was
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Energy diagram Energy Source 3
Energy Source 2
112 (100)
Energy Source 1
10000 (10000)
129 (100)
22 (20)
5614 (5000)
26 (20)
64 (50)
172 (200) 112 (100)
764 (600)
51 (40)
17 (20)
13 (10)
13 (10)
9 (10)
Storages: J Flows: J/day
Emergy diagram 128605 (100000)
3622 (3333)
16843 (15000)
28847 (33333)
Zero
92485 (72222) 138976 (108333)
10371 (8333)
2757974 (2166667)
2885 (3333)
45966 (36111)
Zero
Storages: sej Flows: sej/day
Fig. 8. Average energy and emergy values (Day 500–980) under the pulsing frequency combination of 15–15–15 and of 0 (upper numbers are flows or storages under the pulsed energy sources and lower numbers in the parentheses are flows or storages under the constant energy sources).
empower (r2 = 0.9977 in phase tests and r2 = 0.9963 in frequency tests). 4.4. When two energy sources are involved in the Q production When additional energy source E4 was added to the model (Fig. 4), the energy of P and Q were determined by the frequency combination of J1–E2–E3–E4 (Fig. 11). The matching of frequencies between the two energy sources J1 and E2, which were involved 16.0
Replacement time of Q (day)
transferred to Q and less energy in P dissipated through the heat sink or density dependent pathway. That is, productivity of P increased 12% from 100 J to 112 J, while consumption rate of P for Q production increased 28% from 50 J to 64 J. The increased transfer rate of energy from P to Q eventually increased the energy storage of Q 27% from 600 J to 764 J. Emergy flow patterns were similar to those of energy flows (Fig. 8). Emergy productivity of P increased 12% from 15,000 sej to 16,843 sej, while emergy consumption rate of P for Q production increased 24% from 8333 sej to 10,371 sej. As a result, emergy of Q increased 27%. Because the increased productivity of P and energy transfer rate from P to Q were identified as the major mechanisms of the increased energy storage of Q and total empower under the frequency and phase matching (15–15–15, = 0), replacement times of P and Q were calculated to confirm the mechanisms (Fig. 9). Replacement time indicates the speed of production or consumption of the storages. The replacement times of P and Q were 2 and 15 days, respectively, when the energy sources were constant. Among the tested pulsing frequency combinations, the replacement times of Q were in the range of 14.92 ± 0.13 days, while those of P varied in the range of 1.73 ± 0.39 days. The replacement time of P was greater than 2 days when the pulsing frequency of J1 was different from that of E2, and less than 2 days when the pulsing frequencies of J1 and E2 were matched. Figs. 6 and 7 show the close relationship between Q and total empower. In Fig. 6, the graphs of Q and empower show a similar pattern. In Fig. 7, empower was generally higher in higher Q. To closely examine the relationship, the correlations of P and Q with total empower were tested in Fig. 10. There was no correlation between P and total empower (r2 = 0.0025 in phase tests and r2 = 0.1495 in frequency tests), while Q was strongly correlated with total
15.5 2-2-15 1-2-15 15-15-2
15.0
100-100-100 30-30-30 50-50-50
10-10-10
20-20-20
15-15-15
2-15-15
No pulsing
3-3-3 2-2-2 5-5-5 1-1-1
15-2-15
14.5
14.0 1.0
1.5
2.0
2.5
3.0
Replacement time of P (day)
Fig. 9. Average replacement time (Day 500–1000) of P and Q with variable frequency combinations among the three energy sources, where = 0.
S. Lee / Ecological Modelling 271 (2014) 21–31
29
Fig. 10. Correlations of P and Q with total empower (a) test of phase difference and (b) test of frequency combination (each point is an average of all energy or empower values from Day 500 to 1000 under a tested or frequency combination).
in the P production, was necessary for high Q and total empower. Matching of frequencies between E3 and E4, which were involved in the Q production, however, was not a necessary condition for high Q or total empower. For example, Q was 689 J and total empower was 2.47 × 105 sej/day under the frequency combination of 2–2–2–15. Even if the frequencies of E3 and E4 were matched, P, Q, and total empower under the frequency combinations of 15–2–15–15 and 2–15–15–15 were lower than those under the constant energy sources. The energy storage of Q and total empower were high when the frequency of E3 or E4 was close to the matched frequency of J1 and E2. For example, Q was 689 J and total empower was 2.47 × 105 sej/day under the frequency combination of 2–2–2–15. However, Q was 595 J and empower was 2.08 × 105 sej/day under 2–2–15–15, which were lower than those under the constant energy sources. When the frequencies of E3 and E4 deviated more from those of J1 and E2 (i.e., 2–2–30–30), Q and empower decreased to 504 J and 1.81 × 105 sej/day, respectively. The mechanism of increased empower acquisition under the four pulsed energy sources was similar to that under the three pulsed energy sources. Under the frequency combinations of 15–15–15–15, 2–2–2–2, 5–5–2–2, 2–2–5–5, and 15–15–2–2, the energy of Q was greater and that of P was less than those under the constant energy sources (Fig. 11a). The replacement times of P under these frequency combinations were shorter than that under the constant energy sources (Fig. 11b).
5. Discussion 5.1. ‘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 under the MEP that 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. 5.2. Do pulsed energy sources increase the empower acquisition of an ecosystem? 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
30 (a)
S. Lee / Ecological Modelling 271 (2014) 21–31 700
2-2-2-15 (2.47)
15-15-15-15 (2.47) 2-2-2-2 (2.41)
5-5-2-2 (2.27)
650
2-2-5-5 (2.23) 15-15-2-2 (2.22) No pulsing (2.15)
600 Q (J)
2-2-15-15 (2.08)
550 2-2-30-30 (1.81)
500
15-2-2-15 (1.75)
450
2-15-2-15 (1.64)
15-2-15-15 (1.53) 2-15-15-15 (1.49)
400 100
150
200
250
300
P (J)
(b)
16.0
Replacement time of Q (day)
5-5-2-2
15.5
2-2-5-5 2-2-15-15
15-15-2-2 2-2-30-30 2-15-15-15
15.0
2-2-2-15 No pulsing
15-15-15-15 15-2-15-15
2-15-2-15
2-2-2-2
turnover rate (short replacement time) of P and efficient transfer of energy from P to Q. The faster turnover rate of P may indicate that fast developing primary producers are favored in competition under the pulsed resource environment (Friman and Laakso, 2011; Holt, 2008). Although the frequency matching between J1 and E2 is critical, the test of four energy sources suggested that the energy of Q and total empower may be reduced by pulsed energy sources when the frequencies of the energy sources involved in the Q production (E3 and E4) are far from those of J1 and E2. The energy sources involved in the Q production may need to pulse as frequently as the pulses of the energy sources involved in the P production to accelerate overall production by frequently utilizing the energy storage of P. The results of simulation provide some insight on the supply of resources and ecosystem growth. Seasonal or daily matching of pulsed resources would promote the growth of a natural ecosystem. In the long term, for example, dry season fires regenerate nutrients for the growth of forests during the rainy season. When the peak of the nutrient pulse is matched with the peak insolation and water, the primary productivity will increase to promote the secondary production. In the short term, for example, daily fluctuation of insolation tends to be matched with frequent water supply and nutrient fluctuation from fast consumption and recycling under the warm and humid climate.
15-2-2-15
5.3. Maximum empower and consumer energy
14.5
14.0 1.0
1.5
2.0
2.5
3.0
Replacement time of P (day)
Fig. 11. Average energy values and replacement times of P and Q under variable frequency combinations among the four energy sources, where = 0, (a) energy and total empower and (b) replacement times (Each point is an average of all values from Day 500 to 1000. The four-number combination in each point indicates pulsing frequencies of J1–E2–E3–E4 in days. Numbers in the parentheses indicate total empower (×105 sej/day).).
In the producer–consumer model system, the energy storage in Q was strongly correlated with the total empower acquisition through the system. This may indicate that the energy storage of consumers is an indicator of maximum empower through the system. Cai et al. (2004) quoted the 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. 5.4. An intuitive mechanism
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, R (For the remaining R, see Fig. 2 and Table 1. See Montague, 2007 for the equation and details of remaining energy R in a flowlimited source.). Although the remaining energy 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 complete 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. The analyses of the energy and emergy flows in the system and the replacement times of P and Q revealed that the high Q and total empower can result from the fast
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 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 1.5 c2 , 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 than that of the constant energy sources does. Since the amount of energy flow through the system is determined by the system configuration, however, the simple mathematical fact may not always guarantee that the matching of pulsed energy sources increases 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. 5.5. Practical aspects The results of this study provide a theoretical background for understanding empower acquisition through an ecosystem under pulsed resources. The periodic pulses of resources in the studied
S. Lee / Ecological Modelling 271 (2014) 21–31
model system are probably unrealistic to represent the pulses of a real ecosystem that is generally influenced by both temporally regular and irregular resources. Nevertheless, the results suggest that the matching of pulsed resources possibly promotes the empower acquisition of a system. The methodology used in this study may be applied to specific ecosystem types. For example, the characteristic difference between terrestrial and aquatic ecosystems under the pulsed resource environment (Nowlin et al., 2008) would be better understood by applying the patterns of pulsed resources and parameters observed in those ecosystem types to the simulation model of this study. Once a specific model of an ecosystem is established, one can calculate energy and emergy of variables and flows. Creating and referring to the graphs, such as Figs. 6 and 7, practitioners may restore or manage resource pulsing patterns that result in a designated energy level in a system (e.g., biomass). This practice may be useful for the management of human-caused resource supply (e.g., nutrient overload). 6. Conclusions This study started with the question that if a system could be exposed to either constant external resources or pulsed external resources of the same temporal average intensity, which resources would maximize power acquisition of a system. According to the maximum empower principle, a system is likely to select resources or change to a new configuration that maximizes total empower acquisition. Simulation of a producer–consumer model showed that total empower acquisition of the system and energy gain in the consumer storage can increase under the pulsed energy sources. In particular, the matching of phases and frequencies among the pulsed energy sources involved in the primary production was critical to increase total empower acquisition and consumer energy storage. The primary mechanism was that the matching of pulsed energy sources in phase and frequency promotes energy acquisition of primary producers that are further efficiently transferred for the production of consumers. Energy acquisition of consumers was strongly correlated with total empower acquisition of the system presumably because the consumers are in the high energy hierarchical position controlling the producers thus contributing to the total empower acquisition through the system. Acknowledgements I am grateful to Mark Brown and anonymous reviewers for valuable comments and suggestions on the manuscript.
31
References Brown, M.T., Odum, H.T., Jorgensen, S.E., 2004. Energy hierarchy and transformity in the universe. Ecological Modelling 178, 17–28. Cai, T.T., Montague, C.L., Davis, J.S., 2006. The maximum power principle: an empirical investigation. Ecological Modelling 190, 317–335. Cai, T.T., Olsen, T.W., Campbell, D.E., 2004. Maximum (em)power: a foundational principle linking man and nature. Ecological Modelling 178, 115–119. Campbell, D.E., 1984. Energy filter properties of ecosystems. Ph.D. Dissertation, University of Florida, Gainesville, FL, USA. Chesson, P., 2003. Understanding the role of environmental variation in population and community dynamics: introduction. Theoretical Population Biology 64, 253–254. DeLong, J.P., 2008. The maximum power principle predicts the outcomes of twospecies competition experiments. Oikos 117, 1329–1336. Friman, V., Laakso, J., 2011. Pulsed-resource dynamics constrain the evolution of predator–prey interactions. American Naturalist 177, 334–345. Holt, R.D., 2008. Theoretical perspectives on resource pulses. Ecology 89, 671–681. Lodge, D.J., McDowell, W.H., McSwiney, C.P., 1994. The importance of nutrient pulses in tropical forests. Trends in Ecology and Evolution 9, 384–387. Lotka, A.J., 1922. Contribution to the energetics of evolution. Proceedings of the National Academy of Sciences of the United States of America 8, 147–151. Montague, C.L., 2007. Dynamic simulation with energy systems language. In: Fishwick, P.A. (Ed.), Handbook of Dynamic System Modeling. Chapman & Hall/CRC, Boca Raton, FL, pp. 28.1–28.32. Nowlin, W.H., Vanni, M.J., Yang, L.H., 2008. Comparing resource pulses in aquatic and terrestrial ecosystems. Ecology 89, 647–659. Odum, H.T., 1996. Introduction: emergy and real wealth. In: Environmental Accounting: Emergy and Environmental Decision Making. Wiley, New York, pp. 1–14. Odum, H.T., 1988. Self-organization, transformity, and information. Science 242, 1132–1139. Odum, H.T., Odum, E.C., 2000a. Calibrating models. In: Modeling for all Scales: an Introduction to System Simulation. Academic Press, San Diego, CA, pp. 96–102. Odum, H.T., Odum, E.C., 2000b. Simulating emergy and transformity. In: Modeling for all Scales: an Introduction to System Simulation. Academic Press, San Diego, CA, pp. 151–164. Odum, W.E., Odum, E.P., Odum, H.T., 1995. Nature’s pulsing paradigm. Estuaries 18, 547–555. Ostfeld, R.S., Keesing, F., 2000. Pulsed resources and community dynamics of consumers in terrestrial ecosystems. Trends in Ecology and Evolution 15, 232–237. Patten, B.C., 1995. Network integration of ecological extremal principles: exergy, emergy, power, ascendency, and indirect effects. Ecological Modelling 79, 75–84. Patterson, M.G., 1996. What is energy efficiency? Concepts, indicators and methodological issues. Energy Policy 24, 377–390. Tilley, D.R., 2010. Mathematical revisions to Odum’s dynamic emergy accounting. In: Proceedings of the 6th Biennial Emergy Conference, pp. 583–597. Tilley, D.R., Brown, M.T., 2006. Dynamic emergy accounting for assessing the environmental benefits of subtropical wetland stormwater management systems. Ecological Modelling 192, 327–361. Ulanowicz, R.E., 1997. Ecology, the Ascendent Perspective. Columbia University Press, New York, NY. Vassallo, P., Beiso, I., Bastianoni, S., Fabiano, M., 2009. Dynamic emergy evaluation of a fish farm rearing process. Journal of Environmental Management 90, 2699–2708. Yang, L.H., Bastow, J.L., Spence, K.O., Wright, A.N., 2008. What can we learn from resource pulses? Ecology 89, 621–634.
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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Resource pulses can increase power acquisition of an ecosystem Seungjun Lee ∗ Department of Environmental Engineering Sciences, Center for Environmental Policy, Phelps Lab., University of Florida, Gainesville, FL 32611, USA
a r t i c l e
i n f o
Article history: Available online 11 January 2013 Keywords: Dynamic emergy accounting Maximum empower principle Pulsing paradigm Resource matching Resource pulse Simulation model
a b s t r a c t Pulsing is prevalent in nature. As resource pulse has been recognized as one of the major factors influencing ecosystem structures and processes, it is important to investigate why nature pulses and what benefits an ecosystem obtains from pulsed resources. The main question of this study was that if a system could be exposed to either constant external resources or pulsed external resources of the same temporal average intensity, which resources would maximize power acquisition of a 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 a refined dynamic emergy accounting method. A producer–consumer model system was built and simulated by varying 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 a system, 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 increase total empower acquisition and consumer energy storage. The primary mechanism was that the matching of pulsed resources in phase and frequency promotes energy acquisition of primary producers that is further efficiently transferred for the production of consumers. Energy acquisition of consumers was strongly correlated with total empower acquisition of the system presumably because the consumers are in the high energy hierarchical position controlling the producers thus contributing to the total empower acquisition through the system. © 2012 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Hypotheses on the self-organization of ecosystems Goal functions for the self-organization of ecosystems have long been discussed among ecologists particularly who have studied energetics or system-level properties. Lotka (1922) stressed that a prevailing system tends to increase total available energy flux through the system during natural selection. Later, the hypothesis of the maximum power principle (MPP) proposed by Lotka (1922) has been partially supported and complemented by Odum. Odum proposed the maximum empower principle (MEP) that hypothesizes natural selection goes to a system that maximizes empower, the quality-corrected energy flux (Brown et al., 2004). He suggested empower instead of power as the maximized flow because the quality difference among various energy types (Odum, 1988) has been a problem in defining energy terms, especially when many energy sources of different qualities drive a process (Patterson, 1996).
Although the MPP or MEP has been hypothesized as a goal function for the self-organization of an ecosystem, only a few empirical or modeling studies on the hypothesis have been reported (e.g., Cai et al., 2006; DeLong, 2008). The scarcity of the studies on the MPP or MEP may be attributed to the difficulty in identifying and quantifying available energy flux through complex energy networks (Cai et al., 2004). Meanwhile, Ulanowicz (1997) proposed ascendency as a hypothesis on the development of ecosystems. Ascendency, an index encompassing both qualitative and quantitative aspects of system development, is defined as the multiplication of average mutual information and total system throughput. Although the ascendency includes mutual information as a factor of system development, it appears that both the MPP and ascendency hypotheses agree that a critical component of self-organization in a prevailing system is the total energy flux through the system.
1.2. Pulsing paradigm and resource pulse
∗ Tel.: +1 352 392 2426; fax: +1 352 392 3624. E-mail address: slee@ufl.edu 0304-3800/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2012.11.028
Resource pulse 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
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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 pulse, flood pulse, population oscillation), 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 instance, an ecosystem may better obtain 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 well. 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 the same between a constant external resource and a pulsed one, would the ecosystem draw more energy flux under the matching of the two pulsed external resources than that of the 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. 1.3. Dynamic emergy accounting 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), however, there have been only a few studies related to it (e.g., Tilley, 2010; Tilley and Brown, 2006; Vassallo et al., 2009). As Tilley (2010) discussed, the dynamic emergy accounting method introduced by Odum (1996) conflicts with the concept of emergy so it is necessary to refine the method to accurately simulate the trajectories of emergy or UEV.
frequency combination among the sources. I hypothesized that a system under pulsed external energy sources draws more empower through the system under the 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. 2. Theory 2.1. 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 Fig. 1 as follows: dQ > 0 : dEmQ = EmJ − EmH dT where EmJ = TrJ ·J and EmH = TrQ ·K2·Q·F
(1)
dQ = 0 : dEmQ = 0 dT
(2)
dQ < 0 : dEmQ = TrQ · dQ (3) dT where EmX is the emergy of X, TrX is the transformity (or UEV) of X. 2.2. Inconsistencies in the Odum’s method 2.2.1. The depreciation pathway When dQ/dT > 0, or Q is accumulated, the emergy of the depreciation pathway K1·Q was considered zero in Eq. (1) because heat sink is a necessary process to maintain Q but does not carry emergy. Odum and Odum (2000b), however, stated regarding Eq. (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 Eq. (1). That is, the depreciation pathway K1·Q was regarded as carrying emergy in Eq. (3) but not in Eq. (1). Because heat sink is necessary to maintain Q but does not carry emergy, emergy is not lost through the depreciation pathway.
Energy diagram F
Energy Source
J
Q
K2 Q F
K1 Q
Emergy diagram
F
1.4. Study plan This study aims to investigate how the matching of pulsed external energy sources influences empower (emergy/time) acquisition of an ecosystem using a numerical simulation model. First, I refined Odum’s dynamic emergy accounting method for the simulation of emergy. 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
EmJ
EmQ
EmH
Zero Fig. 1. Odum’s energy and emergy systems diagrams for the dynamic accounting method (Odum and Odum, 2000b).
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2.2.2. When dQ/dT = 0, dEmQ = 0? In the second conditional equation (Eq. (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 us 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. 2.2.3. 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. 2.2.4. 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 of emergy for a state variable can be written by inflows and outflows of emergy without conditional equations. 2.3. A new method in dynamic emergy accounting Tilley (2010) suggested a unified equation for dynamic emergy accounting derived from Fig. 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
TrQ =
EmQ Q
(4)
(5)
Eq. (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 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 Eq. (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
23
K1·Q into a heat sink and an available energy. Eq. (4) can be generalized by the following: dEmQ = TrQ =
X
TrX · X −
Y
TrQ · QY
EmQ Q
(6)
(7)
where X is the input available energy flux of each kind, QY is the output available energy flux of each kind from Q. 3. Methods 3.1. Model system The effects of pulsed energy sources on empower acquisition of an ecosystem were studied using a producer–consumer model. The model (Fig. 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 flowlimited. 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. Energies 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 very large amount, 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 density of energy sources 2 and 3 were defined as E2 and E3, respectively. 3.2. Simulation of the basic model Using the energy and emergy systems diagrams (Fig. 2), a numerical simulation model was built (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. Values for energy flows and storages were roughly estimated by assuming the steady-state of the state variables (P and Q), and the coefficient for each pathway function was back-calculated from the pathway function and its flow value as suggested by Odum and Odum (2000a). Energy values and their transformities (Table 2) were selected by considering the hierarchy among the energy sources (i.e., J1, E2, E3, TrJ1 , TrJ2 , TrJ3 ). It was assumed that the consumption rate from P to Q is half of the production rate of P. The initial steady-state values of P and Q were set at 200 J and 600 J by considering the reasonable energy flow rates through pathways and the replacement times of P and Q. The steady-state model was simulated in 0.1 day intervals for 1000 days (10,000 steps) using R (available at the R project for Statistical Computing, http://www.r-project.org). Energy, emergy, transformity of P and Q, and empower (emergy/time) values from the three energy sources (EmJ1−R , EmJ2 , EmJ3 ) were simulated. 3.3. Definition of the constant and pulsed energy sources Fig. 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 ()
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Energy diagram Energy Source 2 E2
Energy Source 3 E3
J3
J2 FPfb Energy Source 1
J1
FQfb FPcon
P FPpro
Q FQpro
FQdd
FPdd R
FQhs
FPhs
Emergy diagram EmJ3
EmPfb
EmJ1-R + EmJ2
P
EmQfb Q
EmPcon EmPcon + EmJ3 EmPdd
Zero
Zero
EmQdd
Fig. 2. Energy and emergy systems diagrams of the producer–consumer model.
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 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 the following equations: Constant energy source : ES = c
(8)
Table 1 Variables and equations for the producer–consumer model. Variables and equations
Description
Unit
P Q J1 R E2 E3 J1 − R = ES1in·P·R·E2 J2 = ES2in·P·R·E2 J3 = ES3in·P·Q·E3 FPpro = Ppro·P·R·E2 FPfb = Pfb·P·R·E2 FPhs = Phs·P FPdd = Pdd·P FPcon = Pcon·P·Q·E3 FQpro = Qpro·P·Q·E3 FQfb = Qfb·P·Q·E3 FQhs = Qhs·Q FQdd = Qdd·Q TrJ1 TrJ2 TrJ3 TrP = EmP /P TrQ = EmQ /Q EmJ1−R = ES1in·P·R·E2·TrJ1 EmJ2 = ES2in·P·R·E2·TrJ2 EmJ3 = ES3in·P·Q·E3·TrJ3 EmPfb = Pfb·P·R·E2·TrP EmPdd = Pdd·P·TrP EmPcon = Pcon·P·Q·E3·TrP EmQfb = Qfb·P·Q·E3·TrQ EmQdd = Qdd·Q·TrQ d(EmP )/dT = EmJ1−R + EmJ2 − EmPfb − EmPdd − EmPcon d(EmQ )/dT = EmPcon + EmJ3 − EmQfb − EmQdd
Storage of producers Storage of consumers Total flow rate of energy source 1 Flow rate of remaining energy source 1 Density of energy source 2 Density of energy source 3 Inflow rate of captured energy source 1 by the system Inflow rate of energy source 2 drawn by the system Inflow rate of energy source 3 drawn by the system Production rate of P Energy consumption rate of P for autocatalytic feedback Heat sink rate of P Density-dependent death rate of P Rate of P consumption for Q production Production rate of Q Energy consumption rate of Q for autocatalytic feedback Heat sink rate of Q Density-dependent death rate of Q Transformity of energy source 1 Transformity of energy source 2 Transformity of energy source 3 Transformity of P Transformity of Q Inflow rate of emergy from energy source 1 Inflow rate of emergy from energy source 2 Inflow rate of emergy from energy source 3 Emergy flow rate of autocatalytic feedback of P Emergy flow rate of density-dependent death of P Emergy flow rate of P consumption for Q production Emergy flow rate of autocatalytic feedback of Q Emergy flow rate of density-dependent death of Q Emergy change of P per time Emergy change of Q per time
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
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Table 2 Steady-state values of the variables and coefficients for the producer–consumer model. Variable or coefficient
Value a
200 600a 10,000 5000 100 100 1 100 1000 5.00 × 10−5 1.00 × 10−6 8.33 × 10−6 1.00 × 10−6 2.00 × 10−7 0.05 0.1 4.17 × 10−6 3.33 × 10−6 1.67 × 10−6 1.67 × 10−2 1.67 × 10−2
P Q J1 R E2 E3 TrJ1 TrJ2 TrJ3 ES1in ES2in ES3in Ppro Pfb Phs Pdd Pcon Qpro Qfb Qhs Qdd a b c d
Unit
Steady-state energy flowb
Meaning
J J J/day J/day J J sej/J sej/J sej/J J−2 J−2 J−2 × day−1 J−2 J−2 day−1 day−1 J−2 × day−1 J−2 × day−1 J−2 × day−1 day−1 day−1
– – – – – – – – – 5000 100 100 100 20c 10 20 50 40 20d 10 10
– – – – – – – – – Rate of energy source 1 per unit P and energy sources 1 and 2 Rate of energy source 2 per unit P and energy sources 1 and 2 Rate of energy source 3 per unit P and Q and energy source 3 Reproduction proportion of P per unit energy sources 1 and 2 Consumption proportion of P per unit energy sources 1 and 2 Specific rate of heat sink from P Specific rate of density-dependent death of P Predation proportion of P by Q per unit Q and energy source 3 Reproduction proportion of Q per unit P and energy source 3 Consumption proportion of Q per unit P and energy source 3 Specific rate of heat sink from Q Specific rate of density-dependent death of Q
Replacement times of 2 and 15 days were assumed for P and Q, respectively. Steady-state energy flows were roughly estimated. It was assumed that 20% of production is used for feedback in P. It was assumed that 50% of production is used for feedback in Q.
Pulsed energy source : EP = c
sin
2 a
(t − ) + 1
(9)
where a is the frequency (days) of the pulsed energy source, t is the time (day), and (days) is the phase of the function. 3.4. Tests of the effects of pulsed energy sources and analyses of mechanisms The effects of phase difference and frequency combination 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, transformity of P and Q, and empower from the three energy sources were simulated by varying between the pulsed energy sources. For the test of frequency combination, variable frequency combinations among the three pulsed energy sources were applied in the simulation model with = 0. Average flows and storages of energy and emergy in the pathways and state variables were calculated to analyze how the matching of pulsed energy sources in phase and frequency changes the acquisition of empower and energy in the system. Average
replacement times of P and Q were calculated to analyze the energy replacement and transfer rates within the system under the matching of the pulsed energy sources. Replacement time under the steady state of P or Q was calculated by the following equation: Replacement time(day) =
energy storage (J) production rate (J/day)
3.5. Tests of the effects of pulsed energy sources with an additional energy source As more than two pulsed energy sources can be used in the production of consumers (Q) and the matching of them may change the energy and emergy flow patterns in the system, the effects of
Energy Source 3 E3
J3 Frequency of A Phase difference (
2c
A
B
Frequency of B
Energy (J)
Constant resource
Energy Source 4 E4
J4 FQfb = Qfb P Q E3 E4
Pulsed resources
)
(10)
FPcon = Pcon P Q E3 E4 FQpro = Qpro P Q E3 E4
c
0 Time (Day)
Fig. 3. Frequency and phase difference in the pulsed energy sources.
Fig. 4. Part of the energy systems diagrams of the producer–consumer model with the addition of energy source 4.
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frequency combination on the empower acquisition of the system were examined by adding additional constant-force energy source 4, inflowing along with energy source 3 for the production of Q (Fig. 4). The density of energy source 4 was defined as E4, whose value was 10 J and transformity was 10,000 sej/J. The steady-state flow rate of the energy source 4 was 10 J/day. I tested how the matching of frequencies among energy sources involved in the same or different transformation processes (i.e., production of P or Q) affects the empower acquisition of the system by varying the frequency combination among the four energy sources. 4. Results 4.1. Simulation results under the constant energy sources
4.2. Simulation results under the pulsed energy sources 4.2.1. Temporal patterns of the variables The variables of P, Q, EmP , EmQ , and total empower oscillated under the pulsed energy sources. Fig. 5 shows the simulation results under the frequency (days) combinations of 2–2–2 and 2–15–15 among the three pulsed energy sources (i.e., 2–2–2 means the pulsing frequencies of J1, E2, and E3 are 2, 2, and 2 days, respectively. Likewise, 2–15–15 means the frequencies of J1, E2, and E3 are 2, 15, and 15 days, respectively.). When the frequencies of the three energy sources were identical (i.e., 2–2–2), P, EmP , and total empower oscillated with one maximum and one minimum peaks within an oscillation cycle like sine functions. When the frequencies of the three energy sources were different (i.e., 2–15–15), however,
270
800
240
700
210
600
Q (J)
P (J)
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 not to change the simulation results 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.
180 150
400
120
300 520
540
560
580
600
4.5
3.0
4.0
2.5
EmQ ( 106 sej)
EmP ( 104 sej)
500
3.5 3.0 2.5
500
520
540
560
580
600
500
520
540
560
580
600
500
520
540
560
580
600
2.0 1.5 1.0
500
520
540
560
580
600
3.0
Total empower ( 105 sej/day)
Total empower ( 105 sej/day)
500
2.5 2.0 1.5 1.0 0.5 0.0 500
520
540
560
Time (day)
580
600
3.0 2.5 2.0 1.5 1.0 0.5 0.0
Time (day)
Fig. 5. Examples of simulation results under the pulsed energy sources (Black lines: frequency combination of 2–2–2. Gray lines: frequency combination of 2–15–15. Only simulation results from Day 500 to 600 are shown for better representation of the patterns. Pulsing patterns after Day 600 are the same as those from Day 500 to 600.).
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Fig. 7. Average energy of P and Q and total empower under variable frequency combinations among the three energy sources, where = 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).).
The phase matching between J1 and E2 (phase deviation of E3 from J1 and E2) relatively stabilized the energy storage of Q and total empower even under the phase deviation of E3 from J1 and E2 (Fig. 6b and 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. Fig. 6. Average energy of P and Q and total empower under variable 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 (). The dashed lines indicate P, Q, and total empower under the constant energy sources.).
P, 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 steady state (pulsing steady state) after day 500. 4.2.2. 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 (Fig. 6a). 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. The energy storage of Q and total empower were maximal when the phase difference among the pulsed energy sources was near zero (Fig. 6b and c). Q was maximal at 762 J when the phase difference of E3 from the other energy sources was about 4 days. Q was 744 J when the phases of the three energy sources were matched ( = 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 = 0.
4.2.3. 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 (Fig. 7). Simulation was conducted under variable frequency combinations that were selected in consideration of the replacement times of P and Q. I speculated that maximal empower acquisition may occur when the pulsing frequencies of resources are close to the replacement times of P and Q (2 and 15 days). From this point, I simulated the model with variable frequency combinations within 100 days. 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 (i.e., 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). 4.3. Analyses of mechanisms To analyze energy and emergy flow patterns under the pulsed energy sources, average flow rates and storages were calculated from the simulation results (Fig. 8). Under the pulsing frequency combination of 15–15–15 ( = 0), average inflow rates of energy sources 1, 2, and 3 increased from 5000 J, 100 J, and 100 J to 5614 J, 112 J, and 129 J, respectively. More energy accumulated in P was
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Energy diagram Energy Source 3
Energy Source 2
112 (100)
Energy Source 1
10000 (10000)
129 (100)
22 (20)
5614 (5000)
26 (20)
64 (50)
172 (200) 112 (100)
764 (600)
51 (40)
17 (20)
13 (10)
13 (10)
9 (10)
Storages: J Flows: J/day
Emergy diagram 128605 (100000)
3622 (3333)
16843 (15000)
28847 (33333)
Zero
92485 (72222) 138976 (108333)
10371 (8333)
2757974 (2166667)
2885 (3333)
45966 (36111)
Zero
Storages: sej Flows: sej/day
Fig. 8. Average energy and emergy values (Day 500–980) under the pulsing frequency combination of 15–15–15 and of 0 (upper numbers are flows or storages under the pulsed energy sources and lower numbers in the parentheses are flows or storages under the constant energy sources).
empower (r2 = 0.9977 in phase tests and r2 = 0.9963 in frequency tests). 4.4. When two energy sources are involved in the Q production When additional energy source E4 was added to the model (Fig. 4), the energy of P and Q were determined by the frequency combination of J1–E2–E3–E4 (Fig. 11). The matching of frequencies between the two energy sources J1 and E2, which were involved 16.0
Replacement time of Q (day)
transferred to Q and less energy in P dissipated through the heat sink or density dependent pathway. That is, productivity of P increased 12% from 100 J to 112 J, while consumption rate of P for Q production increased 28% from 50 J to 64 J. The increased transfer rate of energy from P to Q eventually increased the energy storage of Q 27% from 600 J to 764 J. Emergy flow patterns were similar to those of energy flows (Fig. 8). Emergy productivity of P increased 12% from 15,000 sej to 16,843 sej, while emergy consumption rate of P for Q production increased 24% from 8333 sej to 10,371 sej. As a result, emergy of Q increased 27%. Because the increased productivity of P and energy transfer rate from P to Q were identified as the major mechanisms of the increased energy storage of Q and total empower under the frequency and phase matching (15–15–15, = 0), replacement times of P and Q were calculated to confirm the mechanisms (Fig. 9). Replacement time indicates the speed of production or consumption of the storages. The replacement times of P and Q were 2 and 15 days, respectively, when the energy sources were constant. Among the tested pulsing frequency combinations, the replacement times of Q were in the range of 14.92 ± 0.13 days, while those of P varied in the range of 1.73 ± 0.39 days. The replacement time of P was greater than 2 days when the pulsing frequency of J1 was different from that of E2, and less than 2 days when the pulsing frequencies of J1 and E2 were matched. Figs. 6 and 7 show the close relationship between Q and total empower. In Fig. 6, the graphs of Q and empower show a similar pattern. In Fig. 7, empower was generally higher in higher Q. To closely examine the relationship, the correlations of P and Q with total empower were tested in Fig. 10. There was no correlation between P and total empower (r2 = 0.0025 in phase tests and r2 = 0.1495 in frequency tests), while Q was strongly correlated with total
15.5 2-2-15 1-2-15 15-15-2
15.0
100-100-100 30-30-30 50-50-50
10-10-10
20-20-20
15-15-15
2-15-15
No pulsing
3-3-3 2-2-2 5-5-5 1-1-1
15-2-15
14.5
14.0 1.0
1.5
2.0
2.5
3.0
Replacement time of P (day)
Fig. 9. Average replacement time (Day 500–1000) of P and Q with variable frequency combinations among the three energy sources, where = 0.
S. Lee / Ecological Modelling 271 (2014) 21–31
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Fig. 10. Correlations of P and Q with total empower (a) test of phase difference and (b) test of frequency combination (each point is an average of all energy or empower values from Day 500 to 1000 under a tested or frequency combination).
in the P production, was necessary for high Q and total empower. Matching of frequencies between E3 and E4, which were involved in the Q production, however, was not a necessary condition for high Q or total empower. For example, Q was 689 J and total empower was 2.47 × 105 sej/day under the frequency combination of 2–2–2–15. Even if the frequencies of E3 and E4 were matched, P, Q, and total empower under the frequency combinations of 15–2–15–15 and 2–15–15–15 were lower than those under the constant energy sources. The energy storage of Q and total empower were high when the frequency of E3 or E4 was close to the matched frequency of J1 and E2. For example, Q was 689 J and total empower was 2.47 × 105 sej/day under the frequency combination of 2–2–2–15. However, Q was 595 J and empower was 2.08 × 105 sej/day under 2–2–15–15, which were lower than those under the constant energy sources. When the frequencies of E3 and E4 deviated more from those of J1 and E2 (i.e., 2–2–30–30), Q and empower decreased to 504 J and 1.81 × 105 sej/day, respectively. The mechanism of increased empower acquisition under the four pulsed energy sources was similar to that under the three pulsed energy sources. Under the frequency combinations of 15–15–15–15, 2–2–2–2, 5–5–2–2, 2–2–5–5, and 15–15–2–2, the energy of Q was greater and that of P was less than those under the constant energy sources (Fig. 11a). The replacement times of P under these frequency combinations were shorter than that under the constant energy sources (Fig. 11b).
5. Discussion 5.1. ‘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 under the MEP that 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. 5.2. Do pulsed energy sources increase the empower acquisition of an ecosystem? 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
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2-2-2-15 (2.47)
15-15-15-15 (2.47) 2-2-2-2 (2.41)
5-5-2-2 (2.27)
650
2-2-5-5 (2.23) 15-15-2-2 (2.22) No pulsing (2.15)
600 Q (J)
2-2-15-15 (2.08)
550 2-2-30-30 (1.81)
500
15-2-2-15 (1.75)
450
2-15-2-15 (1.64)
15-2-15-15 (1.53) 2-15-15-15 (1.49)
400 100
150
200
250
300
P (J)
(b)
16.0
Replacement time of Q (day)
5-5-2-2
15.5
2-2-5-5 2-2-15-15
15-15-2-2 2-2-30-30 2-15-15-15
15.0
2-2-2-15 No pulsing
15-15-15-15 15-2-15-15
2-15-2-15
2-2-2-2
turnover rate (short replacement time) of P and efficient transfer of energy from P to Q. The faster turnover rate of P may indicate that fast developing primary producers are favored in competition under the pulsed resource environment (Friman and Laakso, 2011; Holt, 2008). Although the frequency matching between J1 and E2 is critical, the test of four energy sources suggested that the energy of Q and total empower may be reduced by pulsed energy sources when the frequencies of the energy sources involved in the Q production (E3 and E4) are far from those of J1 and E2. The energy sources involved in the Q production may need to pulse as frequently as the pulses of the energy sources involved in the P production to accelerate overall production by frequently utilizing the energy storage of P. The results of simulation provide some insight on the supply of resources and ecosystem growth. Seasonal or daily matching of pulsed resources would promote the growth of a natural ecosystem. In the long term, for example, dry season fires regenerate nutrients for the growth of forests during the rainy season. When the peak of the nutrient pulse is matched with the peak insolation and water, the primary productivity will increase to promote the secondary production. In the short term, for example, daily fluctuation of insolation tends to be matched with frequent water supply and nutrient fluctuation from fast consumption and recycling under the warm and humid climate.
15-2-2-15
5.3. Maximum empower and consumer energy
14.5
14.0 1.0
1.5
2.0
2.5
3.0
Replacement time of P (day)
Fig. 11. Average energy values and replacement times of P and Q under variable frequency combinations among the four energy sources, where = 0, (a) energy and total empower and (b) replacement times (Each point is an average of all values from Day 500 to 1000. The four-number combination in each point indicates pulsing frequencies of J1–E2–E3–E4 in days. Numbers in the parentheses indicate total empower (×105 sej/day).).
In the producer–consumer model system, the energy storage in Q was strongly correlated with the total empower acquisition through the system. This may indicate that the energy storage of consumers is an indicator of maximum empower through the system. Cai et al. (2004) quoted the 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. 5.4. An intuitive mechanism
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, R (For the remaining R, see Fig. 2 and Table 1. See Montague, 2007 for the equation and details of remaining energy R in a flowlimited source.). Although the remaining energy 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 complete 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. The analyses of the energy and emergy flows in the system and the replacement times of P and Q revealed that the high Q and total empower can result from the fast
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 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 1.5 c2 , 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 than that of the constant energy sources does. Since the amount of energy flow through the system is determined by the system configuration, however, the simple mathematical fact may not always guarantee that the matching of pulsed energy sources increases 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. 5.5. Practical aspects The results of this study provide a theoretical background for understanding empower acquisition through an ecosystem under pulsed resources. The periodic pulses of resources in the studied
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model system are probably unrealistic to represent the pulses of a real ecosystem that is generally influenced by both temporally regular and irregular resources. Nevertheless, the results suggest that the matching of pulsed resources possibly promotes the empower acquisition of a system. The methodology used in this study may be applied to specific ecosystem types. For example, the characteristic difference between terrestrial and aquatic ecosystems under the pulsed resource environment (Nowlin et al., 2008) would be better understood by applying the patterns of pulsed resources and parameters observed in those ecosystem types to the simulation model of this study. Once a specific model of an ecosystem is established, one can calculate energy and emergy of variables and flows. Creating and referring to the graphs, such as Figs. 6 and 7, practitioners may restore or manage resource pulsing patterns that result in a designated energy level in a system (e.g., biomass). This practice may be useful for the management of human-caused resource supply (e.g., nutrient overload). 6. Conclusions This study started with the question that if a system could be exposed to either constant external resources or pulsed external resources of the same temporal average intensity, which resources would maximize power acquisition of a system. According to the maximum empower principle, a system is likely to select resources or change to a new configuration that maximizes total empower acquisition. Simulation of a producer–consumer model showed that total empower acquisition of the system and energy gain in the consumer storage can increase under the pulsed energy sources. In particular, the matching of phases and frequencies among the pulsed energy sources involved in the primary production was critical to increase total empower acquisition and consumer energy storage. The primary mechanism was that the matching of pulsed energy sources in phase and frequency promotes energy acquisition of primary producers that are further efficiently transferred for the production of consumers. Energy acquisition of consumers was strongly correlated with total empower acquisition of the system presumably because the consumers are in the high energy hierarchical position controlling the producers thus contributing to the total empower acquisition through the system. Acknowledgements I am grateful to Mark Brown and anonymous reviewers for valuable comments and suggestions on the manuscript.
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