# Towards a Social-Ecological-Entropy Perspective of Sustainable Exploitation of Natural Resources

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## Abstract

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## 1. Introduction

## 2. What We Mean by Social-Ecological System?

- A social subsystem interacts with its environment through extraction or restoration of ecological resources.
- A social subsystem interacts with another social subsystem by cooperation or competition processes.
- The members of a social subsystem interact with each other by sharing, transmitting, or transferring knowledge.

#### 2.1. Irreversibility in a SES

#### 2.2. The Analogy with Chemical Kinetics

#### 2.3. Bio-Mathematical Models and SEE Novelty

## 3. The Modeling Strategy

- The relations between social subsystems and their ecological surroundings can be treated as energetic transformations;
- The dynamics of these relations respond to irreversible processes;
- The social subsystems can exploit and restore their environment;
- Each social subsystem has an internal structure that modifies the interaction with its environment;
- The internal structure consists of differentiated sectors and there exists a population flux between them;
- The population flux is regulated by some rates that are inherent to the system;
- External agents can modify some rates and others are controlled internally.

#### 3.1. Model in Abstracto

- Knowledge sectors: answer the question of who knows what?
- Knowledge transfer-method: answer the question of who learns from who?
- Characterization parameters: describe which type of and how much knowledge the epistemological community has.
- Control parameters represent when an epistemological community considers that someone already knows the necessary information and can change sector.

#### 3.2. Social-Ecological Entropy Production as Sustainability Criterion

#### 3.3. Types of Intervention

- Natural intervention: the change of environmental conditions. For example, the change of temperature or humidity or a natural disaster that occurs across the natural ecosystem affects the resource or the social subsystems.
- Addition intervention: the increase, decrease, or substitution of elements in the system. For example, the arrival of a new community into a pre-existing system.
- Behavior intervention: the change of control parameters to regulate the behavior of the communities. For example, a change in the number of years of elementary school.

#### 3.4. The Model

#### 3.5. Knowledge Transfer Methods

#### 3.6. Mobility through Sectors: The Mathematical Model

#### 3.7. Relation with Resource

#### Complete Model

## 4. Results and Discussion: The Two Community Case

#### 4.1. Methodology of Simulations

- Select a knowledge-transfer method for ${C}_{1}$ and ${C}_{2}$ substituting in the system (14) the corresponding populations ${X}_{ni}$.
- Obtain the model solutions by fixing the characterization parameters and varying the control parameters of ${C}_{2}$.
- Calculate the entropy production of each solution and classify it.
- Compare the obtained results with steps (1–3) for different knowledge-transfer methods.

#### 4.2. Entropic Threshold

#### 4.3. Comparison of Knowledge Transfer Methods

#### 4.4. D-D

#### 4.5. D-P

#### 4.6. P-P

#### 4.7. P-D

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Nielsen, S.N.; Fath, B.D.; Bastianoni, S.; Marques, J.C.; Müller, F.; Patten, B.C.; Ulanowicz, R.E.; Tiezzi, E.; Jorgensen S, E. A New Ecology: Systems Perspective, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 2020. [Google Scholar]
- Fiscus, D.A.; Fath, B.D. Foundations for Sustainability: A Coherent Framework of Life-Environment Relations; Academic Press: London, UK, 2019. [Google Scholar]
- Motesharrei, S.; Rivas, J.; Kalnay, E.; Asrar, G.R.; Busalacchi, A.J.; Cahalan, R.F.; Cane, M.A.; Colwell, R.R.; Feng, K.; Franklin, R.S.; et al. Modeling sustainability: Population, inequality, consumption, and bidirectional coupling of the Earth and Human Systems. Natl. Sci. Rev.
**2016**, 3, 470–494. [Google Scholar] [CrossRef] [PubMed][Green Version] - Motesharrei, S.; Rivas, J.; Kalnay, E. Human and nature dynamics (HANDY): Modeling inequality and use of resources in the collapse or sustainability of societies. Ecol. Econ.
**2014**, 101, 90–102. [Google Scholar] [CrossRef][Green Version] - Fu, B.; Li, Y. Bidirectional coupling between the Earth and human systems is essential for modeling sustainability. Natl. Sci. Rev.
**2016**, 3, 397–398. [Google Scholar] [CrossRef][Green Version] - Henderson, K.; Loreau, M. How ecological feedbacks between human population and land cover influence sustainability. PLoS Comput. Biol.
**2018**, 14, e1006389. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mayer, A.L.; Donovan, R.P.; Pawlowski, C.W. Information and entropy theory for the sustainability of coupled human and natural systems. Ecol. Soc.
**2014**, 19, 11. [Google Scholar] [CrossRef] - Cardinale, B.J.; Duffy, J.E.; Gonzalez, A.; Hooper, D.U.; Perrings, C.; Venail, P.; Narwani, A.; Mace, G.M.; Tilman, D.; Wardle, D.A.; et al. Biodiversity loss and its impact on humanity. Nature
**2012**, 486, 59. [Google Scholar] [CrossRef][Green Version] - McGinnis, M.D.; Ostrom, E. Social-ecological system framework: Initial changes and continuing challenges. Ecol. Soc.
**2014**, 19, 30. [Google Scholar] [CrossRef][Green Version] - Horbowy, J. The dynamics of Baltic fish stocks based on a multispecies stock production model. J. Appl. Ichthyol.
**2005**, 21, 198–204. [Google Scholar] [CrossRef] - Niiranen, S.; Blenckner, T.; Hjerne, O.; Tomczak, M.T. Uncertainties in a Baltic Sea Food-Web Model Reveal Challenges for Future Projections. AMBIO
**2012**, 41, 613–625. [Google Scholar] [CrossRef][Green Version] - Ferraro, P.J.; Sanchirico, J.N.; Smith, M.D. Causal inference in coupled human and natural systems. Proc. Natl. Acad. Sci. USA
**2019**, 116, 5311–5318. [Google Scholar] [CrossRef] - Asokan, V.A.; Yarime, M.; Onuki, M. A review of data-intensive approaches for sustainability: Methodology, epistemology, normativity, and ontology. Sustain. Sci.
**2020**, 156, 955–974. [Google Scholar] [CrossRef] - Nicolas, C.; Kim, J.; Chi, S. Quantifying the dynamic effects of smart city development enablers using structural equation modeling. Sustain. Cities Soc.
**2020**, 53, 101916. [Google Scholar] [CrossRef] - Osorio, L.A.R.; Lobato, M.O.; Castillo, X.A.D. An epistemology for sustainability science: A proposal for the study of the health/disease phenomenon. Int. J. Sustain. Dev. World Ecol.
**2009**, 16, 48–60. [Google Scholar] [CrossRef] - Liu, J.; Dietz, T.; Carpenter, S.R.; Alberti, M.; Folke, C.; Moran, E.; Pell, A.N.; Deadman, P.; Kratz, T.; Lubchenco, J.; et al. Complexity of coupled human and natural systems. Science
**2007**, 317, 1513–1516. [Google Scholar] [CrossRef] [PubMed][Green Version] - Diamond, J. Collapse: How Societies Choose to Fail or Succeed; Penguin: New York, NY, USA, 2005. [Google Scholar]
- Weiss, H.; Bradley, R.S. What drives societal collapse? Science
**2001**, 291, 609–610. [Google Scholar] [CrossRef][Green Version] - Brauer, F.; Castillo-Chavez, C. Mathematical Models in Population Biology and Epidemiology; Springer: New York, NY, USA, 2010; Volume 40, p. 416. [Google Scholar]
- Strogatz, S. Nonlinear Dynamics and Chaos; Cornell University MAE: Ithaca, NY, USA, 2014. [Google Scholar]
- Haveman, S.P.; Bonnema, G.M.; van den Berg, F.G.B. Early Insight in Systems Design through Modeling and Simulation. Environ. Model. Softw.
**2014**, 28, 171–178. [Google Scholar] [CrossRef][Green Version] - Wonham, M.J.; de Camino-Beck, T.; Lewis, M.A. An epidemiological model for West Nile virus: Invasion analysis and control applications. Proc. R. Soc. Lond. Ser. B Biol. Sci.
**2004**, 271, 501–507. [Google Scholar] [CrossRef][Green Version] - Onyejekwe, O.O.; Tigabie, A.; Ambachew, B.; Alemu, A. Application of Optimal Control to the Epidemiology of Dengue Fever Transmission. J. Appl. Math. Phys.
**2019**, 7, 148–165. [Google Scholar] [CrossRef][Green Version] - Santamaría-Holek, I.; Castaño, V. Possible fates of the spread of SARS-CoV-2 in the Mexican context. R. Soc. Open Sci.
**2020**, 7, 200886. [Google Scholar] [CrossRef] - Acuña-Zegarra, M.A.; Santana-Cibrian, M.; Velasco-Hernandez, J.X. Modeling behavioral change and COVID-19 containment in Mexico: A trade-off between lockdown and compliance. Math. Biosci.
**2020**, 325, 108370. [Google Scholar] [CrossRef] - Njeuhmeli, E.; Schnure, M.; Vazzano, A.; Gold, E.; Stegman, P.; Kripke, K.; Tchuenche, M.; Bollinger, L.; Forsythe, S.; Hankins, C. Using mathematical modeling to inform health policy: A case study from voluntary medical male circumcision scale-up in eastern and southern Africa and proposed framework for success. PLoS ONE
**2019**, 14, e0213605. [Google Scholar] - Dever, G.A. An epidemiological model for health policy analysis. Soc. Indic. Res.
**1976**, 2, 453–466. [Google Scholar] [CrossRef] - Nokes, D.; Anderson, R. The use of mathematical models in the epidemiological study of infectious diseases and in the design of mass immunization programmes. Epidemiol. Infect.
**1988**, 101, 1–20. [Google Scholar] [CrossRef] [PubMed] - Kondepudi, D.; Prigogine, I. Modern Thermodynamics: From Heat Engines to Dissipative Structures: Second Edition; Wiley: Hoboken, NJ, USA, 2014; pp. 1–518. [Google Scholar]
- Banitz, T.; Schlüter, M.; Lindkvist, E.; Radosavljevic, S.; Johansson, L.-G.; Ylikoski, P.; Martínez-Peña, R.; Grimm, V. Model-derived causal explanations are inherently constrained by hidden assumptions and context: The example of Baltic cod dynamics. Environ. Model. Softw.
**2022**, 156, 105489. [Google Scholar] [CrossRef] - Beltrami, E.J. Mathematics for Dynamic Modeling; Academic Press: Cambridge, MA, USA, 1987; p. 277. [Google Scholar]
- Mendoza, C.I. Inhomogeneous Transmission and Asynchronic Mixing in the Spread of COVID-19 Epidemics. Front. Phys.
**2021**, 9, 683364. [Google Scholar] [CrossRef] - Santamaría-Holek, I. Termodinámica Moderna: Teoría de no Equilibrio con Enfoque Multidisciplinario; Trillas: México, Mexico, 2014. [Google Scholar]

**Figure 1.**Schematic representation of the structure of the social-ecological model. Two epistemological communities ${C}^{\left(1\right)}$ and ${C}^{\left(2\right)}$ act on a single ecosystem to exploit the resource $\mathcal{R}$. The interaction of the communities and the resource produces an external dynamic, consisting of the population changes of every subsystem involved (${C}^{\left(1\right)},{C}^{\left(2\right)},\dots ,R$). This is accounted for by the time behavior of the internal structure of each community i composed by actors and sectors with populations ${S}_{a\beta}$ with $a=1,2,\dots ,n$, $\beta =1,2,\dots ,r$ that reflect several degrees and types of knowledge. The internal dynamics consist of the flux of populations among different knowledge sectors. The two epistemological communities may also interact between them. The figure is own creation.

**Figure 2.**(

**a**) Direct (D) and (

**b**) phase-in (P) knowledge-transfer methods. The solid (red) arrows indicate the knowledge flow transfer whereas the dashed (blue) arrows indicate the population flow between the distinct sectors. N is the population of new individuals, A the population of individuals acquiring technical knowledge, E the population of individuals experimenting with the technical knowledge and P the population of individuals able to produce or extract the natural resource, see Table 1. The figure is own creation.

**Figure 3.**Illustration of the different nature of possible solutions in terms of the population dynamics (upper panels) and the corresponding social-ecological entropy production (lower panels). (

**a**) Sustainable: The post-intervention behavior of the entropy production reaches a new periodic behavior with an entropy production inferior to the threshold value, indicated by the horizontal red-dashed line. (

**b**) Exhaust: The population reaches a time-independent value that makes the entropy production null. (

**c**) Catastrophic: The post-intervention behavior leads to an entropy production behavior with larger values than the entropic threshold. The populations take shallow values that are non-compatible, with minimal survival populations. The figure is the authors’ own creation.

**Figure 4.**Four different parametric planes obtained after assuming different knowledge-transfer methods. Blue points correspond to sustainable situations, yellow points to non-sustainable situations by exhaustion and red points indicate non-sustainable catastrophic situations. (

**a**) D-D knowledge-transfer mode, (

**b**) D-P knowledge-transfer mode, (

**c**) P-D knowledge-transfer mode and (

**d**) P-P knowledge-transfer mode. The figure is own creation.

**Figure 5.**Illustration of the different nature of possible solutions in terms of the populations dynamics (upper panels) and the corresponding social-ecological entropy production (lower panels) of the D-P mode of knowledge-transfer method. (

**a**) Sustainable: The post-intervention dynamics has low entropy production rate after the intervention by the second community making this scenario a very resilient one. (

**b**) Sustainable: The post-intervention dynamics is still sustainable but it has a higher entropy production rate. Therefore, it is less resilient after the intervention by the second community than in case (

**a**). (

**c**) Unsustainable dynamics with catastrophic fate. After few oscillations after the intervention the dynamics collapses, surpassing the entropy production rate threshold. See the main text for a detailed discussion of the results. The figure is the authors’ own creation.

**Figure 6.**Illustration of the different nature of possible solutions in terms of the populations dynamics (upper panels) and the corresponding social-ecological entropy production (lower panels) of the P-D mode of knowledge-transfer method. (

**a**) Sustainable: The post-intervention dynamics has low entropy production rate after the intervention by the second community. The dynamics tends to an asymptotic behavior, making this scenario a very resilient one. (

**b**) Sustainable: The post-intervention dynamics is still sustainable but entropy production rate tends to increase with time after the intervention by the second community, indicating a fragile post-intervention behavior. (

**c**) Sustainable: The dynamics is sustainable but shows chaotic oscillations after the intervention. This reduces the capability of predicting future events. See the main text for a detailed discussion of the results. The figure is own creation.

**Figure 7.**Illustration of the different nature of possible solutions in terms of the populations dynamics (upper panels) and the corresponding social-ecological entropy production (lower panels) of the P-P mode of knowledge-transfer method. (

**a**) Sustainable: The post-intervention dynamics has peaks with high entropy production rate after the intervention. The SES can be considered sustainable but fragile, since an unexpected perturbation may induce the dynamics to cross the entropy production rate threshold. (

**b**) Sustainable: The post-intervention dynamics shows an irregular dynamics which is more manifest through the entropy production rate. The long-time behavior seems to decrease the entropy production maintaining the SES with an irregular but sustainable dynamics. (

**c**) Catastrophic: After the intervention, the entropy production rate shows oscillations with increasing amplitude that eventually cross the entropy production rate threshold, indicating that the SES unsustainable. See the main text for a detailed discussion of the results. The figure is own creation.

**Figure 8.**Illustration of the change of the parametric plane in three different times. (

**a**) Corresponds to a maximal time of ${t}_{max}=3000$ time units, (

**b**) ${t}_{max}=5000$ time units and (

**c**) ${t}_{max}=10000$ time units. The figure is own creation.

N | Sector of new individuals in the community or that does not have any relevant knowledge in order to exploit the resource. |

A | Sector of individuals acquiring technical knowledge or learning how to manipulate a resource. |

E | Sector of individuals experimenting with the technical knowledge, that is, acquiring environmental knowledge by interacting with the surroundings or ecosystem. |

P | Sector of individuals able to produce, extract or exploit a resource. |

$\mathit{\gamma}$ | Quality-of-inclusion rate of one sector into another one. |

$\omega $ | Amount of technical knowledge for extraction. |

$\lambda $ | Amount of environmental knowledge for restoration of the resources. |

${\u03f5}_{\alpha}$ | Knowledge-transfer rate of knowledge type $\alpha =\omega ,\lambda $. |

C_{1} | C_{2} |
---|---|

Characterization | |

${\rho}_{1}\to 0.45$ | ${\rho}_{2}\to 0.5$ |

${\gamma}_{1}\to 0.4$ | ${\gamma}_{2}\to 0.4$ |

${\omega}_{1}\to 0.3$ | ${\omega}_{2}\to 0.6$ |

${\lambda}_{1}\to 0.6$ | ${\lambda}_{2}\to 0.3$ |

${\mu}_{1}\to 0.01$ | ${\mu}_{2}\to 0.01$ |

${\kappa}_{1}\to 0.8$ | ${\kappa}_{2}\to 0.8$ |

Control | |

${\u03f5}_{\omega 1}\to 0.3$ | ${\u03f5}_{\omega 2}\to \alpha $ |

${\u03f5}_{\lambda 1}\to 0.3$ | ${\u03f5}_{\lambda 2}\to \beta $ |

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**MDPI and ACS Style**

Michel-Mata, S.; Gómez-Salazar, M.; Castaño, V.; Santamaría-Holek, I.
Towards a Social-Ecological-Entropy Perspective of Sustainable Exploitation of Natural Resources. *Foundations* **2022**, *2*, 999-1021.
https://doi.org/10.3390/foundations2040067

**AMA Style**

Michel-Mata S, Gómez-Salazar M, Castaño V, Santamaría-Holek I.
Towards a Social-Ecological-Entropy Perspective of Sustainable Exploitation of Natural Resources. *Foundations*. 2022; 2(4):999-1021.
https://doi.org/10.3390/foundations2040067

**Chicago/Turabian Style**

Michel-Mata, Sebastián, Mónica Gómez-Salazar, Víctor Castaño, and Iván Santamaría-Holek.
2022. "Towards a Social-Ecological-Entropy Perspective of Sustainable Exploitation of Natural Resources" *Foundations* 2, no. 4: 999-1021.
https://doi.org/10.3390/foundations2040067