J. theor. Biol. (1979) 80, 271293
Total Energy Costs in Ecosystems BRUCE
HANNON
Energy Research Group, Office of ViceChancellor for Research, University of Illinois at UrbanaChampaign, Urbana, IL 61801, U.S.A. (Received 8 March 1978, and in revised form 14 May 1979) A micro and macro ecoiogical system hypotheses and their associated theory of total (direct and indirect) energy cost have been proposed to explain the meaning of system growth and the steady state condition. Models have been proposed which can be used to test the hypotheses. In brief, the hypotheses are that while the components of an ecosystem strive to maximize their total direct and indirect energy storage within the constraints of their production characteristics, the overall system strives to minimize the metabolized energy per unit of stored biomass energy. The steady state result is a system with the largest possible stored biomass under the general constraints of water, air, soil and sunlight availability. Feeding strategies are discussed with the conclusion that a net total energymaximizing feeding strategy may be able to replace the present concepts.
1. Introduction The introduction of economic modeling into ecology has been well summarized by Rapport & Turner (1977). Analogies with economic inputoutput analysis and complex ecosystem structures and flows have been established (Hannon, 1973; Finn, 1976; Patten, Bosserman, Finn & Cale, 1977) and feeding strategies have been discussed in terms of benefits and costs (Schoener, 1971; Rapport & Turner, 1975,1977). An analogy between energy storage maximizing ecological components and profit maximizing industrial firms has been drawn. The establishment of ecological “markets” and supplydemand equilibrium in the ecosystem has also been discussed (Hannon, 1976). In this paper, the concept of direct and indirect or total energy cost is defined and micro and macroecological development forces are proposed. For each component of an ecosystem in any given time period, it is hypothesized that a newlydefined energy is maximally stored with respect to the input and output flows of that component. This is the proposed micro211
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B. HANNON
ecological force. The system is also assumed to operate under a macroecological force. The ecosystem as a whole develops in such a way that the metabolized energy per average unit of system storage is continuously reduced. This energy expenditure reaches a minimum at the system steadystate condition where all incoming energy is used for maintenance and operation and system storage is at the maximum possible level. Using the concept of total energy cost, I hypothesize finally that feeders will organize their inputs according to a maximum net total energy criterion. Should these hypotheses be experimentally confirmed, a simple and consistent view of ecosystem behavior would be established. 2. Theory (A)
DEFINITIONS
A component is a transforming or storage unit for which the complete set of mass or energy inputs and outputs can be specified. The ecosystem is defined as N such components (n living components plus two special components) which are related to each other by these flows. A living component is one which has the capacity for reproduction. Figure 1 shows the energy balance for a typical component in the ecosystem. The energy flow from outside the system into the jth component is defined? as Ej, the sunlight photosynthesized by the system or any other external energy input. Other external energies used by a biological component are solar and geothermal heat energy, the free chemical energy in imported substances such as air, water and critical nutrients and the heat which is embodied (e.g. evaporation and transport of rain water) in physical inputs to a component. These nonphotosynthetic energies could be included in the analysis of this paper but are omitted to simplify the presentation. For a discussion of the means of incorporating these energies into ecosystem analysis, the reader is referred to H. T. Odum (1957). The production input mass or energy flows to each component are called xii and the total output mass or energy flows are called xj. I wish to compute all of the ecosystem energy flows involved in storing energy or, equivalently, exporting energy from storage. To do this, I must define two types of energy flow: direct and indirect. The direct energy flow into the jth component is composed of solar energy and any other caloriometric energy form supplied from outside the system, and the caloriometric values of the Xij. The indirect energy flow is the energy 7 When
the letter
symbol
appears
in boldface
type,
it represents
a vector
or a matrix.
TOTAL
Production
ENERGY
COSTS
IN
lI
inputs
Feedmg and subsistence
J
213
ECOSYSTEMS
iota I output El 5
l , h’fXb), pduction
outputs
CE f I .rI*
balancefor the jth component of an ecosystem. associated with all those flows other than the direct energy flows themselves, which are necessary so that the direct energy flow can take place. As an everyday example, consider the electricity from a wall outlet which is running an electric clock. The direct energy is that energy actually flowing into the electric clock motor. The indirect energy is the energy used in transforming, transmitting and generating the electricity plus the energy used in mining and transporting the power plant fuel and in supplying all the other inputs necessary for generating the directly used electricity. In the same way direct and indirect energy are defined here for the ecosystem. The sum of the direct and indirect energy is called the total energy. The total energy cost, .zj, is the total energy expended per unit of output, xj. The system flows are arranged in matrix form in Fig. 2. There is only one type of total output flow from each component although it may flow to every component in the system. The output flow is composed of two parts: the respiration flow xJ and the production output flow, for example, xjk, the production output flow from thejth to the kth component. The flows (xii, xj) may be stated in terms of energy, mass or a combination of both. When a combination of mass and energy flows are used, only the flows which make up a particular xj need to be stated in the same units. The respiration flow for the jth component is divided into four parts: xi, the feeding flow or the energy spent acquiring inputs; x$‘, the basal metabolism flow needed for subsistence ; x7, the flow into storage ; and x7, the flow exported from system storage. The “metabolized” energy (thermal form only) or “maintenance and operating” energy are xj”+xf. The feeding flow expenditures can be divided into parts such as Iijxij. The dimensionless constant values 1, modify the input flows such that FIG.
1. The
energy
x; = i I,x,. i=l
The effectiveness of the input flow is reduced by the amount of feeding flow required to obtain it.
B. HANNON
274 I
2
Note
that
.
j
.
n
Environment1
Storoge
ond export
'i::j
Ef +
E,” = E,
and
,$lFp
= ,$(x;
tx,')
FIG. 2. Matrix display of the flows through a general ecosystem.
The total system energy input and the energy given off by all components as heat are thought of as coming from and going to an “Environment” component. The introduction of this component allows the inclusion of the metabolism flows (xf and xb) in the matrix. Since these metabolism flows are not the entire respiratory energy flow, the total system energy input must be broken into two parts: the first component (Environment I) has a total output ($(,+,));’
which
is equal
to the combined
th e second component
(Environment
metabolism II)
flow
has a total
which is equal to the total system energy storage and export (i (xS+xe) i 1). The distribution
of the output of the Environment
II
component can only be determined by experiment. Theoretically, these outputs are the excess energy values generated by each of the components and would therefore parallel the distribution of the energy storage vector. If the only energy leaving the flowing portions of the system is exported energy, then the distribution of the output of Environment II would reflect this export vector exactly. Since the total system energy input quantity and distribution is known, the distribution of the energy output from the Environment I component can be determined unambiguously by subtraction. When the flows from the components 1 through n are stated in terms of mass, then the flows from the Environment II component are in mass terms and will not affect the flows from the Environment I component which would then be the total energy input to the system. In this case the
TOTAL
ENERGY
COSTS
IN
ECOSYSTEMS
275
Environment II flows are irrelevant to the theory of total energy cost derived in the next section. This is true since the system outputs (column values) in the matrix for the Environment II component are always zero. In the case where all the system flows are measured in energy terms the Environment II component affects the total energy costs only by affecting the flow values in the Environment I component’s output.
(B)
DEVELOPMENT
OF THE
ENERGY
COST
EQUATIONS
First, I assume that the variable of time is completely implicit. This means that all the processes described by the ensuing theory take place in a unit time. Thus, for example, xij and xj are actually flows even though the units do not need to explicitly contain units of time. The following theory therefore represents a picture of the interaction among the components within the given time period. Because of the difficulty of ecosystem data gathering, this period would most likely be a climatic cycle, such as a year, since most of the flows tend to vary greatly with climatic change. From Fig. 1, the energy balance for the jth component is Ej = ~~~~;
In matrix notation,
.sixij.
(1)
for all components, E = @xX).
(2)
A similar statement for the U.S. economy has been made by Bullard & Herendeen (1975). Here E is the vector of ail direct external energy inputs, the matrix d is a diagonalized vector of the output flows, x is a matrix of the input flows and E is the vector of total (direct plus indirect) energy costs per unit of output flow. To solve for E, the matrix (zi x) must be invertible. To aid in establishing invertibility, the matrix is normalized with respect to A, Eg’
= E(g]il
Xkl)
E’ = &(1G’) or E = E’(IG’)l, where E’ is a row vector in which each term is Ei~i and G = 2xl.
(3)
The total
B. HANNON 276 energy costs, E, exist and are unique if each irreducible submatrix of G’, defined as G’ has elements &u, such that
block diagonal
with the inequality holding for at least onej (Ortega, 1972). By establishing the Environment II component whose row consists of balancing flows and whose column consists only of zeros, the irreducible block matrix has at least one column sum less than one. The E are not affected by the actual row entries of the Environment II component. It is important to understand the nature of the values of E. If the two import components had not been introduced, E would represent the direct energy content of the associated production flows. The G’ matrix, if expressed completely in energy units, would then be identical to the G matrix in Hannon (1973). With the first import component containing the feeding and subsistence energy flows, these flows and the direct energy content of the production flows are combined into the E. Therefore, si represents the total system energy cost per unit of storage in the ith component in the given time period. Equation (3) is also a definition of energy cost. The definition is based on relative or normalized flows and therefore are not necessarily dependent on the absolute value of these flows. To produce the storage function six;, I note that the flow balance requirements for the ith component are (see Fig. 1): N
xi = x; + c Xik k
and therefore, if I combine export and storage flows into x; N Xl
= C
(XX)ij
=
(4)
XT,
since the feeding and subsistence flows are combined into a column (Environment I) of the matrix (see Fig. 2). Consider next the first form of equation (3) in component form Ej = ~ Ei(~  X)ij, c
(5)
The total system energy input is therefore, with equation (4), E totat = ~ Ei ~ j
(aX),
= ~
EiXf.
(6)
TOTAL
ENERGY
COSTS
IN
ECOSYSTEMS
277
From equation (6) it is clear that in the special steady state case where each xs approaches zero, then each &i would approach infinity. This condition may be avoided by including the replacement or “depreciation” flows with the storage flows. A more probable system steady state condition, however, is that the x; are nonzero but
In this case the E are not infinite. From equation (6) it is clear that if I assume that the ai are constant for the time period under study, then I can determine the effect of changes in the x9 on the required total energy input. That is, the additional energy input required by the entire system to replace an additional stored or exported unit of xi would be Ed. The same result could be achieved by the stricter assumption that the E’ and G’ are constant in equation (6). To store or to export one more unit of x; required &i more units of input energy into the system. Thus for a small change in an x4, which does not either affect the other $ or change the component’s production abilities appreciably, the required additional energy flow into the system can be estimated.
(C)
EXAMPLES:
OF ENERGY
COSTS
To demonstrate the application of equations (3) and (6) I choose two wellstudied ecosystems: Silver Springs (Odum, 1957) and Cone Spring (Tilly, 1968). Both models are based on energy flow in aquatic ecosystems. The Silver Springs system is strictly hierarchial while the Cone Spring System contains several large feedback loops. The use of annual data in these examples does not imply that I think annual cycles will not mask important and meaningful variations in the E. Collapsing the data into an annual framework no doubt does conceal some important variations. I am not able to find data for a shorter term, or more interestingly, a series of shorter terms, for a growing ecosystem. The data for the Silver Springs system is balanced and displayed in Hannon (1973). The production matrix for equation (3) is similar but contains two import components as shown in Table 1. The Silver Springs system was in the steady state condition. Therefore, there was no storage during the period of study. The 2,498 kcal/M’/yr is the total export from the system and I have assigned all of it to the producer component. The numbers in the Environment I column are the metabolic rates for each of the living components. The entry in the Environment II row is needed to balance the export flow. The Environment II row entries are
B. HANNON
278
TABLE
1
The production matrix for the Silver Springs ecosystem (kcal rnb2 yr‘)
1 Producer (1) Herbivore (2) Carnivore (3) Tcp carnivore (4) Decomposer (5) I
Component 3
2
4
5
Environment I
Storage and export
0 0 0 0 0 18,312
2,874 0 0 0 0 494
0 382 0 0 0 0
0 0 21 0 0 0
3,455 1,095 46 6 460 0
11,983 1,891 315 15 4,602 0
2,498 0 0 0 0 0
2,498
0
0
0
0
0
 2,498
II
similar to the “value added” quantities in the economic accounting system. Row totals (including export and storage) must equal corresponding column totals. The E vector is [20810,494,0,0,0,0,0] and therefore the E’ vector is [l, 0147,0,0,0,0,0]. Each column entry in the matrix in Table 1 is normalized by the corresponding row total (total output). This process yields the G’ matrix. Following equation (3), the matrix (IG’)l is formed and presented in Table 2. With the data in Table 2, I can produce the energy costs for the system by equation (3). These new costs appear in Table 3. Since the actual distribution of the exported biomass was not known, a second set of energy costs were calculated by assuming that the exported biomass (and the Import II component entries) was distributed over each component in proportion to the component’s standing biomass. These results are also shown in Table 3. The variation in energy cost is obviously quite small under either distribution of the exported quantity. The lack of variation may be due to the TABLE 2
The (I  G’)l
matrix for the Silver Springs production matrix of Table 1
1
2
Component 3
4
5
Environment I
:
8.33 1.35
2.37 8.33
2.37 8.33
2.37 8.33
1.60 8.33
8.33 1.53
: 5 I
O16 0.01 203 7.53
0.01 0.16 207 7.68
0.01 1.16 2.07 7.68
1.01 1.16 2.07 7.68
0.01 0.17 3.14 7.51
0.01 0.18 231 8.56
II
1.00
1QCJ
1ao
10k.l
1.00
1.00
TOTAL
ENERGY
COSTS IN ECOSYSTEMS
279
TABLE 3
The total energy? per unit of stored or exported biomass, E, for the Silver Springs ecosystem
Producer export only Proportional export
1
2
Component 3
8.53 8.54
868 8.31
8.68 8.31
4
5
Import I
8.68 8.31
8.57 8.49
8.56 8.50
t Dimensionless.
particular distribution of metabolism at the steadystate condition or it may be due to the strict hierarchial nature of the data on the Silver Springs system. The narrow variation could also be due to the nature of the total energy cost. The producer component has a high direct energy input but a small indirect energy cost because of its position at the beginning of the system. The top carnivore component has a low direct energy input but a Iarge indirect energy cost because of the position at the end of the food chain. The high metabolic cost of the producer component may be a characteristic of the marine ecosystems where solar energy fixation requires maintenance of position near the water surface. However, the application of equation (3) to a forest ecosystem (Gosz et al., 1978) yields only a 1.4% maximum variation in the E. The extensiveness of the data for this ecosystem also allowed me to add additional sectors for “Heat” and “Evapotranspiration” energy flows. These flows were 38 times the total plant production flow and yet they changed the E associated with the production flows by only about O3%. Note that, if I form E = E’(Z G)’ where G is the normalized production from matrix (Hannon, 1973), than all E = 1.0. To determine the effect of the hierarchial structure on the apparent constancy of the E, I developed the data for the Cone Spring system from detailed work on Tilley’s data by Williams & Crouthamel (1972) (see Table 4). Again, the E are nearly constant for every component. The Cone Spring matrix, although it possesses a few feedback loops, is not divided in a complex manner. It has only four living components and a single input accounts for most of the total input in each component. In any event, one more unit of storage or export from the producer or plant component, for example, can be obtained with 3.02 units of additional system energy input. For an increased unit of storage or export by any of the other three living components would require 2.88 additional units of external energy input.
280
B. HANNON TABLE 4
The production matrix and total energy costs for Cone Spring ecosystem (kcal mm2yr‘) Component 1
2
3
4
5
Environment I
Storage and exnortt
0 0 0
2302.0 0 3400O
0 3971.0 129.0
1814.0 203.0 0
0 0 0
Producers (plants) (1) Detritus (2) Bacteria (3) Detritus feeders (4) Carnivore (5) I
0 0 7719.0
200.0 167.0 0
0 0 0
60.0 0 0
3700 17.0 0
II
34650
635.0:
0
0
0
2.88
2.88
2.88
E§
0 0 285.0 8882.0 0 3579.4 52043 2308.9 0 1600.2 116.6 75.1
3.02
2.88
 4100.0
t Figures adjusted for balance (maximum adjustment 0.2). $ All import to Detritus assumed to be needed for export balance. 3 Dimensionless, calculated from equation (3).
This extra external energy input must be proportional increases in the existing energy inputs. A total component energy efficiency can be defined by multiplying each of the terms in the ith row by G’ by si and then dividing the column sums of the resulting matrix into the total outputs of their respective component. Such a calculation is made for the Cone Spring ecosystem and shown in Table 5. These component efficiencies, which may be interpreted as a measure of the relative ease with which the system produces a particular output, show greater variation than the E. The inverse of these efficiencies are known in economics as the (total) output multipliers. The matrix in Table 5 also shows the origin of the fractions of the total direct and indirect energy which was required by each component. To demonstrate the procedure of using mixed units matrices, I selected the data of Van Hook (1971) on spider and orthopteran populations. This study included intercomponent energy and mass flows. But the data did not include direct solar inputs nor specific numerical statements about the amount stored or exported from the system. As a result the energy source is vegetation and the material leaving the system is due to excretion and mortality. Thus the results of this study and analysis are not immediately comparable to those displayed in Tables 3 and 4.
TOTAL
ENERGY
COSTS
IN
281
ECOSYSTEMS
TABLE 5
The total (direct plus indirect) energy flow from the ith component to the jth componentfor the Cone Spring ecosystem?
1 3roducers (plants) (1) Detritus (2) Bacteria (3) Detritus feeders (4) Zamivore (5) Lmport I Zolumn total Iota1 output/ column total (percent)
Component 3
2
4
860 ( 3.3) 26,799 (62.6) 0 0 0 10,322 (24.1) 15,008 (97.8) 6,658 (945) 0 4,615 (10.8) 336 (2.2) 217 (31) 0 0 25,562 (96.7)
577 (1.3) 482 (1.2) 0
0 0 0
5 0 0 0
Environmerit I 6,946 (305) 9,938 (43.6) 5,302 (23.3)
173 (2.4) 1,067 (95.6) 0 49 (4.4) 0 0
593 (2.6) 0 0
26,422
42,795
15,344
7,048
1,116
22,779
43.4
35.2
34.7
34.6
34.7
33.9
t Numbers in parenthesis are percentages of column totals.
The G’ matrix for the Van Hook energysodium flow data is shown in Table 6 along with the energy costs calculated by equation (3). The E in Table 6 means, for example, that the total system energy flow is 4.15 kcal for each mg of excretion and mortality from the first grazing insect (Melanoplus). The next few additional units of sodium stored or excreted by Melanoplus should each require the same amount of external system energy. The variation of the E in this matrix should reflect the ranking of the component in the system : the higher the ai, the higher the energy quality of a unit of biomass in that component.
(D)
THE
ENERGY
STORAGE
MAXIMIZING
HYPOTHESIS
In economics, consumer utility is assumed to obtain from a monotonically rising function of consumption. Such an assumption is based on demands which could exceed basic physical needs. Consequently, I have not thought it an appropriate criterion to apply to ecosystems analysis. The broader economic concept of selfinterest seems to be a more feasible measure to adopt. For this reason, and for those given in Hannon (1976), I choose energy storage at the measure of component utility. Each component is assumed to maximize its own energy storage with respect to its input and output flows in the specified time period.
B. HANNON
282
TABLE 6
The direct energy (kcal me2 yr‘) and sodium (mg m’ yrI) jlows, amf the total energy cost E, in a grassland ecosystem(Van Hook, 1971)t Component Grazing insects . .
Environment I
Excretion and mortality
1
2
3
4
Predator 5
0 0 0 0 0 9043
0 0 0 0 0 16.30
0 0 0 0 0 22.21
0 r 0 0 0 589
@41 0.15 0.40 0.21 0 0
0 0 0 0 0 0
21.39 3.40 253 088 1.17 0
II
.t
3
t
s
0
0
 29.37
E (kcal mg‘)
4.15
4.59
7.58
5.40
5.60
0
7 Units for all rows are mgmm2 yr’ except for the Import I row which is in kcai mm2 yr‘. 1 These entries represent a distribution of the total excretion and mortality quantity (29.37) which is not important to the determination of the energy costs.
Therefore, I must develop a theory which maximizes sixf for each living component. Since this process will be carried out for each component in the entire system simultaneously, the total storage in the system will contain, directly and indirectly, the total system energy input for this time period. The input flows are related to the output flow through a “production function”. A production function, F, is empirically determined and relates all of the input flows, xv, with the output flow, xj. It can be derived from time series data on the components’ inputs and outputs. These data are used in conjunction with an assumed form of a differential equation in the component flows. The solution of this equation for the specified time period is the production function [see Section (G)]. These functions must account for the local constraints on the system of temperature variation and the availability of soil, water, air and sunlight. A typical production function F’ for ith component Fj = Fj(x,) = 0.
(7)
There are also a variety of timeimplicit forms for production functions in economics such as the CobbDouglas function, the constant elasticity of substitution (CES) function and the translog function (Walters, 1970). The
TOTAL
ENERGY
COSTS
IN
ECOSYSTEMS
283
simplest economic production function describes how the input factors (e.g. labor, capital, energy) combine to determine the quantity of output. On the other hand, biological production functions should be sufficiently complex to demonstrate how the inputs of free chemical energy, solar radiation, biomass (feeding inputs), and the energy embodied in nonbiomass inputs, such as water, combine to form the three outputs: metabolic heat, production biomass, and exported and stored biomass including exported free chemical energy. If the production functions for each component are linear, such as those produced from a linear multiple regression analysis of time series data, then linear programming techniques must be used as described in Hannon (1976). The Lagrangian function V necessary for maximization of total energy storage in the ith component is V = cix; + i AjFj,
(8)
where ;i are the Lagrangian multipliers. The functions in equations (6) and (7) are assumed to have continuous first partial derivatives and the Jacobian of the Fj must have rank q (Aoki, 1971) where 4 is the number of separately given production functions for the ith component. These conditions assure the existence of the 1. Setting the derivatives of equation (8) equal to zero gives (9)
where .si is positive or negative depending on whether Xi is an output or an input respectively. Note that Ei could be a function of the Xi. Therefore equation (9) holds for components which behave competitively or monopolistically. The xs in equation (8) contains the quantity exported from storage, xe. But the xe are independent of the system flows and therefore it is possible for exports to be such that xs is steady. This condition represents a freezing of system development and equation (9) should represent that condition. It is also possible that the xe occur in proportion to the “natural” storage flows as represented by equation (9) and yet system storage would be increasing. I consider such a condition as a rejuvenation from which the system will proceed to develop according to equation (9). If the xe are such as to allow storage to proceed in some but not all components, then equation (9) does not hold. I can only say that this condition represents a system storage rate less than the maximum one.
284
B. HANNON
The energy cost ratio Ei/aj can be shown to be equal to ax,/ax,.~ The term axj/axi is the rate of exchange of xi for x2 When xi is an output and the xi an input to the jth component, the term is defined as the marginal productivity of the ith input to that component. The storage maximizing component will tend to use more of that particular input with the greatest marginal productivity. In steady state systems the substitution among inputs will be nearly zero and when energy is the only flow variable, all marginal productivities will be nearly one. Thus the ratio of any two si must be nearly one. This result helps explain the lack of E variation in the first two examples in Section (c). Equation (9) is an extrema. To find the maximum total energy storage in the specified time period, we find that all eigenvalues of the Hessian Matrix of V must be negative (Aoki, 1971, p. 21). This condition, together with those which assure the existence of the ~j are the necessary and sufficient conditions for the existence of a maximum for the storage function. Equation (9) is in exactly the same form as equation (7) in Hannon (1976) except that here si is the total energy cost as determined by equation (3) while previously the energy cost was considered to be the direct unit energy content only. To prove or disprove either of the storage maximizing hypotheses we must empirically determine production functions and derive a vector of energy costs by equation (9). This vector is compared to the direct energy content costs and the total energy costs of equation (3). If equation (9) can be shown to be correct, then it becomes a definition of the total energy costs, E. In this paper I have assumed that each component has only one type of output or that each output of a component has the same total energy cost. If this were not the case, the matrix of equation (3) would not be invertible. If each of the outputs could be assigned an E, then linear programming techniques could be used to find those E which minimize the residual error. For the purposes of this theory, therefore, it was critically important to be able to define components as having only one type of output. Such a definition means that the output of a component has only one total energy cost. This requirement may mean that certain plants and animals must be divided in components according to the total energy costs of their various outputs. While this is relatively easy in economics because the processes are made by man, it may be difficult in biology, for example, to divide a plant into interconnected components. As the theory now stands, every component contributes positively toward energy storage in the ith component : that is, the whole ecosystem has greater t If the Jacobian determinant is not zero.
TOTAL
ENERGY
COSTS
IN
ECOSYSTEMS
285
energy storage with the ith component than without it. At the steady state condition, the quantity of each component is probably such that overall system energy storage would be less if any component were a different size or had a different flow pattern. The same is probably true for the universe : the rate of entropy formation in the universe is lower with steady state natural systems than with no life on earth.
(E)
FEEDING
STRATEGIES
The theory in the previous sections has been used to define energy cost and posit a maximum energy storage hypothesis. In each case either the flow through the components or certain ratios of the flows must be specified to determine the energy costs. A concept is needed to explain how a component chooses a particular set of inputs that make up the observed total input. In an ecosystem where the individual component flows are small when compared to the total system input flow, it seems feasible to assume that each component has no ability to significantly change any of the energy costs. Therefore, each component could determine its own optimal flow pattern by adjusting its feeding energy expenditures, that is, by adjusting the fraction of energy, 1,, diverted from its own storage, into the pursuit of input energies. In my previous paper (Hannon, 1976) I suggested that an optimum energy feeding strategy would require that the net energy feeding path be followed. Net energy was defined as the direct energy content of a unit of food less the metabolism energy spent acquiring that unit. Feeders establish a feeding path by selecting foods in the order of declining net energy input. In the present paper, the feeding (and subsistence) energy has been incorporated into the energy cost E. While I now assume that the feeders are aware of the direct and indirect energy cost of their potential foods, I also assume they are directly sensitive to the allocation of their own feeding efforts. That is, even though a component’s own feeding energy is embedded indirectly in the energy costs of its inputs, the feeding selection is also governed to some extent by direct feeding efforts. Such an assumption clearly implies the existence of a degree of altruistic behavior of the part of each component. The component is assumed to be sensitive to its own direct feeding requirements (self interest) and to the total system energy cost of its feeding (altruism). To compute the net total energy of the ith input to the jth component, I reduce the ei by the direct and indirect cost of its feeding effort. Thus E; = Ei Ejlij.
(10)
B. HANNON 286 Potential inputs would be ranked in order of decreasing si by these feeders in such a way as to maximize the total net energy input for any given quantity consumed. In other words, these feeders would arrange the inputs according to equation (10) and adjust the quantities consumed by maintaining the least slope on an E’ vs. x curve so as to maximize the area (net total energy intake) under this curve (Fig. 3). Such a ranking should be equivalent to the marginal productivity method of ranking inputs presented in the previous section. Figure 3 depicts the variation of input flows into a component with the net energy costs from equation (10). The output times the net energy cost is the actual measure of net energy input to the component. As shown in Fig. 3, this product rises to a maximum at x  x,ptimum. At this point the component receives the maximum net total energy input. In other words, the incremental total energy input is equal to the incremental total energy exerted for this input, at the optimal input flow. If the hypotheses of optimal feeding and maximum storage are correct, this optimal flow should be the observed or specified input flow. To derive Fig. 3 would require complete knowledge of the flows for all different food availabilities for each component. Such information is not easily attained. However, one can calculate the E for the known flow system and assume that they will remain constant for small changes in the feeding energy expenditures through small changes in the 1. This change in 1 times the associated input flows would be the change in output, Ax. The quantity A.s’x/Ax is an approximation of the slope of the E’X vs. x curve in Fig. 3. If it is positive, more growth will occur. If the slope is near zero no further growth should occur and the optimal flow has been determined. A negative slope would indicate that the component was reducing its size. In general, to determine the E when the feedstocks change or become more difficult to find may require an iterative solution to equations (3) and (10).
t
RG.
3. A composite net total energy feeding relation.
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287
An increase in a specific 1, could mean a decrease in the xy and a corresponding increase in the x!. This change produces a new set of E from equation (3) and then a new set of E’ through equation (10). These new E’ could rearrange the ranking of feed preferences and these changes would produce changes in the x which would produce still another set of E, and so on. Eventually, if a stable feeding pattern could be found, stable values of E could be determined. Schoener (1971) and Rapport & Turner (1975) discussed energymaximizing and timeminimizing feeders. The typical energymaximizing feeder is one which consumes that quantity of a certain input such that net direct energy is a maximum. A timeminimizing feeder obtains a direct energy input quota by feeding in the fastest possible way. As a result of the definition of direct and indirect energy cost given in this paper, the experimental proof or disproof of the existence of energymaximizing feeders is made more difficult. The same point can be made about timeminimizing feeders. One can define the total time spent in feeding in the same sense as total energy cost has been defined : There is a direct and indirect time cost of feeding. Or one could view an individual feeder as having had to spend time in a former period to gain the resources to enable it to feed in the present period. Such a view gives a converging series of incremental times which yield a total time for feeding. With either view, the total time spent for feeding is larger than the direct time spent and it is not clear whether timeminimizing feeders exist in this sense. For a recent review of feeding theories see Pyke, Pulliam & Charnov (1977).
(F)
GROWTH
AND
THE
STEADY
STATE
If true, the maximum storage hypothesis and the optimal feeding strategy given in the previous sections may be viewed as a microecological force on each component. That is, given the number and operational characteristics (production functions) of the components, the microecological force causes them to adjust their flows so as to maximize the, total stored energy in each component, simultaneously. And therefore, within these constraints, the total system energy stored is maximized for the specified time period. But what macroecological force is consistent with this component behavior? I hypothesize that this macro force causes an increase in the maximum energy storage for the entire system. Therefore, the system continuousIy reduces the nonstored respiration energy (the “maintenance and operating” energy) per average unit of present stored total energy in the
288
B.
HANNON
system. In other words, during each successive time period the additions of total stored energy must require diminishing total maintenance and operating energy per average unit stored. The macroecological force may therefore be a generalization of the “inverse sizemetabolic rate law” (Odum, 1975). An examination of the data (Whittaker, 1970; Wiegert & Evans, 1967) shows that in terrestrial ecosystems, mature tree systems show the lowest rate of maintenance and operation energy per unit of biomass (0.022 yr ‘). Large animals such as elephant, kob and deer have much higher rates (3.2,20.1 and 33.2 yr‘, respectively), The rates for sparrows, squirrels and mice are still higher (46, 74 and 207 yrI, respectively). From a study of Ugandan Kob (Buechner & Golley, 1967), increasing average age meant increasing size, which in turn meant declining maintenance and operating energy per unit of mammal. Under the macrohypothesis landbased ecosystems would evolve to increase the amount of stored total energy. The macro hypothesis does not mean that those components with a high rate of maintenance and operating energy per unit biomass will necessarily be eventually eliminited entirely from the ecosystem. For example, it is not possible for a living ecosystem to eliminate Qecomposers even though the rate is very high for this component. The macro hypothesis indicates that the system may minimize system dependence on such a component while the micro hypothesis indicates that this component (and all others) will strive to store the maximum total energy based on its input and output flows. The individual components work within the constraints of their production functions while the system as a whole determines the nature of these functions. Some ecologists might argue that as an ecosystem evolves, dependency on the decomposer sector is enhanced. I cannot find sufficient data to support or refute this position. However, when I compare a prairie ecosystem with its eventual successor, it is easy to realize that the ratio of standing biomass to decomposer biomass is much greater in the forest ecosystem than in the prairie ecosystem. In the latter, virtually the entire standing biomass is converted annually by the decomposer sector while only the leaves and occasional deadfall would require decomposers in the forest ecosystem. The economic parallel to the above concept is centered on the ratio of economic output per unit of capital. The macro force or goal in economics (United States) is to maximize the output per unit of capital (human and mechanical) with the understanding that each sector will maximize profits (micro force). The inverse difference between economic and ecologic macro forces is immediately apparent. The captured solar energy in our food plus the fossil fuel energy directly consumed per unit of energy stored for persons
TOTAL
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in the U.S. is about 380 yr ’ or about 19 times the ratio of comparable sized mammals (kob).f To view the economic system in analogous terms to the above biological theory, we would count as storage the new capital formed in the prescribed time period, both human and mechanical, and set the E such that the direct and indirect solar and fossil energy content of the new capital was maximized. Since eventually the fixed flowrate of energy input (the sun) will be entirely needed for system maintenance and operation, this reduction process eventually stops at an irreducible minimum. All flows then stabilize, net additions to system storage cease, and the steady state would have been reached. At the steadystate condition, the amount of system total energy flow for maintenance and operation per unit of total energy stored in the system is a minimum and the total (direct and indirect) stored energy is a maximum. In other words, the system continues to increase its stored biomass until the total metabolic energy rate is essentially equal to the rate of total energy absorption of solar radiation. This latter rate is the maximum potential supply rate for the system. Throughout the development to the steadystate condition, the system should continue to behave according to equations (3), (9) and (10). The steadystate condition in chemical reactions is that nonequilibrium thermodynamic condition where the mass and energy flows to and from the system are steady. In the steadystate condition, entropy is being formed by the system at the least possible rate (Katchalsky & Curran, 1965). The same principle may hold for ecosystems. If it does then one might say that ecosystems develop in such a way that the rate of entropy formation is continuously reduced. One may be able to go on to rank the components of an ecosystem by the degree to which they deviate from the equilibrium condition. For example, producers may be low rate entropy generators and top carnivores may be relatively high rate generators. The strategy of the ecosystem would be to maximize the quantity of low rate generators and minimize the quantity of the high rate ones. This hypothesis is equivalent to the one given earlier in this section. It also allows one to define living systems as @se systems wherein entropy is decreasing at a specified rate relative to 7 The average person directly uses 55% of the total personal consumption energy. personal consumption energy is 67% of total U.S. energy which is about 75 x lOI5 Btu per year (Hannon, 1975). With these data and assuming that the average U.S. person weighs about 140 pounds and that each pound is equivalent to 2720 Btu (Weigert & Evans, 1967), that the population of the U.S. is 210 million people, that the average food consumption is 3300 kcal per person per day and finally that the food consumed represents onethird of the energy captured in food plants, the direct energy absorbed from the sun and earth per unit of energy stored in an average U.S. person is 350 yr‘.
290
B.
HANNON
their total energy input, while the universe continues to increase in entropy content, although possibly at a rate lower than the rate when no life existed on earth. (G)
A DYNAMIC
THEORY
The theory demonstrated in this paper thus far is static in the sense that time is mentioned only implicitly. Making the theory dynamic is more difficult. Observation of the rate of change of flows in living components sometimes reveals a logistic growth form. For my purposes here, I assume that the general LotkaVolterra’f equation, N
i.
L=
(11)
ai  c b,x,,
xi
j=l
will hold for the total output flow in each component. The xi is the total output rate of the ith component. The term fi is the time rate of change of a flow rate or acceleration of energy (or mass) in the ith component. The use of equation (11) requires that the energy input be a continuous function with continuous partial derivatives. Actual energy input functions do not entirely behave this way unless they are averaged over longer periods, for example, annual cycles. Consequently I am not trying here to model short term system response to energy availability. I am limited here to very general constraints on ecosystem development. The solution to equation (11) is the production function for the ith component and it would be the F of equation (7). Evaluating the coefficients and solving equation (11) is a difficult process requiring a great deal of time series flow data. In the absence of such data, I make a series of assumptions to produce an interesting and potentially useful result. ‘First, I assume that the normalized growth rate ii/xi is proportional to the amount stored in the ith component, xf. This assumption is tantamount to assuming that the a, is the maximum potential supply rate for the ith N
component
and the term
C b,x,
is a composite
demand
rate. The
j=l
assumption
means f. 1 ._
t This form of equation ai, b, > 0, logistic increase: ai < 0, b, = 0, exponential
zz (&x;.
is desirable since a variety ai > 0, b, = 0, exponential decline.
(12) of growth forms can be easily derived: increase ; ai, b, < 0, logistic decline ; and
TOTAL
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ECOSYSTEMS
Since the rightmost term in equation (11) is a demand rate, the b, can be considered unit demand rates and in this sense they are related to the elements of the G’ matrix in equation (3). I assume that the b, are proportional to these elements, g’ij: b, = qigij
= y.
(13) I
With equations (4), (ll), part of storage),
(12) and (13), I have (with export considered as a
$
1
=
__
ai
aiYli

___ Vi Ui)?i
x. ”
which indicates that the storage rate in the ith component varies linearly with respect to the total output flow rate of the ith comp0nent.t This result is shown in Fig. 4, where the effect of cropping and importing disturbances are demonstrated. Effectively, I have assumed that there exists a preferred logistic flow path and cropping is followed by exponential flow (horizontal line), while importing (fertilizer, for example) periods are followed. by a vertical return to the logistical path. Of course, each of these events must take place over the region where the gij are reasonably constant. The result of equation (14) couples the storage and the total output flow rates. Thus the total input rate (or total consumption rate) is a function of the storage rate. In this sense my concept of utility based on storage rate is reconciled with the economic concept of utility based on consumption. In the biological system described above, “utility” is an inverse function of consumption (total output), while in the economic system utility is defined as a monotonically increasing function of consumption. Since the term 1 is really an acceleration of energy or mass, it is similar to the rate of income changes used in economics (Hamberg, 1977). The simple HarrodDomar investmentsavings growth models are similar to equation (11) with b, = 0. The result in economics is the apparent requirement of exponential national income growth.
t Solving equations (12) and (14) simultaneously and integrating additions to storage in the ith component from time r = 0 gives:
where
xp is the value of xi at t = 0.
the resulting
xf gives Si, the
292
B.
c
,
Moxamum storage rate I%711
*...., “..,,
HANNON
_,’ . .. .. ...//”
,Cropping
Total
FIG.
flow
period
rote, x,
4. An example of growth of the ith component.
3. Summary and Conclusions A micro and macroecological system hypotheses and their associated theory of total (direct and indirect) energy cost have been proposed to explain the meaning of system growth and the steady state condition. Models have been proposed which can be used to test the hypotheses. In brief, the hypotheses are that while the components of an ecosystem strive to maximize their total direct and indirect energy storage within the constraints of their production characteristics, the overall system strives to minimize the metabolized energy per unit of stored biomass energy. The steady state result is a system with the largest possible stored biomass under the general constraints of water, air, soil and sunlight availability. Feeding strategies were discussed with the conclusion that a net total energymaximizing feeding strategy may be able to replace the present concepts. Validation of these hypotheses must await experimental results. But if proven correct, they could be of substantial aid in guiding the cropping of ecosystems and in quantifying the effects of air and water pollution and the effects of soil depletion, among other possibilities. Perhaps the most important result though would be to sharpen our understanding of the similarities and differences between ecological and economic systems. The ecological system has developed under the force of a constant (cyclical) external energy flow input. As such, the undisturbed system eventually reaches a steady state. Modern economic systems have developed with the same uniform flow of energy from the sun plus a rate of energy flow from a finite storage system. This additional rate has been limited only by our technological ingenuity. But eventually all such finite energy storage will
293 approach exhaustion. During this transition the lessons of ecosystem development should be exceedingly useful to the human society. TOTAL
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I wish to thank Ahmed Sameh, James Rebitzer, William Ruesink and Robert Costanza for their many helpful discussions. REFERENCES AOKI, M. (1971). Introduction to Optimization Techniques. New York: MacMillan. BUECHNER,H. & GOLLEY, F. (1967). In Secondary Productivity of Terrestrial Ecosystems (W. Petrusewicz, ed.), Ch. 2.2.1, p. 243. Warsaw, Poland: Polish Academy of Sciences. BULLARD, C. & HERENDEEN,R. (1975). Proc. IEEE, 63, 484, March 1975. FINN, J. (1976). J. theor. Biol. 56, 363. GOSZ, J. R., HOLMES, R. T., LIKENS, G. E. & BORMANN, F. H. (1970 Sci. Am. 238,96. HAMEIERG, D. (1977). In Modern Economic Thought (S. Weintraub,ed.), Ch. 17, P* 333. Philadelphia, Pa: University of Pennsylvania Press. HANNON, B. (1973).J. theor. Biol. 41, 535. HANNON, B. (1975). Science 189, 95. HANNON, B. (1976). J. theor. Biol. 56, 253. KATCHALSKY, A. & CURRAN,P.(1965). NonEquilibrium Thermodynamics in Biophysics, p. 229. Cambridge, Mass.: Harvard University Press. See also: PRIGOGINEZ, I. (1955). Introduction to Thermodynamics of Irreversible Processes, p. 74. Springfield, Illinois: C. C. Thomas. ODUM, E. (1975). Ecology. New York: Holt, Rinehart, Winston, ODUM, H. T. (1957). Ecological Monographs 27, 55. ORTEGA, J. (1972). Numerical Analysis, p. 109. New York: Academic Press. PATTEN, B., BOSSERUAN,R., FINN, J. & CAIE, W. (1976). In Systems Analysis and Simula,tion in Ecology (B. Patten, ed.), Ch. 17, p. 457: New York: Academic Press. PIKE, G., P~LLIAM, H. L CHARNOV,E. (1977). Q. Rev. Biol. 52, 137. RAPPORT, D. & TURNER, J. (1975). Ecofogy 56,942. RAPPORT, D. & TURNER, J. (1977). Science 195, 367. SCHOENER,T. (1971). Annual Review of Ecology and Systematics, vol. 2, p. 369. TILLY, L. (1968). Ecological Monographs, Vol. 38, Pt. 2 (Spring), p. 169. VAN HOOK, R. I., Jr. (1971). Ecological Monographs, Vol. 41, Pt. 1 (Winter), p. 1. WALTERS, A. (1970). An Introduction to Econometrics, p. 319. New York: Norton and Co. WHITTAKER, R. (1970). In Communities and Ecosystems, Ch. 4. London: Macmillan Co. WIEGERT, R. & EVANS, F. (1967). In Secondary Productivity of Terrestrial Ecosystems (W. Petrnsewicz, ed.), Ch. 3.2.1, p. 499. Warsaw, Poland: Polish Academy of Sciences. WILLIAMS, M. & CROUTHAMIZL,D. (1972). System Analysis of Cone Spring. Unpublished Report for Institute for Ecology, University of Georgia, Athens, Georgia. (See also Patten et al., 1976, p. 573.)