# The Role of Energy Return on Energy Invested (EROEI) in Complex Adaptive Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

_{1}and L

_{2}. We assume that level 1 is at higher potential energy than level 2, so that energy flows from L

_{1}to L

_{2}but not the reverse (rabbits do not eat foxes, not normally at least). These stocks are normally measured in terms of population; that is, the number of individuals. However, what moves from one stock to the other is not individual rabbits (prey), but the metabolic energy they provide to foxes (predator). Therefore, a more general version of the model sees the L

_{1}(prey) and L

_{2}(predator) stocks as energy stocks. Energy comes into the system from a non-quantified stock (grass) represented as a small cloud in the SD graphical representation. This energy accumulates into the L

_{1}stock (rabbits), and it gradually moves to the L

_{2}stock (foxes), but some of it is lost to the environment in the form of waste heat. We will see that it is this view that allows us to use the LV model to define the EROEI of the energy transfer between the stocks.

_{1}, k

_{2}, k

_{3}, and η as fixed parameters:

_{1}/dt = k

_{1}L

_{1}− k

_{2}L

_{1}L

_{2}

_{2}/dt = ηk

_{2}L

_{1}L

_{2}− k

_{3}L

_{2}

_{1}and L

_{2}can be measured in energy units. The k

_{1}and k

_{3}parameters are measured in units of [time]

^{−1}; that is, as frequencies. Their values are proportional to how fast the system replenishes or empties its stocks. k

_{2}has the dimension of [time]

^{−1}∗ [energy]

^{−1}and it is proportional to the rate of interaction of the two stocks. Note that k

_{2}is the dimension of the inverse of an “action” (energy ∗ time). It is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived through the principle of the least action. As recently discussed by Sharma and Annila, the second law of thermodynamics can be understood in terms of an equation of motion [30]. According to Annila, the natural process (energy dissipation in trophic levels) moves following the steepest descents of the potential energy landscape by equalizing differences in energy via various transport, transformation, and dissipative processes, e.g., diffusion, heat flows, electric currents, and chemical reactions [31].

_{1}and L

_{2}stocks can be easily determined by numerical methods. In the form written above, the two stocks undergo an unending series of oscillations as a function of time (Figure 2). This behavior makes the LV system an example of a biological clock. It can be shown that it has a frequency equal to (1/2π) (k

_{1}ηk

_{2})

^{1/2}[32].

_{1}and L

_{2}in phase space, where the oscillations are represented by a closed trajectory (Figure 3). In this version of the model, all the cycles are identical and superimposable.

_{1}parameter is set to 0. It means that the higher potential stock is not replenished. We call this version the “Single Cycle Lotka–Volterra” model (SCLV). These are the equations for the model:

_{1}/dt = −k

_{2}L

_{1}L

_{2}

_{2}/dt = ηk

_{2}L

_{1}L

_{2}− k

_{3}L

_{2}.

_{2}stock and for the flow from L

_{1}and L

_{2}(with the dL

_{1}/dt labeled as “Production”). Instead, L

_{1}decreases monotonically.

_{2}) can be defined as the ensemble of resources used to exploit the resource (L

_{1}) and therefore it can be labeled as “capital” in order to align the terminology of the model with the commonly used nomenclature in economics.

_{2}= 0 (extinction of the predators) but not necessarily for L

_{1}= 0. In the case of an economic system, it means that the resource will not be completely exploited before the capital stock goes to zero.

## 3. Results: Modeling the EROEI Parameter

_{out})/(E

_{in}) was defined by Charles Hall in 1979 as EROI or EROEI (energy return on energy invested) [1,2,3,38]. More explicitly, it is the energy gained by a stock divided by the energy lost by the stock.

_{out})/(E

_{in}) ratio.

_{1}(resource) stock into the L

_{2}(capital) stock (dL

_{1}/dt or −k

_{2}L

_{1}L

_{2}). In the same way, the “expended” energy is the energy that the L

_{2}stock loses; that is, the flow out of the L

_{2}stock (k

_{3}L

_{2}).

_{1}stock is not the same thing as the flow into the L

_{2}stock. Because of the second law of thermodynamics, one unit of energy resource creates much less than one unit of capital energy. The loss for the predator/prey trophic level in vertebrate biological chains is called the “Lindemann efficiency” and it is often estimated as ca. 10%, with a high degree of uncertainty. Recent data [39] indicate that it can be as low as 1%. This feature is considered in the LV/SCLV model by means of the η parameter, which is always <1.

_{2}(ηk

_{2}L

_{1}L

_{2}) divided by the flow of energy out of L

_{2}, (k

_{3}L

_{2}). Specifically:

_{2stocks}= ηk

_{2}L

_{1}/k

_{3}

_{1}, in the multicycle LV model, it oscillates periodically in proportion to L

_{1}. Note also that EROEI is directly proportional to three factors: the efficiency of the transformation, η; the transformation rate factor, k

_{2}; and the amount of the resource, L

_{1}. The EROEI is also inversely proportional to k

_{3}, the factor that describes how fast capital disappears because of depreciation (definable also as maintenance, the term dL

_{2}/dt). Note that the L

_{2}stock does not appear in the formula. In the LV model, foxes are assumed to chase rabbits at a rate independent from their number. Of course, in the real world, economies of scale occur, and some predators do band together to hunt their prey. The formula could be modified making it explicitly proportional to L

_{2}. However, in the present study, we will remain with the basic model assumptions.

_{2}, describes both activities. However, in many cases, the two activities can be separated, especially in the case of the human economy. For instance, the oil industry is an energy producer, but oil wells do not directly spawn other oil wells. Society dedicates a stock of resources and energy to generate a complete industrial sector that provides the oil industry with materials, equipment, and human power. This differentiation can be accounted for in the model by adding another stock. This version of the model is the result of a common procedure in system dynamics, where systems can be complexified by adding more stocks and more interactions among them. For instance, the “World3” model used in one of the first dynamic studies of the world system, the Limits to Growth [40], consisted of five main stocks of capital and resources. Other more recent models include renewable energies [41] and use larger numbers of parameters (see, e.g., the MEDEAS world model [42]). In the present case, we aim at keeping the model simple to be able to use it to determine the EROEI in an explicit form. This determination is more difficult to do univocally in a more complex model. Nevertheless, we use this three-stock model as an example to outline how the procedure can be expanded.

_{2}) stock is “society”, the entity that grows on the exploitation of the resources (Equation (7)). It grows proportionally to the flow of resources it obtains from the resource stock (Equation (6)). η

_{12}represents the efficiency of transformation of resources into an increase of societal capital. At the same time, the societal capital does not directly exploit resources, but it must allocate some capital to the third stock, L

_{3}, that aggregates the “producers”—that fraction of the stocks that directly exploits natural resources (Equation (8)). The resulting three equations are (still in the single cycle assumption):

_{1}/dt = −k

_{2}L

_{1}L

_{2}

_{2}/dt = η

_{12}k

_{2}L

_{1}L

_{2}L

_{3}− k

_{3}L

_{2}L

_{3}

_{3}/dt = η

_{23}k

_{3}L

_{2}L

_{3}− k

_{4}L

_{3}

_{2}stock, is proportional to the producer stock, L

_{3}, to the amounts of resources available and to the whole society that provides a market pull for the production: the efficiency of this transformation is represented by η

_{23}. The flow of resources into the producer stock is proportional to the societal stock and to the producer stock itself in terms of compensating capital depreciation.

_{2}stock (societal stock), which is equal to η

_{12}k

_{2}L

_{1}L

_{2}L

_{3}. The denominator is given by the outflow from L

_{2}(k

_{3}L

_{2}L

_{3}). The resulting EROEI is the same as it was defined for the simpler two-stock system:

_{3stocks}= η

_{12}k

_{2}L

_{1}L

_{2}/k

_{3}

_{out}/E

_{in}ratio. Nevertheless, we believe that this version is the least arbitrary choice.

_{2stocks}= ηk

_{2}L

_{1}/k

_{3}.

_{1}= k

_{3}EROEI

_{2stocks}/ηk

_{2}. Substituting these values in the LV equations, we have that:

_{1}/dt = (1/η) EROEI

_{2stocks}(k

_{1}k

_{3}/k

_{2}− k

_{3}L

_{2})

_{2}/dt = L

_{2}k

_{3}(EROEI

_{2stocks}− 1)

_{2stocks}parameter oscillates, too, since it is proportional to the L

_{1}stock.

_{1}is set to zero and the equations become:

_{1}/dt = −(1/η) k

_{3}L

_{2}EROEI

_{SCLV}

_{2}/dt = L

_{2}k

_{3}(EROEI

_{SCLV}− 1)

_{1}stock declines monotonically with time since all terms in the first equation are positive. Since the EROEI

_{SCLV}and L

_{1}are directly proportional to each other, the EROEI

_{SCLV}will also decline monotonically along the production cycle. Neither needs to go to zero but will stabilize in the long run at values > 0. The second equation shows that the growth of the L

_{2}stock is exponential when we have EROEI

_{SCLV}>> 1, which may happen during the initial phases of growth. The L

_{2}stock reaches a maximum for EROEI

_{SCLV}= 1, then it declines when EROEI

_{SCLV}< 1. The exergy that flows into L

_{2}, ηk

_{2}L

_{1}L

_{2}is conventionally termed “production”, especially in the case of the oil industry. This quantity can be expressed in terms of the EROEI

_{SCLV}parameter as k

_{2}L

_{2}EROEI for the SCLV system.

_{SCLV}according to the SCLV model. We also show the related parameter known as “Net Energy”, which is equal to E

_{out}-E

_{in}and is related to EROEI

_{SCLV}by the formula: Net Energy = (EROEI

_{SCLV}− 1)∙E

_{in}. Note that the Net Energy is negative when EROEI

_{SCLV}< 1.

_{2}/dt and setting it to zero for the SCLV case, considering the expression EROEI

_{SCLV}= ηk

_{2}L

_{1}/k

_{3}, we find that the maximum production occurs when EROEI

_{SCLV}= k

_{2}L

_{2}/k

_{3}+ 1. All the terms in this expression are positive, so the production curve will peak and start declining before the EROEI of extraction has become lower than one. In other words, production will peak for an EROEI

_{SCLV}> 1.

_{SCLV}from the formula EROEI

_{SCLV}= ηk

_{2}L

_{1}/k

_{3}. The practical feasibility of this method depends on the availability of data expressed in suitable energy units, which is rare. In the present study, we are not going into this subject, limiting ourselves to general considerations on the effect of the EROEI parameter on systems that can be described using the LV model.

## 4. Discussion

#### 4.1. Biological Systems

#### 4.2. Epidemics

_{1}= 0). In the SIR approach, the efficiency parameter is taken as unity; that is, η = 1. This does not mean that viruses and bacteria do not obey the laws of thermodynamics, just that in this system, the entropy factor can be neglected.

_{t}is often utilized in epidemiology. It is equal to the number of new infections divided by the fraction of infected people. Specifically, it is the same as the “net reproduction rate” parameter in population biology. The SIR equations show that R

_{t}is given by the number of new infections divided by the number of recoveries (or deaths) (k

_{3}I). Determining Rt for ongoing epidemics requires complex procedures [58,59] and this necessity often clouds the issue of what Rt really is. However, on the basis of the definitions given in this paper, Rt is conceptually and mathematically the same thing as the energy return for energy invested (EROEI), expressed by the same formula, except for the different notation and the lack of the η efficiency parameter. An infection spreads for R

_{t}> 1 and declines for R

_{t}< 1. For R

_{t}= 1, the curve of the number of infected cases reaches a maximum, which is sometimes defined as the “herd immunity” point.

#### 4.3. Economic Systems

#### 4.4. Socioeconomic Systems

_{2}stock. This eventually leads to decline.

_{s}), where f

_{s}stands for the “societal factor” and it indicates the fraction of resources that are not re-invested into more resource production. We can call “f

_{s}“ the “societal disposable income fraction”. In this case, we can modify the equations of the single cycle (SCLV) model as:

_{1}/dt = −k

_{2}L

_{1}L

_{2}

_{2}/dt = η(1 − f

_{s}) k

_{2}L

_{1}L

_{2}− k

_{3}L

_{2}.

_{2}L

_{1}/k

_{3}:

_{1}/dt = −(1/η) k

_{3}L

_{2}EROEI

_{2}/dt = L

_{2}k

_{3}((1 − f

_{s}) EROEI − 1)

_{s}factor does not affect the EROEI of the exploitation of whatever resource a civilization is relying on, but it affects the growth rate of the L

_{2}stock (dL

_{2}/dt). If, for instance, f

_{s}is equal to 1, even if the EROEI is much larger than one, all the production is used for building societal capital. In this case, dL

_{2}/dt = 0 and society does not grow. There are historical cases in which a society is believed to have declined as the result of excessive military expenses; that is, having dedicated too much of the produced resources to non-producing assets. This may have been the case, for instance, for the Soviet Union in the 1980s [71,72] even though this is unlikely to be the only factor involved.

_{s}to a value small enough that growth can be maintained or, at least, decline is slowed. In the modern Western society, we may be seeing this effect as the result of depletion of the high EROEI fossil resources [70]. The consequence is that economic growth is maintained at the expense of downsizing or the elimination of services, such as universal health care, state pensions, public schools, and more. Fossil fuels allowed the production of many more goods and services by requiring only 20 percent (1850, coal) or even 10 percent or less of all economic activity to be required to run the rest of the economy [67,69]. Alternatively, for example, in England in 1500, about half of all economic activity was dedicated to obtaining the energy (food, fodder, wood) necessary to run society, with much less left over for amenities [73]. A similar phenomenon may have taken place for the decline of the Roman Empire, generated by the progressive depletion of the mineral resources it was dependent upon [74].

_{s}” factor is frayed with uncertainties. For instance, we might say that the large budget allocated to military expenses in the US is an example of an unnecessary burden on society. On the other hand, it might be argued that, according to the so-called “Carter Doctrine” [75], without such expenses, the US could not access the production of fossil resources in regions such as the Middle East. In such an interpretation, a large fraction of the US military budget should be factored in the calculation of the EROEI of fossil fuels, which would be consequently reduced, perhaps well below unity. This does not detract from the general observation that a declining EROEI may be the main factor involved in civilization decline or collapse.

## 5. Conclusions

_{t}parameter to be unveiled in epidemiologic studies to the energy return on energy invested (EROI or EROEI) parameter in the biophysical approach to energy economics [78], which, to our knowledge, had not been pointed out in the scientific literature before.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**An interpretation of the 2-stock trophic chain model according to the conventions of system dynamics.

**Figure 4.**Single-cycle two-stock model (SCLV): Resources = L

_{1}; Capital = L

_{2}; Production = dL

_{1}/dt.

**Figure 6.**Example of the dissipative 3-stock system: the collapse of a complex society (from Bardi et al. [67]).

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**MDPI and ACS Style**

Perissi, I.; Lavacchi, A.; Bardi, U. The Role of Energy Return on Energy Invested (EROEI) in Complex Adaptive Systems. *Energies* **2021**, *14*, 8411.
https://doi.org/10.3390/en14248411

**AMA Style**

Perissi I, Lavacchi A, Bardi U. The Role of Energy Return on Energy Invested (EROEI) in Complex Adaptive Systems. *Energies*. 2021; 14(24):8411.
https://doi.org/10.3390/en14248411

**Chicago/Turabian Style**

Perissi, Ilaria, Alessandro Lavacchi, and Ugo Bardi. 2021. "The Role of Energy Return on Energy Invested (EROEI) in Complex Adaptive Systems" *Energies* 14, no. 24: 8411.
https://doi.org/10.3390/en14248411