Oil Production, Net Energy, and Capital Dynamics: A System-Coupled Lotka–Volterra Approach

Abstract

Net energy—defined as the energy remaining after accounting for the energy required for resource extraction and processing—shapes the fundamental physical constraints of energy systems. Although the extended Energy Return on Investment (EROIext) incorporates extraction, refining, transportation, and end-use infrastructure, its long-term structural dynamics remain underexplored. This study applies a Single-Cycle Lotka–Volterra (SCLV) model to examine interactions between resource stock, capital accumulation, and EROIext in the global petroleum system. The model is calibrated using historical data from 1965 to 2012 to explore structural trajectories under simplified assumptions. Results indicate that production peaks endogenously around 2041 within the model framework, while EROIext declines and falls below unity by 2081 under the assumed structural relationships. These years represent model-derived structural outcomes rather than deterministic forecasts. Capital stock reaches its maximum at the same energetic threshold (EROIext = 1), marking an internally generated transition in the resource–capital system. An entropy-based indicator is introduced as a thermodynamic proxy mirroring the decline in energetic efficiency within the modeled subsystem. These findings show how energetic reinvestment constraints generate endogenous peak and threshold behavior in resource-dependent systems. The analysis offers a structural perspective on interactions between depletion, capital accumulation, and net energy under simplified thermodynamic assumptions. These results provide insights into long-term structural constraints of the oil system, which may inform energy planning and policy discussions under conditions of declining net energy availability.

1. Introduction

Energy policy debates increasingly require a comprehensive understanding not only of the quantity and cost of energy resources, but also of the net energy delivered to society and its long-term systemic implications. In line with the emerging consensus within the Net Energy Analysis (NEA) community, net energy is defined as the energy supplied to society in the form of energy carriers after subtracting the energy invested for the production and distribution of those energy carriers [1]. The capacity of societies to sustain economic and institutional functions depends in part on this net energy surplus. When net energy is high, surplus energy can support innovation, infrastructure, and governance; when it declines, a greater share of economic activity must be devoted to energy production itself, potentially influencing macroeconomic performance [2,3,4].

A central metric for quantifying net energy is the Energy Return on Investment (EROI), defined as the ratio between the total energy returned and the total energy invested to accomplish the conversion over the entire life cycle of the system under study [5]. The interpretation of EROI depends critically on the system boundary adopted. Following the stage-based taxonomy clarified by [5], EROI can be defined at (i) the primary stage (extraction level), (ii) the point-of-use stage (including refining and distribution), and (iii) the useful stage (including end-use conversion efficiency; see also [6]).

Historically, fossil fuels—particularly coal and conventional oil—exhibited high primary-stage EROI. Empirical analyses indicate that the primary-stage EROI of conventional oil has declined over time [7,8,9]. A comprehensive reinterpretation of peak oil dynamics from a net-energy perspective was provided by [1], who emphasized the potential divergence between gross production peaks and net energy availability under declining EROI. Engineering-based assessments of global oilfields have quantified long-term trends in extraction-stage EROI [10,11], while macroeconomic coupling approaches have explored interactions between energy return and economic output [12,13]. More recent analyses have examined the relationship between EROI and profitability in unconventional production systems [14].

The observed decline in primary-stage EROI is broadly consistent with structural changes in global oil production, including the progressive shift from high-productivity conventional reservoirs toward more technically complex and energy-intensive sources such as deepwater, tight oil, and oil sands [10,15,16]. These structural shifts provide a physical context for observed long-term EROI trends. Recent studies have also explored technological approaches to improving monitoring and operational efficiency in petroleum processing systems [17]. Although global proven reserves have not uniformly decreased, changes in resource composition and productivity imply increasing capital and energy requirements per unit of output.

Energy inputs can be disaggregated following [18] into direct energy used in extraction and processing, indirect energy embodied in capital goods and materials, and auxiliary energy requirements. Explicit specification of these components is essential for ensuring comparability across studies. Prior to the recent clarification of stage-based taxonomy, several studies adopted broader boundary definitions often described as “extended EROI,” incorporating infrastructure energy, indirect inputs, and multi-product system interactions [19,20,21].

The extended EROI (EROIext) adopted in this study builds on this earlier extended-boundary tradition while aligning its definitions explicitly with the contemporary taxonomy proposed by [5]. Conceptually, EROIext corresponds to the point-of-use stage, as it accounts for energy inputs required to deliver refined energy carriers to society. It differs from narrower point-of-use definitions by explicitly incorporating infrastructure-related indirect energy investments, including capital embodied in extraction, refining, transport, and distribution systems. It does not extend to the useful-stage boundary, as end-use conversion efficiencies are treated separately. The numerator corresponds to the gross energy content of refined petroleum products supplied to society within the defined boundary.

A substantial body of literature has incorporated EROI into long-term projections and macroeconomic frameworks. Dynamic biophysical and stock-flow consistent models have examined interactions between declining net energy and economic structure [22,23,24,25,26]. Integrated global modeling efforts have explored transition feasibility under biophysical constraints [27,28,29], while system-wide analyses have evaluated surplus energy availability and infrastructure requirements in low-carbon transitions [30,31,32]. These studies demonstrate that net energy constraints can influence macroeconomic trajectories under certain structural conditions. At the same time, EROI represents only one biophysical indicator among many determinants of societal outcomes; institutional arrangements, technological innovation, distributional dynamics, and policy frameworks also play critical roles. Similar EROI levels may therefore correspond to different economic and social trajectories. Numerical modeling has become an essential tool across a wide range of energy-related engineering problems, including reservoir stimulation, drilling optimization, and enhanced recovery processes. Recent studies have applied numerical and data-driven approaches to improve energy extraction efficiency, fracture behavior prediction, and resource utilization [33,34,35]. These approaches highlight the importance of integrating physical processes with computational modeling frameworks in understanding energy system performance. However, most existing studies focus on engineering-scale processes while fewer studies address system-level energetic efficiency and its thermodynamic implications.

Despite these advances, certain analytical combinations remain underexplored. In particular, while dynamic EROI modeling and macroeconomic frameworks have been developed, fewer studies explicitly couple extended-boundary EROI definitions with entropy-based thermodynamic interpretation within a unified Lotka–Volterra-type resource–capital dynamic structure applied specifically to the global petroleum system. The present study focuses on this integration rather than on asserting that EROI alone determines societal stability.

From a thermodynamic perspective, energy systems operate by importing concentrated (low-entropy) energy carriers and exporting degraded (high-entropy) outputs [36,37]. Incorporating entropy analysis provides an additional physical lens through which long-term net energy trajectories can be interpreted, without implying deterministic relationships between EROI and social outcomes.

This study applies a Single-Cycle Lotka–Volterra (SCLV) model to the global petroleum system. Petroleum resources and the capital required for extraction, refining, transport, and utilization are represented as coupled energy stocks. EROIext is explicitly defined at the point-of-use stage with an extended boundary including both direct and indirect infrastructure-related energy inputs. The model is calibrated using historical data from 1965 to 2012 and projected through 2100. By integrating entropy analysis into this dynamic framework, we examine the co-evolution of extended EROI, capital accumulation, and resource productivity within a physically coupled system.

This integrated framework contributes in three principal ways. First, it models oil production, capital investment, EROIext, and entropy generation as interdependent trajectories within an explicit dynamic structure. Second, it identifies dynamic thresholds, including the projected year when EROIext falls below unity within the defined boundary. Third, it provides a thermodynamically informed interpretation of petroleum’s long-term net energy role under changing resource conditions.

Ultimately, the results underscore the importance of evaluating energy systems not solely by production volume or monetary cost, but by their capacity to sustain net energy flows over time within explicit physical and thermodynamic boundaries. In doing so, this study complements existing macro-level EROI modeling efforts by focusing on the dynamic integration of extended EROI accounting and entropy-based interpretation within a unified petroleum resource–capital framework.

2. Methods

The petroleum energy production system is modeled using the Single-Cycle Lotka–Volterra (SCLV) model proposed by [38]. Additionally, entropy is introduced into the SCLV framework based on the approach of [37], and its dynamics are analyzed accordingly. This section first describes the basic structure of the SCLV model. It then explains how entropy is formulated using the parameters of the SCLV model. Finally, it details the data and fitting techniques used to model the petroleum energy production system.

2.1. SCLV Model

2.1.1. Lotka–Volterra (LV) Model

The Lotka–Volterra (LV) model, originally developed by [39,40], describes predator–prey interactions through a system of coupled differential equations. Although the LV model is typically expressed in terms of population sizes, it can also be interpreted in terms of energy flows: energy is transferred from the prey ($L_1$) to the predator ($L_2$). Thus, in a general LV framework, $L_1$ and $L_2$ are treated as energy stocks, and the model describes the flow of energy between them (see Figure 1).

In the LV energy flow diagram, the prey ($L_1$ stock) receives energy from the external environment. This energy accumulates in $L_1$ and is gradually transferred to the predator ($L_2$ stock). During this transfer, a portion of energy is dissipated as waste heat to the environment. The energy that reaches $L_2$ also accumulates but diminishes over time due to consumption and dissipation. For example, in a rabbit–fox ecosystem, rabbits gain energy from grazing (non-explicit environmental stock), and foxes gain energy by preying on rabbits. However, not all energy is transferred directly; some is lost due to hunting costs and metabolic processes (waste heat in Figure 1).

The energy exchange between the two stocks is represented by the following set of differential equations:

$${\displaystyle \frac{d{L}_1}{dt}}=k_1{L}_1-k_2{L}_1{L}_2$$

(1)

$${\displaystyle \frac{d{L}_2}{dt}}={\eta k}_2{L}_1{L}_2-k_3{L}_2$$

(2)

The model incorporates four constants: $k_1$, representing the energy growth rate of the prey; $k_2$, the energy decay rate of the prey; $k_3$, the energy decay rate of the predator; and $\eta$, the energy transfer efficiency, which is the inverse of waste heat loss. Although the Lotka–Volterra (LV) model is seldom used in modern ecological applications due to its simplicity [41], it has found application in economics, particularly in fisheries modeling [38,42]. In these contexts, the prey is conceptualized as a “resource,” while the predator represents the “capital” required to extract or utilize that resource—a perspective adopted in the present study.

The use of a Lotka–Volterra (LV) formulation in this study should not be interpreted as a literal representation of oil systems as predator–prey ecological processes. Real-world oil systems are strongly mediated by technological innovation, market structures, geopolitical factors, and policy interventions. Classical critiques of simplified ecological models have long warned against conflating mathematical tractability with empirical validity. In particular, ref. [41] emphasized that models such as the logistic and Lotka–Volterra equations, while heuristically useful, often lack sufficient empirical validation when transferred to complex real-world systems. Following this cautionary perspective, the LV framework employed here is used as an abstract representation of energy–capital interdependence under biophysical constraints rather than as a predictive description of oil market dynamics. Accordingly, the model should be interpreted as highlighting structural energetic feedbacks rather than capturing the full institutional, technological, and political complexity of contemporary petroleum systems.

2.1.2. Description of the SCLV Model

In our model, the resource stock ($L_1$) corresponds to oil reserves, while the capital stock ($L_2$) represents the capital required to exploit these reserves, including the energy used for extraction, refining, transportation, and the infrastructure needed for petroleum utilization. Since fossil fuel resources such as oil regenerate at an extremely slow rate compared to the consumption and capital accumulation rates, the energy input rate from the environment ($k_1$) is assumed to be zero. The simplified Lotka–Volterra model under this condition is referred to as the Single-Cycle Lotka–Volterra (SCLV) model [38]. The governing equations of the model are as follows:

$${\displaystyle \frac{d{L}_1}{dt}}=-k_2{L}_1{L}_2$$

(3)

$${\displaystyle \frac{d{L}_2}{dt}}={\eta k}_2{L}_1{L}_2-k_3{L}_2$$

(4)

The model is characterized by three key parameters: $k_2$, representing the rate of conversion from resources to capital; $k_3$, denoting the degradation or depreciation rate of capital; and $\eta$, the efficiency of energy conversion. The term $d{L}_1$/$dt$ describes the rate of decline of the resource stock per unit time, which can be interpreted as the energy production rate.

The SCLV model has been used to describe production systems of slowly regenerating resources such as whales or gold [38]. Ref. [43] also applied it to petroleum systems by treating oil discoveries as resources and exploratory drilling as capital, demonstrating its relevance for oil production modeling. In this framework, EROI can be defined as the ratio of energy obtained by capital to the energy consumed by capital:

$$EROI={\displaystyle \frac{\eta k_2}{k_3}}{L}_1$$

(5)

Equation (5) expresses the energy return on investment (EROI) as a function of system efficiency ($\eta$), resource productivity ($k_2$), energy input requirements ($k_3$), and a scaling factor ($L_1$), thereby integrating both technical and resource-dependent parameters into a unified formulation. Because the capital stock in the model includes infrastructure required for petroleum utilization, the calculated EROI corresponds to EROIext.

2.2. Entropy and Its Integration with the Model

Ref. [36] approached the energy production system from a thermodynamic perspective using the theory of dissipative structures. This subsection explains the theoretical basis and formulation of entropy in our SCLV-based framework.

2.2.1. Energy Production System and Dissipative Structure Theory

Dissipative structure theory, established by [36], incorporates open-system dynamics into the second law of thermodynamics. The second law of thermodynamics states that in isolated systems, entropy inevitably increases over time, eventually leading to disorder and equilibrium. In contrast, open systems that exchange energy and matter with the environment may experience spontaneous emergence of ordered structures (dissipative structures) if the inflow of negative entropy surpasses a certain threshold. In open systems, energy dissipation does not simply lead to decay; rather, under continuous energy inflow, it can drive the emergence of new ordered structures that stabilize the system far from equilibrium. A system must meet several conditions to exhibit dissipative structures: it must be open, non-equilibrium, nonlinear, and highly sensitive to fluctuations [44]. The entropy level must also be regulated, such that the total entropy change in the system is less than or equal to zero.

While originally a concept from physics, dissipative structures have been applied in ecology and social science. For example, ref. [45] considered ecosystems—composed of biota and their environment—as dissipative structures that maintain order through exchanges of energy and matter. Likewise, socioeconomic systems, which are subsystems within ecosystems, consume low-entropy resources and expel high-entropy waste, thereby maintaining complex societal structures [44]. As an energy-intensive subsystem, the energy production system continuously exchanges energy and materials with both society and nature. It is inherently dynamic due to factors such as technological progress and new resource discoveries, and its behavior is shaped by nonlinear feedback mechanisms. Thus, it can be treated as a thermodynamic open system to which dissipative structure theory can be applied [46].

According to [37], the energy production system increases internal entropy by consuming low-entropy inputs from the socioeconomic system. Meanwhile, it reduces external entropy by transforming natural resources into usable low-entropy energy, thereby providing a net reduction in societal entropy.

2.2.2. Linking EROI and Entropy

To discuss the relationship between EROI and entropy, ref. [37] developed an analytical diagram of an energy production system based on a Carnot heat engine (Figure 2). In Figure 2, the socio-economic system, the energy production system, and resources are regarded as a single system. Let $Q_1$ be the heat content of the materials and energy initially supplied to the energy resource development system, $S_1$ the corresponding entropy, $Q_2$ the heat contained in the waste heat and waste materials after the supplied materials and energy have been consumed, and $S_2$ the corresponding entropy. The useful work available for utilizing the energy resources is denoted as $W_1$. The heat content of the energy products is $Q_3$ and their entropy is $S_3$. When the energy products are transferred to and consumed in the socio-economic system, the useful work utilized for human purposes is $W_2$, while the waste heat and thermal energy contained in the waste are $Q_4$ and the corresponding entropy is $S_4$.

The total change in entropy of the system is given by

$$\Delta S=S_2-S_1+[{-(S}_4-S_3)]$$

(6)

By multiplying this by the system temperature T, the entropy change is expressed in terms of heat. Then, applying the first law of thermodynamics yields

$$T\Delta S=W_1-W_2$$

(7)

If $W_1=Q_1{\alpha }_1,W_2=Q_3{\alpha }_2$ ($\alpha$ is the ratio of useful value to total energy), then

$$T\Delta S=Q_1{\alpha }_1-Q_3{\alpha }_2$$

(8)

If we simplify the EROI expression and assume that the energy quality coefficient is equal to the ratio of useful value to total energy, then

$$EROI={\displaystyle \frac{{{\alpha }_{2}Q}_3}{{{\alpha }_{1}Q}_1}}\iff {{\alpha }_{2}Q}_3=EROI\ast {{\alpha }_{1}Q}_1$$

(9)

Utilizing the relationship whereby the proportion of net energy equals the reciprocal of EROI, EROI is reformulated as:

$$EROI={\displaystyle \frac{-\Delta S+\Delta S_1}{\Delta S_1}}$$

(10)

where $\Delta S_1=W_1/T$, and $W_1$ represents the network invested for utilizing the energy resource, with $\Delta S_1$ denoting the change in environmental entropy when $W_1$ is released as heat.

The greater the entropy reduction achieved by the energy production system, the larger the EROI becomes. Conversely, when the entropy change due to the energy production system is positive, the EROI falls below 1. Thus, within an energy production system considered as a thermodynamic open system, there exists a strong connection between EROI and entropy.

2.2.3. Integration with the SCLV Model

By combining the above relationship between EROI and entropy with the relationship between EROI and other parameters in the SCLV model, entropy can be expressed in terms of the parameters of the SCLV model. As an assumption for this combination, it is presumed that the EROI derived from the entropy perspective, like how the EROI derived from the SCLV model is EROIext. The validity of this combination will be discussed in the Section 4.

In the SCLV model, the EROI was given by

$$EROI={\displaystyle \frac{\eta k_2}{k_3}}{L}_1$$

(11)

Combining this with the relationship between EROI and entropy yields:

$${\displaystyle \frac{-\Delta S+\Delta S_1}{\Delta S_1}}={\displaystyle \frac{\eta k_2}{k_3}}{L}_1$$

(12)

Rearranging gives:

$${\displaystyle \frac{\Delta S}{\Delta S_1}}=1-{{\displaystyle \frac{\eta k_2}{k_3}}L}_1$$

(13)

Thus, the ratio of total entropy change to the change in environmental entropy when $W_1$ is released as heat ($\Delta S/\Delta S_1$) can be expressed in terms of the parameters of the SCLV model. The entropy ratio ($\Delta S/\Delta S_1$) can be interpreted as the expected entropy reduction effect when energy is supplied to a given system. For example, for a certain system, if $\Delta S/\Delta S_1=6.0$, and energy $x$ causes an entropy change of $s$ when released as heat, then if that energy is invested into the petroleum system, the total entropy decreases by $-6.0s$. The ratio $\Delta S/\Delta S_1$ can be regarded as an indicator of the efficiency of entropy reduction in a system and has characteristics similar to the EROI, which represents the efficiency of an energy production system. However, because entropy is more conceptual than EROI, the entropy ratio can be considered a value that accounts for a more holistic effect.

2.3. Modeling the Petroleum Production System

The SCLV model of the energy production system is applied to the case of petroleum production. The differential equations of the SCLV model are as follows:

$${\displaystyle \frac{d{L}_1}{dt}}=-k_2{L}_1{L}_2$$

(14)

$${\displaystyle \frac{d{L}_2}{dt}}={\eta k}_2{L}_1{L}_2-k_3{L}_2$$

(15)

Here, $L_1$ represents resources; $L_2$ represents the capital required for the production, transportation, and utilization of energy; and $k_2$, $k_3$, and $\eta$ are constants. Modeling was conducted using data from 1965 to 2012. First, we describe how $L_1$ and $L_2$ were defined, and then explain how the constants were fitted using these values.

2.3.1. Resource Stock ($L_1$)

Because the $L_1$ stock does not increase within the model framework, $L_1$ is defined as the “total recoverable reserves,” representing the sum of proved, probable, and expected additional reserves. The proved reserves refer to the amount of crude oil confirmed to be recoverable with current technology; the probable reserves refer to the amount of crude oil expected to be discovered and recoverable with current technology; and the expected additional reserves refer to the increase in recoverable crude oil due to advances in recovery technology.

The annual total recoverable reserves from 1965 to 2012 were determined as follows. First, the total recoverable reserves in 1965 were calculated by subtracting the cumulative petroleum production up to 1965 from the ultimately recoverable reserves (cumulative production + proved reserves + probable reserves + expected additional reserves) published in [47]. The cumulative production up to 1965 was obtained from the data presented in the paper [48]. For the years after 1966, the total recoverable reserves for each year were defined as the total recoverable reserves in 1965 minus the cumulative production up to the previous year. Annual production data were taken from BP Statistical Review [49,50,51].

2.3.2. Capital Stock ($L_2$)

The $L_2$ stock represents the capital required for producing, refining, transporting, and utilizing energy. The capital for production and refining was determined using [48], while the capital for transportation and utilization was estimated with reference to [4]. The methods for obtaining each are described below.

Ref. [48] calculated the standard EROI (EROIst) for global oil production over the period 1860–2012 using macroeconomic data such as energy prices and GDP. We use the subset covering 1965–2012 (Figure 3). In line with conventional usage, EROIst is defined at the wellhead/field boundary, i.e., excluding refining, long-distance transportation, and end-use infrastructure. Accordingly, the input energy of EROIst covers exploration, drilling, lifting, and field operations, while refining and downstream processes are excluded. To estimate the energy requirements for production and refining, we divided EROIst values by annual petroleum production.

As with the $L_1$ stock, annual production values were taken from BP Statistical Review data [51] (MMbbl = million barrels).

$$\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t} \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}(\mathrm{M}\mathrm{M}\mathrm{b}\mathrm{b}\mathrm{l})={\displaystyle \frac{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}\left(\mathrm{M}\mathrm{M}\mathrm{b}\mathrm{b}\mathrm{l}\right)}{\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}\mathrm{s}\mathrm{t}}}$$

(16)

Ref. [4], in considering the EROIext for the United States, referred to the energy required for transportation and utilization. According to [4], when automobiles are assumed to be the primary means of energy transportation, the total of (1) the energy required for transportation and (2) the energy equivalent of the capital needed to use this transportation—such as vehicle maintenance costs and road maintenance costs—amounts to 64% of the energy produced domestically in the United States. We assume automobiles to be the primary means of energy transportation and further assume that, in any given year, 64% of the energy produced is allocated to transportation and utilization capital. This amount was included in the $L_2$ stock.

The energy flow diagram of the SCLV model developed in this study is shown in Figure 4. The process of energy transfer from the $L_1$ stock to the $L_2$ stock represents the movement of energy used for petroleum production as well as energy required to form the capital necessary for petroleum utilization. In the process of energy loss from the $L_2$ stock, the energy consumed in petroleum production and in maintaining the facilities required for petroleum production and utilization is released to the external system in forms that are unusable within the $L_2$ stock, such as heat. Energy that moves directly from the $L_1$ stock to the external system includes waste heat that could not be extracted as usable energy during production, as well as energy invested in sectors unrelated to the capital involved in petroleum production.

2.3.3. Parameter Fitting

Modeling was performed using the above data. The methodology is described below. First, the SCLV model was discretized using a finite difference approximation. The discretized differential equations are as follows:

$$L_1\left(t+1\right)=\left(1-\Delta t{k}_2{L}_2\left(t\right)\right)L_1\left(t\right)$$

(17)

$$L_2\left(t+1\right)=\left(1+\Delta t\eta k_2{L}_1\left(t\right)-\Delta t{k}_3\right)L_2\left(t\right)$$

(18)

Here, $L_1\left(t\right)$ and $L_2\left(t\right)$ represent the resource amount (MMbbl) and capital amount (MMbbl) in year $t$, respectively. The time step ($\Delta t$) was set to one year. Using this discretized SCLV model, parameter fitting was performed. The fitting was conducted using the least-squares method. First, the squared errors between the resource and capital amounts calculated by the model and those obtained from the data were computed. Next, to match the order of magnitude between resource and capital values, the squared errors for resources and capital were each normalized by their respective maximum values.

Subsequently, the sum of the normalized squared errors was calculated:

$$\mathrm{S}\mathrm{u}\mathrm{m} \mathrm{o}\mathrm{f} \mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{s}=\mathsf{\Sigma }{\displaystyle \frac{\mathrm{S}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{s}}{\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m} \mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{s}}}+\mathsf{\Sigma }{\displaystyle \frac{\mathrm{S}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{c}\mathrm{a}\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}}{\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m} \mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{c}\mathrm{a}\mathrm{p}\mathrm{i}\mathrm{t}}}$$

(19)

Finally, the parameters ($k_2$, $k_3$, and $\eta$) were fitted so as to minimize the sum of squared errors. The fitting was performed using the Generalized Reduced Gradient (GRG) method implemented in Excel Solver.

To assess the sensitivity of model outcomes to parameter variations, we vary $k_2$, $k_3$, and $\eta$ by ±10% and evaluate their effects on $L_1$, $L_2$, and production. Results are presented as confidence bands in the subsequent figures.

3. Results

The results are presented in three interrelated parts. First, the baseline data on resource and capital stocks are introduced to provide the empirical foundation of the analysis. Second, the outcomes of fitting the SCLV model to the observed data are reported, allowing for parameter estimation and model validation. Finally, the time-series evolution of EROIext and the entropy ratio, derived from the calibrated model, is analyzed to highlight their dynamic interplay over the study period.

3.1. Baseline Data for Resource and Capital Stocks

The “Original” blue solid line in Figure 5a shows the baseline data for resource stock ($L_1$), while the “Original” blue solid line in Figure 5b depicts the capital stock ($L_2$), and that in Figure 5c represents production ($d{L}_1/dt$). At the start of the observation period in 1965, the total recoverable oil reserves amounted to 4,586,591 million barrels. As capital for petroleum utilization increased, annual production grew, leading to a steady decline in reserves. By 2012, the total recoverable reserves had decreased to 3,469,680 million barrels. The RMSE values are 1.79 × 10⁵ for $L_1$, 2.38 × 10³ for $L_2$, and 6.39 × 10³ for production. These correspond to relative errors of approximately 4–5% for $L_1$, 10–15% for $L_2$, and about 20% for production, indicating moderate agreement between the model and historical data. The corresponding R² values are 0.71 for $L_1$, 0.53 for $L_2$, and –0.73 for production. The negative R² for production reflects the model’s simplified representation of short-term production variability rather than a failure to capture the long-term structural trend. The model is designed to capture long-term structural dynamics rather than precise historical matching, and therefore qualitative agreement is considered the primary evaluation criterion.

3.2. Fitting Results of the SCLV Model

Using the baseline data, the parameters of the SCLV model were fitted, with the resulting values and total squared error summarized in

Table 1. Fitting results of the SCLV model using baseline data. The table reports the estimated parameter values—including $k_2$, $k_3$, and $\eta$—together with the sum of squared errors, which serve as the basis for the subsequent sensitivity analyses.

${\mathit{k}}_2$	${\mathit{k}}_3$	$\mathit{\eta }$	Sum of Squared Errors
2.9397 × 10−7	1.8271 × 10−2	3.2883 × 10−2	26.1245

. The modeled time evolution of the energy quantities for the resource stock, capital stock, and production is represented by the “Model” red solid lines in Figure 5a, Figure 5b and Figure 5c, respectively.

To assess the model’s accuracy, the modeled values of L₁ (resource stock), L₂ (capital stock), and production (dL₁/dt) were compared with their corresponding baseline data. For resource stock and production, the discrepancy between the simulated and observed values tends to increase over time. In contrast, for capital stock, the difference is pronounced between 1965 and 1980, but the modeled values subsequently align closely with the baseline data.

The model was then employed to project the future dynamics of resource stock, capital stock, and production through 2100 (Figure 6). Resource stock declines at an accelerating rate until production reaches its peak around 2042, after which, depletion slows. The capital stock increases steadily until reaching a maximum in 2081, followed by gradual contraction. Production exhibits a bell-shaped trajectory, peaking around 2042 within the model simulation. This peak year represents a structural outcome of the calibrated SCLV framework rather than a predictive forecast of future production.

Structurally, depletion is governed by the nonlinear interaction term $k_2{L}_1{L}_2$. Production is defined as $Production=k_2{L}_1{L}_2,$ indicating that both remaining reserves and accumulated capital positively influence output. The bell-shaped production curve arises from the dynamic coupling between resource depletion and capital accumulation.

EROIext is proportional to the remaining resource stock:

$${\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}}_{ext}={\displaystyle \frac{\eta k_2}{k_3}}{L}_1.$$

(20)

Hence, its decline mirrors the structural evolution of L₁.

The capital stock evolves according to:

$${\displaystyle \frac{d{L}_2}{dt}}=\eta k_2{L}_1{L}_2-k_3{L}_2,$$

(21)

representing the balance between reinvestment from production and capital dissipation. The projected temporal evolution of L₂ is shown in Figure 7, which exhibits a bell-shaped pattern that peaks in 2081.

Setting this equal to zero yields the condition:

$${\displaystyle \frac{\eta k_2}{k_3}}{L}_1=1,$$

(22)

which corresponds to ${\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}}_{ext}=1$. In the calibrated model, this threshold occurs around 2081. This value should be interpreted as a model-derived energetic threshold under the assumed parameterization rather than a deterministic prediction of future oil-system dynamics. The peak coincides with the capital stock peak.

Similarly, the production peak condition in the SCLV structure can be expressed as

$${\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}}_{ext}=1+{\displaystyle \frac{k_2}{k_3}}{L}_2,$$

(23)

and the minimum deviation from this condition occurs in 2042, corresponding to the projected oil production peak.

3.3. Sensitivity Analysis

A sensitivity analysis was conducted for the model developed in this study. The parameters $k_2$, $k_3$, and $\eta$ were individually varied by ±10%, and the resulting $L_1$ stock, $L_2$ stock, and production were calculated and plotted as error ranges (Figure 8). The percentage used for estimating the $L_2$ stock affects only the calibration stage and does not directly alter the structural parameters of the SCLV model. Therefore, although this assumption introduces some uncertainty into the L2 calibration, its influence on the overall system dynamics is limited.

According to Figure 8a,d,g, the $L_1$ stock shows low sensitivity to $\eta$, but its sensitivity to $k_2$ and $k_3$ increases over time. When the error rates in 2100 were calculated for variations in $k_2$, $k_3$, and $\eta$, the error rate for $\eta$ was 10%, whereas those for $k_2$ and $k_3$ were 47% and 43%, respectively. In terms of qualitative behavior, for variations in any of the coefficients, the rate of decline increased over time, but around 2040, the rate of decline began to decrease.

According to Figure 8b,e,h, the $L_2$ stock exhibits high sensitivity to $k_2$ from 2000 to 2080, after which, the sensitivity decreases toward 2100. Sensitivity to $k_3$ and $\eta$, on the other hand, increases year by year. When the error rates in 2100 were calculated, variations in $k_2$, $k_3$, and $\eta$ yielded error rates of 7%, 27%, and 60%, respectively. Qualitatively, in all cases, the $L_2$ stock increases over time, reaches a peak, and then decreases. Changes in the timing of the peak position due to parameter variations are summarized in

Table 2. Sensitivity analysis of the peak year of oil production under different parameter settings. The table compares the baseline SCLV model with cases where the parameters $k_2$, $k_3$, and $\eta$ are varied by ±10%. The results indicate the corresponding shifts in the timing of the oil production peak relative to the baseline model.

	Model	${\mathit{k}}_2$ × 1.1	${\mathit{k}}_2\times$ 0.9	${\mathit{k}}_3$ × 1.1	${\mathit{k}}_3$ × 0.9	$\mathit{\eta }$ × 1.1	$\mathit{\eta }$ × 0.9
Year of Peak	2042	2035	2028	2049	2047	2047	2043
Difference from Model		7 years	14 years	7 years	5 years	5 years	1 year

. A ±10% change in $k_2$, $k_3$, or $\eta$ resulted in a shift of approximately 10–20 years in the peak position.

According to Figure 8c,f,i, production shows gradually increasing sensitivity to $k_2$ from 1965 onward, but sensitivity decreases by 2050, after which, it increases again. Sensitivity to $k_3$ increases annually, peaking around 2040 and then decreasing. Sensitivity to $\eta$ also increases annually but remains almost constant from around 2040 onward. When the error rates in 2100 were calculated, variations in $k_2$, $k_3$, and $\eta$ yielded error rates of 43%, 17%, and 57%, respectively. Qualitatively, similar to the capital stock, production increases over time, reaches a peak, and then decreases for variations in any of the parameters. Changes in the timing of the production peak due to parameter variations are summarized in

Table 3. Sensitivity analysis of the peak year of capital stock under different parameter settings. The baseline SCLV model is compared with cases where the parameters $k_2$, $k_3$, and $\eta$ are varied by ±10%. The results show how these parameters changes shift the timing of the capital stock peak relative to the baseline model.

	Model	${\mathit{k}}_2$ × 1.1	${\mathit{k}}_2$× 0.9	${\mathit{k}}_3$ × 1.1	${\mathit{k}}_3$ × 0.9	$\mathit{\eta }$ × 1.1	$\mathit{\eta }$ × 0.9
Year of Peak	2081	2071	2062	2093	2088	2094	2089
Difference from Model		10 years	19 years	12 years	7 years	13 years	8 years

. Compared with $\eta$ and $k_3$, $k_2$ exhibits higher sensitivity with respect to the peak position.

While the present sensitivity analysis focuses on ±10% perturbations of key parameters to evaluate local stability of the model behavior, wider parameter ranges (e.g., ±20–30%) could further influence quantitative outcomes such as the timing of production and capital peaks, as well as the EROIext = 1 threshold. The current results therefore illustrate local sensitivity around the calibrated parameter set rather than a full exploration of parameter uncertainty. A more comprehensive sensitivity analysis over broader parameter ranges is an important direction for future work.

3.4. Time Evolution of EROIext and Entropy Ratio

The entropy ratio $\Delta S/\Delta S_1$ was computed from the fitted parameters using the following relationship:

$${\displaystyle \frac{\Delta S}{\Delta S_1}}=1-{{\displaystyle \frac{\eta k_2}{k_3}}L}_1$$

(24)

By substituting the fitted coefficients, the trajectories of EROIext (blue solid line in Figure 9) and the entropy ratio (ΔS/ΔS₁) (red solid line in Figure 9) from 1965 to 2100 were calculated. In 1965, the EROIext was approximately 2.5, but it had decreased to about 2.0 by 2010. The decline continues thereafter, falling below 1.0 in 2081 and reaching around 0.79 by 2100. The threshold EROIext = 1 has structural significance within the SCLV framework. Setting ${\displaystyle \frac{d{L}_2}{dt}}=0$ in the capital stock equation yields

$${\displaystyle \frac{{\eta k}_2}{k_3}}{L}_1=1$$

(25)

which corresponds exactly to the condition EROIext = 1. Thus, the capital stock reaches its maximum at the moment when net energy from oil becomes unity. In the calibrated model, this occurs in 2081, coinciding with the peak of the capital stock shown in Figure 7.

Regarding the entropy ratio, the results indicate a continuous increase from 1965 to 2100. The entropy ratio represents the expected entropy reduction effect when energy is introduced into a system, where a smaller value corresponds to a larger reduction effect. Therefore, it can be interpreted that the entropy reduction effect of the oil production system steadily diminishes over the period from 1965 to 2100. By 2081, the value becomes greater than zero, indicating that investing energy into the oil production system is expected to increase entropy rather than reduce it.

4. Discussion

This section interprets the structural dynamics of the resource stock (L₁), capital stock (L₂), production, and $\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}\mathrm{e}\mathrm{x}\mathrm{t}$ within the SCLV framework. The emphasis is placed on endogenous peak formation, energetic thresholds, and structural interpretation rather than predictive forecasting.

4.1. Structural Interpretation of Peak Dynamics

The modeled production peak occurs before half of the ultimate recoverable resource has been depleted, which differs from the classical symmetric logistic formulation proposed by [52]. In the SCLV framework, peak timing is not determined solely by geological depletion but by the nonlinear interaction term $k_2{L}_1{L}_2$, which couples remaining resources with capital intensity. As capital accumulates, depletion accelerates; as the resource base declines, the interaction weakens, generating an endogenous turning point.

This structural behavior contrasts with reserve-fraction-based peak interpretations and indicates that energetic reinvestment dynamics can modify peak timing independently of simple depletion ratios. While additional dissipative mechanisms—such as increasing system complexity—could in principle accelerate capital decline, such effects are not explicitly modeled and remain potential extensions rather than demonstrated outcomes. This interpretation is consistent with previous applications of SCLV-type models to resource systems [38] while extending the framework to incorporate extended EROI dynamics.

4.2. Model Fit and Projection Scope

The SCLV model reproduces the broad qualitative co-evolution between resource depletion, capital accumulation, and production, but it does not perfectly replicate observed historical trajectories. As shown in Figure 5, divergences between modeled and observed trends become more pronounced in the later portion of the calibration period. These discrepancies reflect structural simplifications of the model, including the assumption of a fixed resource base and the exclusion of technological change and unconventional resource expansion. These factors have played an increasingly important role in shaping global oil production in recent decades. While this divergence affects short- to medium-term accuracy, the model is designed to capture long-term structural dynamics rather than detailed historical matching. In addition, the capital stock formulation follows an energy-consistent approach. Production and refining capital are derived from EROIst and annual production (Equation (16)), while transportation and utilization capital are parameterized as a fixed share (64%) of produced energy. This simplification ensures consistency within the energy accounting framework, although it does not capture regional variability.

The framework assumes a fixed ultimate recoverable resource base and excludes endogenous reserve growth arising from technological innovation, new discoveries, unconventional resource development, and price-induced reclassification. The model assumes that if current structural relationships are maintained, the production peak may be reached around 2041. It also omits market-mediated feedback, capital reallocation, substitution dynamics, and policy interventions that have significantly shaped global oil production patterns in recent decades.

Accordingly, projections extending to 2100 should not be interpreted as forecasts. Instead, the reported peak year and energetic thresholds represent structural trajectories implied by the assumed parameters of the SCLV model. The long-time horizon is employed to illustrate asymptotic behavior and energetic threshold conditions under simplified structural assumptions. The resulting trajectories describe the implications of the estimated parameters within a stylized resource–capital system rather than deterministic predictions of future production levels.

4.3. Structural Lag Between Production and Capital Peaks

As illustrated in Figure 6, production reaches its maximum in 2042, whereas capital stock peaks in 2081, producing an approximately four-decade lag. The temporal lag between production and capital peaks is a structural property of the model and arises from the reinvestment condition governing capital accumulation (Equations (22) and (23)). It is therefore not an emergent empirical phenomenon but a direct consequence of the model formulation. Capital continues to expand as long as reinvested energy exceeds dissipation and reaches its maximum when $\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}\mathrm{e}\mathrm{x}\mathrm{t}=1.$ At this threshold, energetic reinvestment becomes insufficient to sustain further expansion of capital stock.

This structural outcome differs from classical logistic peak models in which peak timing is directly tied to depletion fractions. In the SCLV framework, asynchronous turning points arise naturally from energetic feedbacks between resource stocks and capital stocks. Although the model abstracts from demand, price, and substitution mechanisms, it demonstrates that energetic constraints alone can generate temporally separated peaks in coupled resource–capital systems.

4.4. Relation to Previous Net Energy Modeling Approaches

Net energy research has advanced substantially through life-cycle assessment (LCA), environmentally extended input–output (EEIO) analysis, and hybrid methodologies. These approaches provide increasingly refined empirical estimates of EROI across multiple system boundaries and have, in some cases, been incorporated into macroeconomic or transition modeling frameworks.

Previous empirical EROI assessments using LCA and EEIO approaches [53,54] have focused on boundary specification and measurement accuracy. The present framework instead emphasizes structural feedback mechanisms within a reduced dynamical system.

The present study does not aim to replace these methods. Instead, it introduces a simplified dynamical representation in which energetic reinvestment conditions produce endogenous peak and threshold behavior. A distinctive feature of this framework is that the coincidence between the capital peak and the condition $\mathrm{E}\mathrm{R}\mathrm{O}\mathrm{I}\mathrm{e}\mathrm{x}\mathrm{t}=1$ follows directly from the governing equations rather than from imposed assumptions about reserve exhaustion. The model therefore offers a structural complement to empirical estimation studies by highlighting internal energetic feedback mechanisms within a reduced-form dynamical system.

4.5. Entropy Representation Within the Model Framework

The entropy ratio introduced in this study follows the same structural trajectory as EROIext, as shown in Figure 9. Within the SCLV formulation, it provides an alternative thermodynamic representation of declining energetic efficiency. The entropy index used in this study should be interpreted as a thermodynamic proxy rather than a directly measurable physical entropy. It reflects the relative increase in energetic dissipation associated with declining EROI in the modeled oil production system.

However, the entropy variable is defined strictly within the boundaries of the two-stock production subsystem and does not quantify societal entropy, institutional stability, or broader social order dynamics. It should therefore be interpreted as a thermodynamic proxy describing internal dissipation processes rather than as a comprehensive indicator of social outcomes. The alignment between the entropy threshold and the capital peak reinforces the internal consistency of the formulation but does not extend beyond the modeled subsystem.

4.6. Limitations and Future Extensions

The structural simplifications of the SCLV framework impose clear limitations on interpretation. The model assumes a fixed ultimate recoverable resource base and does not incorporate endogenous reserve growth, technological learning, unconventional resource expansion, or market-mediated adjustments. Price feedback, substitution effects, and policy interventions are excluded, and no explicit socioeconomic stock governs energy allocation decisions. The percentage used to estimate the transportation and utilization capital share (64%) is based on a U.S.-focused study and may not fully represent global variations. Although this assumption affects only the calibration stage, further sensitivity analysis (e.g., 50–70%) would be useful to assess its impact more rigorously. Parameter estimation in Equation (19) is performed using a nonlinear least-squares approach implemented in Excel Solver. In addition, the sensitivity analysis is limited to ±10% parameter variations, which primarily capture local stability around the calibrated parameter set. Broader parameter ranges (e.g., ±20–30%) could further influence quantitative outcomes such as peak timing and the EROIext threshold, and represent an important direction for future work. While this approach is sufficient to capture the structural behavior of the system, formal uncertainty quantification (e.g., confidence intervals based on the Hessian matrix) and comparison with alternative optimization algorithms were not conducted and remain subjects for future work. Furthermore, calibration relies on historical data up to 2012, limiting empirical alignment with more recent developments such as shale expansion and accelerated renewable deployment. Consequently, recent structural changes in global oil production, particularly the rapid expansion of shale oil production in the United States, are not explicitly represented in the model. These developments may affect the short- to medium-term dynamics of oil production but do not fundamentally alter the structural mechanisms represented in the SCLV framework.

These constraints restrict the model to structural analysis rather than predictive forecasting. Future extensions could introduce endogenous reserve growth, technological change, renewable energy stocks, and price–capital feedback mechanisms while preserving thermodynamic consistency. Expanding the framework to include an explicit socioeconomic subsystem would allow for a more realistic representation of energy allocation dynamics and facilitate exploration of more complex transition pathways.

5. Conclusions

This study employed a two-stock SCLV (System Coupled Lotka–Volterra) framework to examine the long-term dynamics of oil production from a thermodynamic perspective. By explicitly modeling the nonlinear interaction between resource stock (L₁) and capital stock (L₂), the analysis demonstrates how depletion, capital accumulation, and extended energy return on investment (EROIext) co-evolve within a structurally coupled system.

The results indicate that the production peak emerges endogenously from the interaction term governing resource extraction rather than from a fixed depletion fraction of ultimate recoverable resources. Furthermore, the capital stock reaches its maximum later than production, and this turning point coincides with the condition EROIext = 1. This threshold represents the moment at which reinvested energy becomes insufficient to sustain further capital expansion within the modeled subsystem. The alignment between the capital peak and the energetic threshold arises directly from the governing equations and constitutes a central structural feature of the model.

The primary contribution of this study lies in embedding EROIext within a dynamic resource–capital interaction framework. Rather than treating EROI as a static performance indicator, the model shows how energetic reinvestment constraints can generate endogenous turning points in resource-dependent systems. In this formulation, peak timing and structural transitions are not imposed exogenously but follow from the internal thermodynamic dynamics of the system.

Several limitations must be acknowledged. The model adopts a simplified two-stock structure and does not incorporate endogenous reserve growth, technological progress, price mechanisms, renewable substitution, or policy intervention. Climate constraints and environmental externalities are also not explicitly represented. Accordingly, the projections presented here should not be interpreted as forecasts but as structural trajectories implied by the assumed resource–capital dynamics under simplified conditions.

Moreover, while EROI provides insight into physical and thermodynamic constraints of energy systems, it is not a comprehensive metric for policy evaluation. The energetic thresholds identified in this study describe internal subsystem dynamics and must be interpreted alongside broader economic, environmental, and social considerations within multi-dimensional assessment frameworks.

Future research could extend the present framework by incorporating renewable energy stocks, technological learning effects, endogenous reserve growth, or explicit socioeconomic decision-making structures. Such extensions would allow for examination of how energetic constraints interact with transition pathways in more complex system configurations.

In summary, this study reframes resource depletion not solely as a function of remaining geological reserves, but as a dynamic process governed by declining reinvestable net energy. Within the SCLV framework, thermodynamic constraints can, within the SCLV formulation, generate endogenous structural turning points in resource-dependent energy systems, independent of externally imposed peak assumptions.