Determining the fundamental dynamics of global sustainability in the Anthropocene from a vulnerability perspective

Abstract

This paper conducts a detailed quantitative study of the indicated long-term levels, nature and dynamics of sustainability at the global spatial scale over time, within the context of the vulnerability of the environment-human relationship and system. In this study, the Sustainability Dynamics Framework is applied to the results of the Notre Dame Global Adaptation Initiative index for the period 1995–2018. The study indicates that the obtained S-values at the global spatial scale throughout the specified period were consistent with very weak unsustainability occurring. Through statistical analysis, the study indicates that the triumvirate of influencing factors (Population, GDP, Greenhouse Gases) have a significant influence and impact individually and cumulatively upon obtained S-values. The study then determines the net change in obtained E, HNI and S-values, and consequently evaluates the dynamics of the rate of change of the Earth System (dE/dt), as defined in the Anthropocene Equation theory. To support this, Fourier analysis of net change of S-values is applied for the first time within the SDF application methodology, using eight time periods of 16 years’ duration over the period 1996–2018. The analysis indicates continuous overlapping cycles of 7 years’ duration, consisting of a lag time of 3 years leading to an impact period of 4 years. The Fourier analysis potentially indicates clearly for the first-time humanity’s footprint signal upon the environment-human relationship and system in the Anthropocene. The study concludes that humanity’s actions and inactions have resulted in the indicated unsustainability and vulnerability of the environment-human relationship and system. As a consequence, there is a potential realistic risk of systemic collapse unless fundamental changes occur in respect to the global environment-human relationship.

Introduction

Overview

The Earth System consists of the two fundamental planetary spheres of the environment/ecosphere and the anthroposphere (human society, activities and needs). These operates as a complex coupled dynamic system and relationship at all spatial-temporal scales. This simply means that the environment and the anthroposphere at all spatial-temporal scales interacts, influences and impacts within themselves and upon one another to differing degrees of significance, intensity and magnitude.

The Earth System is currently in the geological epoch of the Anthropocene (Crutzen, 2002; Crutzen and Stoermer, 2000; Steffen et al., 2007; Waters et al., 2016; Zalasiewicz et al., 2008). This epoch reflects the fundamental impacts and influences that the anthroposphere at all spatial-temporal scales have had, and are having, upon the operations and processes of the environment/ecosphere at all spatial-temporal scales.

The nature and operation of the Earth System, both conceptually and in reality, is a mainstream theme of current research and literature. Therefore, in order to have a fundamental understanding of the Earth System, it is necessary and appropriate to have an understanding of what is meant by the environment/ecosphere and the anthroposphere.

Defining the environment

The components of the environment (aka the ecosphere) are generally agreed as consisting of the primary planetary sub-spheres of the atmosphere, biosphere, hydrosphere, ¹ and the lithosphere (e.g. Boumans et al., 2002; Gaffney and Steffen, 2017; Lovelock, 1972, 1988, 2009, 2016; Lovelock and Margulis, 1974; NASA, 1986; Phillips, 2010, 2020; Rockström et al., 2009; Rotmans et al., 1990; Schellnhuber, 1998, 1999, 2001; Smith et al., 2020). Within these four primary planetary sub-spheres are sub-systems of various levels of complexity—for example, plate tectonics, hydrological cycle, glaciers; climatic zones, erosion and weathering, carbon cycle, nitrogen, cycle, soil formation, ecological niches etc. Critically, all of the four primary sub-spheres and their component sub-systems are interconnected and interdependent upon each other for their operation and well-being. Disruptions, degradation and/or destruction within a sub-system can and often results in impacts and feedbacks upon the other environmental sub-systems and reciprocal impacts upon the anthroposphere.

Defining the Anthroposphere

The anthroposphere can be defined as the part of the Earth System which has been made or modified by humans (Kuhn and Heckelei, 2010). Therefore, it fundamentally represented all of human society, activities and needs at all spatial-temporal scales. As a result, the anthroposphere is multi-faceted reflecting the different aspects and stages of development of humans over time. However, the anthroposphere tends to be represented within the literature by a specific component or combination of components in defining and evaluating human-based activities and/or behaviours, and their influence and impacts. This is dependent upon whether a reductionist or holistic perspective is adopted. It should also be noted that unlike in the case of other species, humans tend to be more individualistic in nature and character, as noted for example by Quilley (2011), Sturm (2020), Bossel (1996), Morrow (2017), Gumilev (1973) and Muntean (2020). This is influenced by their upbringing, life experiences, societal norms and characteristics etc., and based upon the human characteristics of, for example: free will; desire to progress and thrive; the ability to innovate, develop and advance; and the ability to communicate and choose through rational and sophisticated means. It is therefore commonplace within the literature for the anthroposphere to be conceived, assessed, or modelled in respect to specific activities, behaviours and/or needs.

The anthroposphere can consequently be considered as any and all aspects of human society, activities and needs at any specified spatial-temporal scale, including its outcomes, influences and impacts upon itself or on the environment. The components of the anthroposphere would therefore include: population; economic activities and needs; resource extraction & use; production & consumption (societal and economic); politics & governance; equity and equality; energy production and consumption; pollution; housing; welfare; degradation and destruction (environmental and anthropospheric); culture & heritage; employment; science and technology; poverty; infrastructure; transportation; norms, behaviours and attitudes (individual and collective); waste generation, storage and recycling; agriculture; war and conflicts; education; law and justice; peace; quality of life; health and well-being (physical, mental, emotional and spiritual) etc. (e.g. Alcamo, 1994; Boumans et al., 2002; Donges et al., 2017a, 2017b; Forrester, 1971; Gaffney and Steffen, 2017; Meadows et al., 1972, 1992, 2004; Pearce and Atkinson, 1998; Phillips, 2010, 2020; Schellnhuber, 1998, 1999, 2001; United Nations, 2015; World Commission on Environment and Development [WCED], 1987).

Defining the earth system

The Earth System has been referred to within the literature by the following alternate terms: (i). ‘coupled human-environment systems’ (CHES) or ‘coupled human and natural system(s)’ (CHANS) (e.g. An et al., 2014; Ferraro et al., 2019; Liu et al., 2007; Wang et al., 2018); (ii). ‘socio-ecological system(s)’ (SES) (e.g. Banos-González et al., 2016; Domptail and Easdale, 2013; Gotts et al., 2019; Martínez-Fernández et al., 2021; Ostrom, 2009); (iii). ‘human-environment systems’ (e.g. Bauch et al., 2016; Manson, 2008; Mehren et al., 2018; Stokols et al., 2013; Turner et al., 2003; or (iv). ‘environment-human system(s)’ (e.g. Gechter and Fass, 2018; Kassaye et al., 2022; Phillips, 2020; Verstraete et al., 2011; Vladimirovna-Bakaeva et al., 2017). Irrespective of how the Earth System may be referred to, it is fundamentally the coupled dynamic system and relationship between the environment and the anthroposphere.

The fundamental definitions of the Earth System and its component sub-systems outlined are extremely abstract in nature, and which does belie its true complexities as a coupled dynamic system and the co-evolutionary relationship between the environment and the anthroposphere. Defining the dynamics of the Earth System in order to understand the interactions and relationship between the environment and the anthroposphere has been a major theme within the literature, particularly in respect to the Anthropocene. At the most fundamental level, both Schellnhuber (1998) and Gaffney and Steffen (2017) have sought to mathematically define the changing and current dynamics of the Earth System over time. In respect to Schellnhuber (1998) and his work on Earth System Analysis theory, he describes the present-day dynamic relationship as follows:

$$\stackrel{.}{\mathbf{N}}=F_2\left(\mathbf{N},\mathbf{A};\mathbf{t};\mathbf{M}\left(\mathrm{t}\right)\right),$$

$$\stackrel{.}{\mathbf{A}}=G_2\left(\mathbf{N},\mathbf{A};\mathbf{M}(t)\right)$$

(1)

Glossary: N - Ecosphere; A - Anthroposphere; t - Time variable (t = 0); F₁, G₁ – Function

(Schellnhuber, 1998, p.40)

The ecosphere (N) (aka the environment) and the anthroposphere (A) are in a strongly coupled relationship, with A acting as the same scale as N, and therefore has become a significant driving force in the co-evolutionary development of the Earth System (Schellnhuber, 1998). This simply means that humanity and its activities (A) are equivalent or potentially greater in force and impact than N, so that it is a dominant influence of change within the Earth System. Due to humanity’s realization of its influence and impacts upon the Earth System, it has consciously developed M(t), which is the chosen management strategy(ies) adopted to develop a co-evolutionary N-A relationship at all spatial-temporal scale – in other words, sustainability. Fan et al. (2021) influenced by Schellnhuber’s work on Earth System Analysis has applied it using statistical physics as an approach to develop new insights and perspectives concerning the complex nature of Earth System dynamics.

Gaffney and Steffen (2017), also influenced by Schellnhuber’s work, were more direct in expressing the dynamic relationship of the Earth System in the Anthropocene, which they stated as follows:

• (i). The rate of change of the Earth System (dE/dt) before and upto the Anthropocene, (E) was:

$$\frac{dE}{dt}=f\left(A,G,I,H\right)$$

(2)

• (ii). dE/dt since the beginning of the Anthropocene has been:

$$\frac{dE}{dt}=\underset{A,G,I\to 0}{f(H)}$$

(3)

Where H, based on Hibbard et al. (2006) can be defined as:

$$H=f\left(P,C,T\right)$$

(4)

and

$$T=f\left(En,K,Pe\right)$$

(5)

Glossary:  E - Earth System; A – Astronomical forcing; G – Geophysical forcing;

I – Internal dynamics of the Earth system; H - Human activity;

P - Population; C - Consumption; T – Technology;

En - Energy; K - Knowledge; Pe - Political Economy

(Gaffney and Steffen, 2017, pp.54–55,57–58)

Gaffney and Steffen (2017) stated that in respect to equation (3) which can be termed as the ‘Anthropocene Equation’, H has become the dominant driving force in the rate of change of the Earth System (dE/dt), succeeding the natural forces of A, G, I as shown in equation (2). The relevance and importance of these equations of Schellnhuber (1998) and Gaffney and Steffen (2017) shall become clear later in the paper, in respect to the Sustainability Dynamics Framework (re: Phillips, 2020).

Another common approach in Earth System conceptualization is the use of adaptive system theory, typically in respect to the use of cycles and loops to illustrate and demonstrate the operational processes, influences, impacts and feedbacks occurring (e.g. Abram and Dyke, 2018; Donges et al., 2017a, 2021a, 2021b; Holling, 2001; Wang et al., 2018). A simple example of a feedback loop between two Earth System subsystems was provided by Donges et al. (2021a) as A → B and B → A, and which can be consequently described as A ↻ B. A more complex interaction loop between subsystems was exemplified by Donges et al. (2021a) as A → B → C → A, which results in 11 possible feedback loops occurring. These two examples are relatively simplistic in nature, but do adequately illustrate the basic principles of adaptive system theory in relation to the conceptualization and evaluation of the Earth System. And yet, it is not only the conceptualization of the nature and dynamics of the Earth System which is a main theme of the current literature. It is also how the current nature and dynamics of the Earth System influences, impacts and assessed in respect to vulnerability and sustainability.

Evaluating vulnerability

Vulnerability was defined by Turner et al. (2003) as the degree to which the environment–human system is harmed due to its exposure to hazard(s). Turner (2010) states there is a synergy between vulnerability and sustainability, climate change and global environmental change. This is because vulnerability reflects the sensitivity and coping capabilities of either the environment or anthroposphere, and consequently the environment-human system as a whole (Turner, 2010). The literature on vulnerability of environment-human system at a specified spatial-temporal scale is abundant in breadth and scope of approach. Some examples include: the spatial distribution of vulnerability in Liupan mountain region, China (Han et al., 2020); index of social system’s vulnerability to ecosystem services changes at the local scale (Berrouet et al., 2019); country-based index assessment of social-ecological status (Estoque and Murayama, 2017); and vulnerability assessment of ecosystem services to changes in land use (Metzger et al., 2006).

Conceptualizing and evaluating sustainability

In respect to sustainability, then it can be stated the potential issues are more challenging, as there is a multitude of definitions, conceptualizations and applications within the current literature. These are dependent upon whether a reductionist or holistic perspective is adopted, and whether it is qualitative (i.e. word-based/indicators), or quantitative (i.e. mathematical/numerical) in nature. Therefore, this potentially gives four basic categories of how sustainability can be conceptualized and assessed – Reductionist-Qualitative; Reductionist-Quantitative; Holistic-Qualitative; and Holistic-Quantitative. With this in mind, Table 1 shows some examples of the current literature using the four categories stated.

Table 1.

Some categorized examples of the current literature on sustainability concepts and applications.

Category	Examples – author(s) and topic(s)
Reductionist–Qualitative	Goodland and Daly (1996) – Defining environmental sustainability Guzzo et al. (2022) – Systems-based approach to examine Circular Economy van Griethuysen (2002) – Evolutionary economic approach Stark and Lindow (2017) – Dynamic model to examine Circular Economy
Reductionist–Quantitative	Pearce and Atkinson (1998) – Weak and Strong Sustainability using capital theory approach Chichilnisky (1997) – Definition & concept from environmental economics perspective Krysiak (2006) – Model of economic production & consumption and weak sustainability Svirezhev and Svirejeva-Hopkins (1998) – Concept of sustainable biosphere Ozcan and Bozoklu (2021) – Sustainability of the per capita ecological balance of 27 OECD countries Somogyi (2016) – Quantitative framework for environmental sustainability
Holistic–Qualitative	United Nations (2015) – Sustainable Development Goals (SDGs) Manderson (2006) – Framework to examine multi-contextual applications of sustainability Reed et al. (2022) – Multi-sector dynamics of human and Earth systems Hay et al. (2014) – Sustainability cycle and loop models Turnbull et al. (2021) - Empirical model to conceptualize sustainability
Holistic–Quantitative	Neuman and Churchill (2011) – General process model using first and second thermodynamics laws Ursino (2019) – Dynamic models of socio-ecological system and sustainability outcomes van Den Bergh and Nijkamp (1991) – Dynamic ecological-economic model Wenisch et al. (2011) – Multi-dimensional approach using sustainability logic Dragicevic (2020) – Concentric framework for sustainability assessment Phillips (2010) – Mathematical Model of Sustainability (MMS) Phillips (2020) – Sustainability Dynamics Framework (SDF)

In respect to Table 1, two of the stated concepts are relevant in respect to the later application of the Sustainability Dynamics Framework (SDF) – these are the Rules of Weak and Strong Sustainability (Pearce and Atkinson, 1998) and the Mathematical Model of Sustainability (MMS) (Phillips, 2010, 2015, 2016). In respect to the MMS, this will be addressed in more detail in the Methodology section in specific regard to the operationalization of the SDF. In relation to the Rules of Weak and Strong Sustainability (RWSS) developed by Pearce and Atkinson (1998), these are a set of fundamental mathematical rules to define weak and strong sustainability using the capital theory approach, which originated from the economics-based disciplines.

Total capital (K) consists of two fundamental types: (i). natural; and (ii). anthropogenic. Natural capital (K_N) are resources which are either renewable or non-renewable in nature, and which reflects the goods and services produced and consumed, and their by-products necessary to support human life. Anthropogenic capital (K_A) was a term developed by Phillips (2020) in developing the Sustainability Dynamics Framework, which assimilates and consists of the three human-based capital components defined by Pearce and Atkinson (1998) – human (K_H – i.e. labour), manufactured (K_M – i.e. tools, machines, buildings, infrastructure etc.), and social (K_S – i.e. education, culture and knowledge etc.).

At the highest level of abstraction, the level of flow from natural capital to anthropogenic capital (K_N → K_A), in the form of resources and services, consequently determines whether weak or strong sustainability occurs. The mathematical rules of weak and strong sustainability, based on the flow of capital is shown in Figure 1.

Figure 1.

The rules of weak and strong sustainability, after Pearce and Atkinson (1998).

Indicators and modelling

In respect to the assessment of vulnerability and sustainability, as well as the Earth System, the most common approaches used are indicators and/or modelling. In relation to Earth System, there is an abundance of models within the literature from general models to specific systemic models. Some examples of Earth System models include: (i) the Earth3 model (Goluke et al., 2018; Randers et al., 2018, 2019), which evaluates potential future scenario-based outcomes of the implementation of the UN Sustainable Development Goals in respect to the Planetary Boundaries framework (Rockström et al., 2009; United Nations, 2015); (ii) The World3 model (Herrington, 2021; Meadows et al., 1972, 1992, 2004; Pasqualino et al., 2015), which is the basis for the Limits to Growth model; (iii) Integrated modelling of the dynamics of ecosystem services, for example, Multiscale Integrated Model of Ecosystem Services (MIMES) (Boumans and Costanza, 2007; Boumans et al., 2015; Oliveira et al., 2022), and the Global Unified Metamodel of the Biosphere (GUMBO) model (Boumans et al., 2002); (iv) the Human and Nature Dynamics (HANDY) model (Motesharrei et al., 2014); and (v) the Analysis, Integration and Modelling of the Earth System (AIMES) approach (Schimel et al., 2015).

In respect to indicators, the Earth System, vulnerability and sustainability have a plethora of indicator-based frameworks available or being developed. Some of the more well-known examples in respect to indicator-based frameworks include: (i) the UN Sustainable Development Goals (SDGs) United Nations, 2015); (ii) Planetary Boundaries (Dearing et al., 2014; Häyhä et al., 2016; Rockström et al., 2009); (iii) the Notre Dame Global Adaptation Initiative (ND-GAIN) index (Chen et al., 2015; University of Notre Dame, 2021); (iv) The Environmental Vulnerability Index (EVI) (South Pacific Applied Geoscience Commission [SOPAC], 2005). Some other examples include: Cochran et al. (2020), Kavvada et al. (2020) and Lehmann et al. (2022), all of which use an earth observations approach for reporting the success in implementing the SDGs; Usubiaga-Liaño and Ekins (2021) who developed a strong environmental sustainability index (SESI) for countries; and Wu et al. (2021) who proposed a planetary boundaries-based approach for evaluating the environmental footprints.

Context and purpose of paper

What the brief discourse of the conceptualization, application and assessment of the Earth System, vulnerability and sustainability has briefly indicated is that there is currently no singular approach which can potentially represent and evaluate the fundamental level(s), nature and dynamics of the Earth System and the co-evolutionary relationship (aka sustainability / unsustainability) at any specified spatial-temporal scale. Furthermore, how and why the vulnerability of the Earth System and co-evolutionary relationship is potentially influenced and impacted upon over time. The use of separate approaches to the question of the outcomes in respect to the Earth System, vulnerability and sustainability has potential significant consequences for all. Specifically, separate bespoke approaches can lead to disjointed understanding, policy and management occurring, as there is a failure to account for the integrated and complex nature of the environment-human system and relationship at any specified spatial-temporal scale. Therefore, whilst it is generally accepted within the literature that the Earth System, vulnerability and sustainability are dynamically integrated; there is however very little within the literature which directly determines and evaluates all of these adopting a consistent and holistic quantitative approach. It is for the reasons stated that this paper has been developed to potentially contribute towards advancing knowledge on the fundamental and critical questions and issues outlined in respect to the global spatial scale.

Therefore, this paper determines and evaluates the indicated long-term levels, nature and dynamics of sustainability at the global spatial scale. This is contextualized in respect to the vulnerability of the environment-human system. Consequently, the objective of this study to provide a detailed understanding of the indicated long-term nature of the global environment-human relationship, and its potential and actual impacts and consequences. The intended outcome of the study is to advance understanding of the indicated levels, nature and dynamics of the global environment-human system and relationship in the Anthropocene, in respect to long-term sustainability and vulnerability. Furthermore, it is the study’s intention to further advance and contribute towards the development of a conducive and meaningful global co-evolutionary environment-human relationship and system.

By the approach adopted in this study, it is intended to further contribute towards the fulfilment of the observation of Phillips (2010, p127) that: ‘The crux of sustainable development is the understanding of the fundamental dynamic relationships between the environment and human’. Therefore, how humanity influences and impacts upon the environment at the global spatial scale, consequently determines the nature and potential outcomes of the global environment-human relationship over time.

Methodology

Overview

This study adopts a revised methodology based on Phillips (2022), in order to apply the Sustainability Dynamics Framework (SDF) (Figures 2–4; re: Phillips, 2020). The SDF is a holistic conceptual and operational framework which evaluates two fundamental aspects of sustainability at any specified spatial-temporal scale(s). These are: (i). the obtained indicated level(s) and nature of sustainability (S) occurring within a defined S-value range of [−1.000 = S = +1.000]; and (ii). the potential influences and interactions occurring upon sustainability (based on Phillips, 2020, 2022). As Phillips (2020) states, the SDF achieves this by mathematically integrating and expanding upon four environment-human system and relationship theories - These are: (i). Rules of Weak and Strong Sustainability (RWSS) (Pearce and Atkinson, 1998); (ii). Earth System Analysis (ESA) (Schellnhuber, 1998); (iii). the Anthropocene Equation (AE) (Gaffney and Steffen, 2017); and (iv). the Mathematical Model of Sustainability (MMS) (Phillips, 2010, 2015, 2016).

Figure 2.

Overview of the finalized sustainability dynamics framework (SDF), after Phillips (2020): (a) shows the dynamics of sustainability and unsustainability; and (b) shows the operational dynamic boundaries of sustainability and unsustainability.

Figure 3.

A revised simple conceptualization of the fundamental dynamic relationships of sustainability and unsustainability, as described by the SDF, in terms of E, HNI and S and dE/dt (after and adapting Phillips, 2020).

Figure 4.

The conceptualization of the key fundamental aspects of the Sustainability Dynamic Framework (SDF), after and adapting Phillips (2020): (a) describes the key dynamic definitions, principles and impacts; (b) describes the key dynamics of rate of change.

In order to apply the SDF and its methodology, a suitable quantitative-based assessment is required. In respect to this study, the results of the Notre Dame Global Adaptation Initiative (ND-GAIN) index (University of Notre Dame, 2021) are used.

The ND-GAIN index is an assessment framework which evaluates the vulnerability and readiness of countries to climate change. The index uses 45 core indicators to evaluate the vulnerability and readiness of 192 UN countries, evaluated annually over the period 1995–2018 (Chen et al., 2015; University of Notre Dame, 2021). Vulnerability indicators are evaluated on a 0–1 scale representing high-low vulnerability, and readiness indicators are evaluated on a 0–1 scale representing low-high readiness (Chen et al., 2015). The data used for the indicators were required to meet the following criteria, as stated by Chen et al. (2015: 5): (1). Available for a high proportion of United Nations countries; (2). Time-series so that changes and trends in country vulnerability and readiness can be tracked. Indicators with data from 1995 to the present are preferred; (3). Freely accessible to the public; (4). Collected and maintained by reliable and authoritative organizations that carry out quality checks on their data; and (5). Are transparent and conceptually clear.

Aggregated category scores for countries are also determined, based upon the obtained scores of the component indicators which are in one of six categories, as shown in Table 2. It is the aggregated vulnerability scores of the six categories which are used to apply the SDF. The aggregate score of each category operates on the same 0–1 value range for vulnerability indicators.

Table 2.

ND-GAIN vulnerability categories and indicators, after and adapting University of Notre Dame (2021). (MMS parameter categorization in brackets, re: Figure 5, Step 2).

Indicator category and MMS parameter	Indicator
Ecosystem services (E)	Projected change of biome distribution
	Projected change of marine biodiversity
	Natural capital dependency
	Ecological footprint
	Protected biome
	Engagement in international environmental conventions
Food (HNI)	Projected change of cereal yields
	Projected population change
	Food import dependency
	Rural population
	Agriculture capacity
	Child malnutrition
Human habitat (HNI)	Projected change of warm periods
	Projected change of flood hazard
	Urban concentration
	Age dependency ratio
	Quality of trade and transport infrastructure
	Paved roads
Health (HNI)	Projected change of deaths from climate change induced diseases
	Projected change in vector-borne diseases
	Dependency on external resource for health services
	Slum population
	Medical staff
	Access to improved sanitation facilities
Infrastructure (HNI)	Projected change of hydropower generation capacity
	Projected change of sea level rise impacts
	Dependency on imported energy
	Population living under 5 m above sea level
	Electricity access
	Disaster preparedness
Water (E)	Projected change of annual runoff
	Projected change of annual groundwater recharge
	Fresh water withdrawal rate
	Water dependency ratio
	Dam capacity
	Access to reliable drinking water

The reasons for using the aggregated category vulnerability scores were two-fold. Firstly, the aggregate scores indicate the cumulative level of vulnerability occurring in respect to each category. Consequently, this reflects the combined impacts of the category indicators in relation to the category overall. Secondly, the aggregated scores provided a suitable and sufficient quantity and quality of data to apply the SDF in a reasonably timely and efficient approach. Specifically, due to the large sample of countries used over the specified period (1995–2018), it was deemed appropriate to use the annual aggregated category scores instead of the individual annual indicator scores, because it provided a rapid but robust quantity and quality of data in order to apply the SDF.

Methodological stages

The study applied a four-stage approach adapting a revised methodology developed in Phillips (2022). The stages and their processes were as follows.

Stage 1 – Data Collation & Organization The data utilized to apply the SDF was sourced from the ND-GAIN website (re:University of Notre Dame, 2021) and the World Bank databank website (re: World Bank, 2020). Both the ND-GAIN and World Bank websites operate on an open-source basis meaning that the data is freely available to all.

In respect to the ND-GAIN data, the aggregate category scores were extracted and placed within a single Microsoft Excel 2019 file. Then the data was examined and countries with missing category scores in any or all years were removed. Therefore, only countries with 100% data in all categories and years were used to eliminate and/or minimize potential biases and errors in the next methodological stages. Consequently, out of a potential 192 UN countries, a total of 139 countries with 100% data in all years formed the global sample, as shown in Table 3.

Table 3.

The data sample of 139 countries used in determining obtained E, HNI and S-values, in alphabetical order.

Country Sample (in alphabetical order)
Albania	Georgia	Oman
Algeria	Germany	Pakistan
Angola	Ghana	Panama
Antigua and Barbuda	Guinea-Bissau	Paraguay
Argentina	Haiti	Peru
Armenia	Honduras	Philippines
Australia	Hungary	Poland
Bahrain	Iceland	Portugal
Bangladesh	India	Qatar
Belarus	Indonesia	Romania
Belgium	Iran, Islamic Republic of	Russian Federation
Benin	Iraq	Samoa
Bolivia, Plurinational State of	Ireland	Saudi Arabia
Bosnia and Herzegovina	Israel	Senegal
Botswana	Italy	Serbia
Brazil	Jamaica	Sierra Leone
Brunei Darussalam	Japan	Singapore
Bulgaria	Jordan	Slovenia
Cambodia	Kazakhstan	South Africa
Cameroon	Kenya	Spain
Canada	Korea, Republic of	Sri Lanka
Cape Verde	Kuwait	Sudan
Chile	Latvia	Suriname
China	Lebanon	Sweden
Colombia	Libyan Arab Jamahiriya	Switzerland
Congo	Lithuania	Syrian Arab Republic
Congo, the Democratic Republic of	Luxembourg	Tanzania, United Republic of
Costa Rica	Madagascar	Thailand
Cote d’Ivoire	Malaysia	Timor-Leste
Croatia	Malta	Togo
Cuba	Mauritania	Trinidad and Tobago
Cyprus	Mauritius	Tunisia
Czech Republic	Mexico	Turkey
Denmark	Moldova, Republic of	Ukraine
Djibouti	Mongolia	United Arab Emirates
Dominican Republic	Montenegro	United Kingdom
Ecuador	Morocco	United States
Egypt	Mozambique	Uruguay
El Salvador	Myanmar	Uzbekistan
Eritrea	Namibia	Venezuela, Bolivarian Republic of
Estonia	Nepal	Viet Nam
Ethiopia	Netherlands	Yemen
Fiji	New Zealand	Zambia
Finland	Nicaragua
France	Nigeria
Gabon	Norway

In regards to the World Bank data, this obtained 11 indicators for the 139 countries over the same specified period of 1995–2018, in order to conduct the statistical analyses in Stage 3. The 11 indicators were organized in three core categories, as shown in Table 4.

Table 4.

The 11 indicators of potential influencing factors upon obtained levels and nature of sustainability at the global spatial scale, based on the triumvirate of potential influencing factors in Phillips (2022).

Population	GDP	Greenhouse gases (GHG)
Population, total	GDP (current US$)	Total greenhouse gas emissions (kt of CO2 equivalent)
Population growth (annual %)	GDP growth (annual %)	Total greenhouse gas emissions (% change from 1990)
Population density (people per sq. km of land area)	GDP per capita (current US$)	CO2 emissions (kt)
	GDP per capita growth (annual %)	CO2 emissions (metric tons per capita)

The 11 indicators from the World Bank database were chosen as potential factors upon the levels, nature and dynamics of sustainability occurring at the global spatial scale. This was based upon the findings of Phillips (2022), which identified a potential triumvirate of negative influencing factors consisting of population, GDP and greenhouse gases (GHG) indicators. In respect to this study, the potential influencing factors within the triumvirate were examined in relation to their potential degree and nature of influence upon sustainability from a vulnerability perspective. Therefore, this potentially will indicate the influence of the triumvirate, individually and cumulatively, upon the nature of the global environment-human system and relationship over a specified long-term period (1995–2018).

Stage 2 – Obtained E, HNI and S-values The first SDF application phase was to determine indicated values for the Environment (E), Human Needs and Interests (HNI) and Sustainability (S) at the global spatial scale for each year over the specified period of 1995–2018. This involved the adaptation and application of the SDF component theory of the Mathematical Model of Sustainability (MMS) (re: Phillips, 2010, 2015, 2016), and which is the means the SDF is operationalized (Phillips, 2020). The adapted MMS application methodology for the ND-GAIN results is shown in Figure 5. This determined indicated E, HNI and S-values and S-levels for all years within the specified period, using Microsoft Excel 2019 to calculate the values. Obtained E and HNI-values occur within a defined range of [0.000 =x = 1.000], and obtained S-values occur within a defined range of [−1.000 = S = +1.000] (Phillips, 2016, 2020). The obtained S-values indicate the level of sustainability occurring at the specified spatial-temporal scale. The S-level indicates the nature of sustainability occurring at the specified spatial-temporal scale, and was determined by using the S-level value ranges in Step 6, Figure 5.

Figure 5.

MMS application methodology to ND-GAIN aggregate category scores, based on and adapting Phillips (2022, 2020).

Stage 3 – Statistical Analysis This stage consisted of two statistical analysis tests – (i). Pearson correlation co-efficient test (two-tailed); and (ii). simple linear multiple regression analysis, both conducted using Microsoft Excel 2019.

The Pearson test evaluated the presence of a statistically significant relationship between obtained S-values and each of the 11 potential influencing factors, as shown in Table 3. Tests were conducted at n = 18, n = 22 and n = 24, dependent upon the available data for each potential influencing factor. Significance (p) was determined with a suitable critical value table at the appropriate degrees of freedom (d.f. = n–2), and entered within the Excel spreadsheet.

The multiple regression analysis was applied in order to indicate the degree of potential association between obtained S-values and the triumvirate categories (Population, GDP and Greenhouse Gases) individually and cumulatively. The results were obtained and tabulated within Microsoft Excel 2019. In the case of the Population and GDP categories, this was conducted for the period 1995–2018. In respect to the Greenhouse Gases (GHG) and the overall triumvirate categories, this was conducted for the period 1995–2012. This was because of missing World Bank indicator data for some of the countries in respect to the GHG category from 2013 onwards, as shown in Table 7. Therefore, to ensure consistency and minimize potential error and bias, the GHG and Triumvirate categories were evaluated for the period 1995–2012.

Stage 4 - Dynamic Analysis The dynamic analysis involved the objective determination and evaluation of the indicated dynamics of global sustainability occurring over the period 1995–2018. The analysis applied the stated dynamic principles and definitions in Figures 2 and 4 to the results obtained in Tables 4 to 11 and Figures 6 and 7. By this approach, the fundamental dynamic interactions and influences upon obtained levels and nature of sustainability occurring at the specified spatial-temporal scale were indicatively determined.

Table 5.

Total aggregate category scores and obtained E-values at the global spatial scale for the period 1995–2018. (All values positive unless indicated as negative).

Year	Ecosystem services		Water		Total	Maximum Total	E−Value
	Total score	Change in total score from previous year	Total score	Change in total score from previous year
1995	64.70792	–	49.5581	–	114.266	278	0.411
1996	64.68522	−0.0227	49.58315	0.025056	114.2684	278	0.411
1997	64.60656	−0.07866	49.61332	0.030166	114.2199	278	0.411
1998	64.47696	−0.12961	49.64058	0.027258	114.1175	278	0.410
1999	64.38328	−0.09368	49.66717	0.02659	114.0504	278	0.410
2000	64.34679	−0.03649	49.69006	0.022897	114.0369	278	0.410
2001	63.72915	−0.61764	49.68213	0.00793	113.4113	278	0.408
2002	63.73678	0.007628	49.66351	−0.01862	113.4003	278	0.408
2003	63.39644	−0.34033	49.65073	−0.01277	113.0472	278	0.407
2004	63.04608	−0.35036	49.61295	−0.03778	112.659	278	0.405
2005	62.67498	−0.3711	49.57217	−0.04078	112.2471	278	0.404
2006	62.67498	0	49.53323	−0.03895	112.2082	278	0.404
2007	62.67498	0	49.4914	−0.04183	112.1664	278	0.403
2008	62.67485	−0.00013	49.43509	−0.05631	112.1099	278	0.403
2009	62.66133	−0.01351	49.37521	−0.05988	112.0365	278	0.403
2010	62.32048	−0.34085	49.31868	−0.05653	111.6392	278	0.402
2011	62.31505	−0.00543	49.25149	−0.0672	111.5665	278	0.401
2012	62.22409	−0.09096	49.1944	−0.05708	111.4185	278	0.401
2013	62.33534	0.111251	49.16652	−0.02788	111.5019	278	0.401
2014	62.00065	−0.33469	49.10931	−0.05721	111.11	278	0.400
2015	61.94167	−0.05898	49.06518	−0.04412	111.0069	278	0.399
2016	61.91771	−0.02397	49.00768	−0.0575	110.9254	278	0.399
2017	61.92258	0.004872	48.95606	−0.05162	110.8786	278	0.399
2018	61.92782	0.005246	48.95606	0	110.8839	278	0.399
Overall Net Change (1995−2018)	−2.78009		−0.60203		–	–	−0.012

Table 6.

Total aggregate category scores and obtained HNI-values at the global spatial scale for the period 1995–2018 (all values positive unless indicated as negative).

Year	Food		Habitat		Health		Infrastructure
	Total score	Change in total score from previous year	Total score	Change in total score from previous year	Total score	Change in total score from previous year	Total score	Change in total score from previous year
1995	63.98429034	−	73.19811992	−	62.65633025	−	51.45648335	−
1996	63.96928804	−0.015002295	73.10887052	−0.089249405	62.73958652	0.083256276	51.335116	−0.121367345
1997	63.6960081	−0.273279947	73.00839285	−0.100477669	63.1704211	0.430834578	51.23373169	−0.101384319
1998	63.43575972	−0.260248373	72.89824121	−0.110151639	63.31519399	0.144772893	51.08103691	−0.152694777
1999	63.25566445	−0.180095277	72.77729705	−0.120944161	63.59543856	0.280244564	50.9846529	−0.09638401
2000	62.82042967	−0.435234779	72.61343238	−0.163864662	63.74186526	0.146426698	50.81510798	−0.169544917
2001	62.75834249	−0.062087175	72.42906171	−0.184370678	63.62621677	−0.115648484	50.6771786	−0.137929377
2002	62.64625609	−0.112086399	72.19899569	−0.230066015	63.38583244	−0.240384328	50.49401922	−0.183159383
2003	62.4050525	−0.241203589	72.1161273	−0.082868393	63.0708347	−0.314997745	50.32469592	−0.169323304
2004	62.07894292	−0.32610958	71.94795379	−0.168173507	63.68524991	0.614415207	50.20352522	−0.121170699
2005	61.69793547	−0.381007453	71.82510268	−0.122851108	63.74581094	0.060561035	50.12884416	−0.074681055
2006	61.29156156	−0.406373914	71.68538382	−0.139718868	63.95759475	0.211783807	49.95087153	−0.177972637
2007	61.31907787	0.02751631	71.59824922	−0.087134591	63.49930974	−0.458285003	49.74306243	−0.207809098
2008	61.24149853	−0.077579337	71.42012025	−0.178128973	63.3248106	−0.174499148	49.75906998	0.016007555
2009	61.5091679	0.267669367	71.22102834	−0.19909191	63.04373348	−0.281077114	49.15367869	−0.605391293
2010	61.13905253	−0.370115367	71.02481817	−0.196210175	62.87558462	−0.168148862	49.03781003	−0.115868661
2011	61.04510047	−0.093952065	70.76684713	−0.257971037	62.82227456	−0.053310056	48.69373421	−0.344075817
2012	60.80450014	−0.240600329	70.50332044	−0.263526691	62.39872316	−0.423551407	48.53204975	−0.161684464
2013	60.87220044	0.067700301	70.53236444	0.029043999	62.2549698	−0.143753357	48.36692566	−0.165124086
2014	60.59266974	−0.2795307	70.57578863	0.043424196	62.2794386	0.024468799	48.42165588	0.05473022
2015	60.74543412	0.152764387	70.66077002	0.084981385	62.01960172	−0.259836883	48.15285895	−0.268796936
2016	60.88981426	0.144380136	70.75780086	0.097030842	61.81621269	−0.203389031	47.88193236	−0.27092659
2017	60.81279433	−0.077019929	70.91497609	0.157175228	61.75844761	−0.05776508	47.66074718	−0.221185178
2018	60.49879953	−0.313994801	71.07739177	0.162415683	61.73187835	−0.026569255	47.66074718	0
Overall Net Change in Total Score (1995−2018)	−3.485490808		−2.120728151		−0.924451897		−3.795736171
Year	HNI
	Raw Total		Max. Total		Raw HNI		Final HNI
1995	251.2952239		556		304.7047761		0.548030173
1996	251.1528611		556		304.8471389		0.548286221
1997	251.1085537		556		304.8914463		0.548365911
1998	250.7302318		556		305.2697682		0.549046346
1999	250.6130529		556		305.3869471		0.549257099
2000	249.9908353		556		306.0091647		0.550376196
2001	249.4907996		556		306.5092004		0.55127554
2002	248.7251034		556		307.2748966		0.552652692
2003	247.9167104		556		308.0832896		0.554106636
2004	247.9156718		556		308.0843282		0.554108504
2005	247.3976933		556		308.6023067		0.55504012
2006	246.8854116		556		309.1145884		0.55596149
2007	246.1596993		556		309.8403007		0.557266728
2008	245.7454994		556		310.2545006		0.558011692
2009	244.9276084		556		311.0723916		0.559482719
2010	244.0772653		556		311.9227347		0.561012113
2011	243.3279564		556		312.6720436		0.562359791
2012	242.2385935		556		313.7614065		0.564319076
2013	242.0264603		556		313.9735397		0.564700611
2014	241.8695529		556		314.1304471		0.564982819
2015	241.5786648		556		314.4213352		0.565505999
2016	241.3457602		556		314.6542398		0.565924892
2017	241.1469652		556		314.8530348		0.566282437
2018	240.9688168		556		315.0311832		0.566602847
Overall Net Change in Total Score (1995−2018)	–		–		–		0.018572675

Table 7.

Final obtained E, HNI and S-values (3 d.p.) and indicated S-levels at the global spatial scale 1995–2018. (All values positive unless indicated as negative).

	E		HNI		S
Year	E-value	Change in E-value from previous year	HNI-value	Change in HNI-value from previous year	S-value	Change in S-value from previous year	S-level
1995	0.411	–	0.548	–	−0.137	–	−VW
1996	0.411	0.000	0.548	0.000	−0.137	0.000	−VW
1997	0.411	0.000	0.548	0.000	−0.138	0.000	−VW
1998	0.410	0.000	0.549	0.001	−0.139	−0.001	−VW
1999	0.410	0.000	0.549	0.000	−0.139	0.000	−VW
2000	0.410	0.000	0.550	0.001	−0.140	−0.001	−VW
2001	0.408	−0.002	0.551	0.001	−0.143	−0.003	−VW
2002	0.408	0.000	0.553	0.001	−0.145	−0.001	−VW
2003	0.407	−0.001	0.554	0.001	−0.147	−0.003	−VW
2004	0.405	−0.001	0.554	0.000	−0.149	−0.001	−VW
2005	0.404	−0.001	0.555	0.001	−0.151	−0.002	−VW
2006	0.404	0.000	0.556	0.001	−0.152	−0.001	−VW
2007	0.403	0.000	0.557	0.001	−0.154	−0.001	−VW
2008	0.403	0.000	0.558	0.001	−0.155	−0.001	−VW
2009	0.403	0.000	0.559	0.001	−0.156	−0.002	−VW
2010	0.402	−0.001	0.561	0.002	−0.159	−0.003	−VW
2011	0.401	0.000	0.562	0.001	−0.161	−0.002	−VW
2012	0.401	−0.001	0.564	0.002	−0.164	−0.002	−VW
2013	0.401	0.000	0.565	0.000	−0.164	0.000	−VW
2014	0.400	−0.001	0.565	0.000	−0.165	−0.002	−VW
2015	0.399	0.000	0.566	0.001	−0.166	−0.001	−VW
2016	0.399	0.000	0.566	0.000	−0.167	−0.001	−VW
2017	0.399	0.000	0.566	0.000	−0.167	−0.001	−VW
2018	0.399	0.000	0.567	0.000	−0.168	0.000	−VW
Net Change (1995−2018)	−0.012		0.019		−0.031		–
% Net Change (1995−2018)	−2.960		3.389		22.437		–

Table 8.

Pearson test results for determining the presence and significance of the relationship between obtained S-values and potential triumvirate-based influencing factors at the global spatial scale. (a). shows the Population category; (b). shows the GDP category; and (c). shows the GHG category. (All values positive unless indicated as negative) (a), (b), (c).

Year	S-value	Population
		Population, total	Population growth (annual %)	Population density (people per sq. km of land area)
1995	−0.137	39,303,704.86	1.40	145.72
1996	−0.137	39,862,597.00	1.37	148.59
1997	−0.138	40,419,274	1.36	151.28
1998	−0.139	40,969,027	1.36	154.08
1999	−0.139	41,509,766	1.35	155.86
2000	−0.140	42,044,278	1.36	159.67
2001	−0.143	42,573,908	1.33	162.42
2002	−0.145	43,101,034	1.35	164.17
2003	−0.147	43,627,968	1.44	164.39
2004	−0.149	44,157,213	1.38	166.46
2005	−0.151	44,689,413	1.50	169.28
2006	−0.152	45,226,271	1.56	172.35
2007	−0.154	45,765,748	1.58	176.19
2008	−0.155	46,312,442	1.58	180.45
2009	−0.156	46,857,536	1.52	184.09
2010	−0.159	47,398,784	1.42	186.79
2011	−0.161	47,929,214	1.36	189.60
2012	−0.164	48,447,613	1.32	192.51
2013	−0.164	48,994,908	1.30	195.08
2014	−0.165	49,546,328	1.28	197.48
2015	−0.166	50,097,463	1.25	200.01
2016	−0.167	50,652,019	1.22	202.73
2017	−0.167	51,202,012	1.19	204.91
2018	−0.168	51,737,793	1.16	207.41
n		24	24	24
d.f.		22	22	22
r		−0.9922	0.3616	−0.9902
p		<0.01	<0.10	<0.01
Statistically Significant?(Yes / No)		Yes	No	Yes

Year	S-value	GDP
		GDP (current US$)	GDP growth (annual %)	GDP per capita (current US$)	GDP per capita growth (annual %)
1995	−0.137	2.1569E + 11	3.84	7077	2.38
1996	−0.137	2.2036E + 11	4.99	7296	3.64
1997	−0.138	2.1946E + 11	4.18	7138	2.83
1998	−0.139	2.1913E + 11	3.06	7015	1.74
1999	−0.139	2.2733E + 11	2.84	7199	1.53
2000	−0.140	2.3491E + 11	4.23	7326	2.89
2001	−0.143	2.3367E + 11	3.12	7205	1.79
2002	−0.145	2.4261E + 11	3.34	7607	1.98
2003	−0.147	2.7242E + 11	4.20	8914	2.81
2004	−0.149	3.0722E + 11	6.06	10,276	4.55
2005	−0.151	3.3282E + 11	5.39	11,283	3.84
2006	−0.152	3.6084E + 11	5.97	12,305	4.34
2007	−0.154	4.0665E + 11	5.96	14,085	4.31
2008	−0.155	4.4594E + 11	3.82	15,561	2.22
2009	−0.156	4.2285E + 11	−0.39	13,578	−1.89
2010	−0.159	4.6316E + 11	4.25	14,476	2.75
2011	−0.161	5.1462E + 11	3.50	16,145	2.22
2012	−0.164	5.2665E + 11	4.07	16,011	2.77
2013	−0.164	5.4144E + 11	3.42	16,433	2.07
2014	−0.165	5.5626E + 11	3.24	16,537	1.93
2015	−0.166	5.2339E + 11	2.82	14,532	1.54
2016	−0.167	5.3145E + 11	2.94	14,509	1.67
2017	−0.167	5.6579E + 11	3.48	15,422	2.25
2018	−0.168	6.0157E + 11	3.17	16,455	1.95
n		24	24	24	24
d.f.		22	22	22	22
r		−0.9825	0.2085	−0.9487	0.1960
p		<0.01	<0.50	<0.01	<0.50
Statistically Significant?(Yes / No)		Yes	No	Yes	No

Year	S-value	GHG
		Total greenhouse gas emissions (kt of CO2 equivalent)	Total greenhouse gas emissions (% change from 1990)	CO2 emissions (kt)	CO2 emissions (metric tons per capita)
1995	−0.137	266,701.25	2.86	155,883	5.14
1996	−0.137	268,153.73	3.57	160,412	5.24
1997	−0.138	297,306.66	10.11	161,014	5.27
1998	−0.139	296,021	40.37	159,855	5.14
1999	−0.139	274,453	29.99	162,407	5.06
2000	−0.140	272,158	32.40	167,290	5.21
2001	−0.143	271,884	33.73	168,526	5.25
2002	−0.145	290,804	37.51	172,228	5.20
2003	−0.147	299,503	42.25	181,073	5.35
2004	−0.149	305,973	42.46	189,675	5.38
2005	−0.151	315,925	47.61	196,257	5.45
2006	−0.152	325,289	48.76	203,718	5.61
2007	−0.154	335,679	55.63	207,185	5.63
2008	−0.155	319,668	55.07	215,575	5.64
2009	−0.156	317,646	46.19	210,444	5.38
2010	−0.159	332,626	55.93	224,659	5.50
2011	−0.161	351,784	60.85	232,969	5.43
2012	−0.164	352,046	62.87	237,321	5.50
2013	−0.164	−	−	238,507	5.38
2014	−0.165	−	−	240,270	5.33
2015	−0.166	−	−	235,339	5.13
2016	−0.167	−	−	233,621	5.04
2017	−0.167	−	−	−	−
2018	−0.168	−	−	−	−
n		18	18	22	22
d.f.		16	16	20	20
r		−0.9215	−0.8793	−0.9890	−0.3316
p		<0.01	<0.01	<0.01	<0.20
Statistically Significant?(Yes / No)		Yes	Yes	Yes	No

Table 9.

The degree of association (R²) between obtained S-values and the triumvirate (individual categories and cumulatively).

	Triumvirate category			Triumvirate (1995–2012)
	Population (1995–2018)	GDP (1995–2018)	GHG (1995–2012)
Multiple R	0.993558055	0.986184508	0.992621085	0.997805455
R Square	0.987157608	0.972559884	0.985296619	0.995615726
Adjusted R Square	0.985231249	0.966783018	0.980772502	0.98757789
Standard Error	0.001344244	0.00201598	0.001228214	0.000987212
Observations	24	24	18	18

Table 10.

Results of the F-test of the cumulative categories of the triumvirate (all values positive unless indicated as negative).

	df	SS	MS	F	Significance F
Population
Regression	3	0.002777964	0.000925988	512.4474309	4.48877E-19
Residual	20	3.61398E-05	1.80699E-06
Total	23	0.002814104
GDP
Regression	4	0.002736885	0.000684221	168.3542259	1.49602E-14
Residual	19	7.72193E-05	4.06418E-06
Total	23	0.002814104
GHG
Regression	4	0.001314139	0.000328535	217.7876004	9.07201E-12
Residual	13	1.96106E-05	1.50851E-06
Total	17	0.001333749
Triumvirate
Regression	11	0.001327902	0.000120718	123.8661442	3.71061E-06
Residual	6	5.84752E-06	9.74587E-07
Total	17	0.001333749

Table 11.

Results of the t-test of the influencing factors in respect to the individual and cumulative categories of the triumvirate (all values positive unless indicated as negative).

	t Stat	p-Value
Population
Intercept	−1.943780779	0.066126677
Population (total)	−1.558584651	0.134778725
Population growth (annual %)	−2.021555761	0.056808061
Population density (people per sq. km of land area)	−1.049464184	0.306481363
GDP
Intercept	−12.24894547	1.8302E-10
GDP (current US$)	−5.238185009	4.68312E-05
GDP growth (annual %)	−2.044031575	0.055062908
GDP per capita (current US$)	2.214365193	0.039226954
GDP per capita growth (annual %)	2.023815862	0.057283305
GHG
Intercept	−7.609392666	3.85143E-06
Total greenhouse gas emissions (kt of CO2 equivalent)	0.648338581	0.528052315
Total greenhouse gas emissions (% change from 1990)	−1.984277172	0.068746765
CO2 emissions (kt)	−9.04159904	5.71503E-07
CO2 emissions (metric tons per capita)	0.512737134	0.616735969
Triumvirate
Intercept	−0.327227305	0.754611184
Population (total)	−1.49695122	0.185047806
Population growth (annual %)	0.466264854	0.657473352
Population density (people per sq. km of land area)	1.230121604	0.264690716
GDP (current US$)	−0.850387396	0.427735094
GDP growth (annual %)	−0.866804099	0.419358531
GDP per capita (current US$)	1.190621037	0.278766563
GDP per capita growth (annual %)	0.92034327	0.392892448
Total greenhouse gas emissions (kt of CO2 equivalent)	0.127932471	0.902382605
Total greenhouse gas emissions (% change from 1990)	0.01056109	0.991916025
CO2 emissions (kt)	−0.522617153	0.619968286
CO2 emissions (metric tons per capita)	−0.706126704	0.50658855

Figure 6.

Graphical trends of obtained E, HNI and S-values at the global spatial scale for the period 1995–2018.

Figure 7.

Residual plots for individual and combined categories of the triumvirate of potential influencing factors. The x-axis is the predicted residual, and the y-axis is the actual residual.

Discussion of results

Obtained S-values and S-levels

Based on the obtained global E and HNI-values in Tables 5 and 6 respectively, the obtained S-values and indicated S-levels at the global spatial scale over the period 1995–2018 are shown in Table 7 and Figure 6.

As Table 7 and Figure 6 indicates, the obtained S-values over the specified period shows a gradual consistent decline from −0.137 (1995) to −0.168 (2018). The indicated S-levels throughout the specified period were consistent with very weak unsustainability (-VW) occurring, based on the S-level value ranges in Figure 5 (Step six). The decline in obtained S-values over the specified period is due to obtained E-values consistently decreasing, and obtained HNI-values consistently increasing. Therefore, the occurrence of very weak unsustainability is due to obtained HNI-values being greater than obtained E-values throughout the specified period. This is consistent with equations (4) and (7) in Figure 2a and b for the conditions for unsustainability to occur.

The obtained indicated levels and nature of global sustainability over the specified period is dependent upon the obtained E and HNI-values. As Tables 5 to 7 indicate, obtained E-values gradually and consistently decreased from 0.411 (1995) to 0.399 (2018), whilst obtained HNI-values gradually and consistently increased from 0.548 (1995) to 0.567 (2018). This represents a percentage change in the obtained E and HNI-values over the specified period of −2.960% and +3.389% respectively. The changes in obtained E and HNI-values resulted in obtained S-values changing by −0.031 over the specified period, representing a percentage change of +22.437% (Table 7). Therefore, this indicates that the percentage net and annual incremental change in obtained E and HNI-values have had a significant impact upon obtained S-values over the specified period. The net change trends in obtained E, HNI and S-values potentially indicates the dynamic interactions occurring at the global spatial scale over the specified period, and which will be specifically discussed in the dynamic analysis section.

Determination of potential influencing factors

The results of the potential influencing factors upon obtained S-values are shown in Table 8. Of the 11 potential factors evaluated, there were seven which had obtained r-values that were highly statistically significant at p < 0.01. In respect to the triumvirate influencing factor categories, two each were in the Population and GDP, and three were in the Greenhouse Gases (GHG) category (see: Table 8). The seven potential significant influencing factors, as indicated in Table 8, were as follows: Total population; Population density; GDP (current US$); GDP per capita (current US$); Total greenhouse gas emissions (kt of CO₂ equivalent); Total greenhouse gases (% change from 1990); and CO₂ emissions (kt). The obtained r-values of the seven factors were within the range of [−0.9922 = r = −0.8783] at p < 0.01, indicating a very strong negative correlation with obtained S-values. Therefore, this indicates that as levels of the influencing factors increase over time, then obtained S-values reciprocally decrease. The strength of the indicated relationships between obtained S-values and the seven factors strongly suggest a clear and definitive influence and interaction upon sustainability and the environment-human system and relationship at the global spatial scale over time. This is a significant and potentially a primary reason for the occurrence of unsustainability indicated in Table 7. However, it is appropriate to consider the degree of cumulative influences of the triumvirate categories (Population, GDP and GHG) and the triumvirate as a whole, before any final determination.

Determining the cumulative influences

Overview The influences of the triumvirate categories on an individual and cumulative basis was determined using simple linear multiple regression analysis. The results of the regression analysis are provided in separate tables and a figure, reflecting the components of the analysis and to assist in their discussion. The results obtained were as follows. The obtained R²-values are shown in Table 9. The results of the F-test are shown in Table 12, and the t-test results are shown in Table 11. Table 12 shows the results of the coefficients and confidence intervals, and finally Figure 7 shows the plots of the residuals.

Table 12.

Coefficients and confidence intervals of the influencing factors in respect to the individual and cumulative categories of the triumvirate. (All values positive unless indicated as negative).

	Coefficients	Standard error	Lower 95%	Upper 95%
Population
Intercept	−0.02293246	0.011797863	−0.047542372	0.001677452
Population (total)	−1.75914E-09	1.12868E-09	−4.11352E-09	5.95241E-10
Population growth (annual %)	−0.005910911	0.002923942	−0.012010146	0.000188324
Population density (people per sq. km of land area)	−0.000235586	0.000224483	−0.000703849	0.000232676
GDP
Intercept	−0.105808188	0.008638147	−0.123888036	−0.087728339
GDP (current US$)	−1.34918E-13	2.57567E-14	−1.88828E-13	−8.10089E-14
GDP growth (annual %)	−0.013714679	0.006709622	−0.027758079	0.000328721
GDP per capita (current US$)	2.01571E-06	9.10288E-07	1.10455E-07	3.92096E-06
GDP per capita growth (annual %)	0.013716641	0.006777613	−0.000469066	0.027902349
GHG
Intercept	−0.101285022	0.013310526	−0.130040666	−0.072529378
Total greenhouse gas emissions (kt of CO2 equivalent)	2.03283E-08	3.13544E-08	−4.74088E-08	8.80654E-08
Total greenhouse gas emissions (% change from 1990)	−6.42855E-05	3.23974E-05	−0.000134276	5.70491E-06
CO2 emissions (kt)	−3.10716E-07	3.43651E-08	−3.84957E-07	−2.36474E-07
CO2 emissions (metric tons per capita)	0.001545137	0.003013507	−0.004965149	0.008055423
Triumvirate
Intercept	−0.013234321	0.040443817	−0.112196776	0.085728134
Population (total)	−4.175E-09	2.789E-09	−1.09995E-08	2.64944E-09
Population growth (annual %)	0.005619976	0.012053183	−0.023873102	0.035113053
Population density (people per sq. km of land area)	0.000639122	0.00051956	−0.000632195	0.001910438
GDP (current US$)	−7.97708E-14	9.38052E-14	−3.09304E-13	1.49762E-13
GDP growth (annual %)	−0.005327559	0.006146209	−0.02036679	0.009711672
GDP per capita (current US$)	2.08781E-06	1.75354E-06	−2.20296E-06	6.37857E-06
GDP per capita growth (annual %)	0.005774699	0.006274505	−0.009578462	0.02112786
Total greenhouse gas emissions (kt of CO2 equivalent)	4.03667E-09	3.15531E-08	−7.3171E-08	8.12443E-08
Total greenhouse gas emissions (% change from 1990)	5.77545E-07	5.46861E-05	−0.000133235	0.00013439
CO2 emissions (kt)	−1.43154E-07	2.73917E-07	−8.13405E-07	5.27098E-07
CO2 emissions (metric tons per capita)	−0.005744711	0.008135524	−0.02565162	0.014162199

Obtained R²-values As shown in Table 9, all of the categories and the triumvirate had obtained R²-values within the range of [0.9726 ⩽ R² ⩽ 0.9956] (4 d.p.). The highest obtained R²-value was for the triumvirate, and the lowest obtained R²-value was for the GDP category. In respect to the obtained R²-values for the triumvirate component categories, the indicated degree of association with obtained S-values is very strong. In relation to the triumvirate, the obtained R²-value indicates a near-total degree of association with obtained S-values. Therefore, the triumvirate component categories and the triumvirate overall have a very strong to near-total influence upon the level and nature of sustainability occurring at the global spatial scale over time. As indicated earlier in respect to Table 8, the influence of potential influencing factors is primarily negative in nature upon obtained S-values.

Therefore, it is reasonable to conclude that the cumulative influence of the triumvirate, both in respect to individual categories and collectively, is negative in nature. Furthermore, the results indicate that the triumvirate categories individually and cumulatively have long-term detrimental impacts and influences upon the level and nature of the environment-human relationship and system. The obtained R²-values results, and the Pearson test results previously, have potentially established the triumvirate of influencing factors as a primary driving force upon the environment-human system and relationship. As a consequence, this detrimentally influences and determines sustainability outcomes at the global and all other spatial scales over time in the Anthropocene. This will be further determined in respect to the other components of the multiple regression analysis.

F-test and t-test Results: As Table 10 indicates, the obtained F-values for the individual and cumulative categories of the triumvirate were in the range of (123.8861 ⩽ F ⩽ 512.4474) (4 d.p.) at a significance level (p) of between (3.7106×10⁻⁶ ⩽p ⩽4.4888 × 10⁻¹⁹) (4 d.p.). Therefore, the null hypothesis that there is no relationship between obtained S-values and the individual and cumulative triumvirate categories can be rejected.

In respect to the t-test results shown in Table 11, this indicates only two potential influencing factors were statistically significant. These were GDP (current US$) in the GDP category, and CO₂ emissions (kt) in the GHG category. The t-value of GDP was −5.2381 (4 d.p.) with a significance value (p) of 4.6831 × 10⁻⁵ (4 d.p.), and CO₂ emissions t-value was −9.0416 (4 d.p.) with a P-value of 5.7150 × 10⁻⁷ (4 d.p.). Therefore, in these two specific instances, the null hypothesis of no relationship between GDP and CO₂ emissions with obtained S-values can be rejected, and accepted in relation to the other potential influencing factors.

The results of the t-test of the significant majority of the potential influencing factors being statistically insignificant may be due to the nature of the data in the individual and cumulative triumvirate categories. This consequently potentially means that the t-test is an inappropriate test to determine their significance. Therefore, any final determination will be based upon the totality of the data obtained, and objective judgements concerning their validity and relevance.

Coefficients and Confidence Intervals: As Table 12 indicates, all but one coefficient in the individual triumvirate categories (Population GDP, GHG) were within the 95% confidence intervals. The only coefficient outside the confidence intervals of the individual categories was Total greenhouse gas emissions (% change from 1990). In respect to the cumulative category (Triumvirate), five of the 11 coefficients were outside the 95% confidence intervals. The six coefficients within the confidence intervals were population growth, population density, GDP growth, GDP per capita, GDP per capita growth and CO₂ emissions. The coefficients results obtained indicate the following.

Firstly, that the influence of the potential influencing factors upon obtained S-values is significant when in conjunction with the other factors within the same category, for example, Population – population (total), population density and population growth. This potentially magnifies their impact and influence at the global spatial scale upon obtained S-values individually and cumulatively within the category.

Secondly, that the potential factors of population growth, population density, GDP growth, GDP per capita, GDP per capita growth and CO₂ emissions have the most significant influence and impact upon obtained S-values than the other factors in the triumvirate, in respect to the individual and cumulative categories. Therefore, the six factors stated are potentially the most significant determinative influences upon the levels of sustainability at the global spatial scale. These factors potentially represent the driving force for the rapid and increasing exploitation of environmental resources and services by through rapidly increasing anthropogenic activities and needs. As a consequence, this has resulted in the continuing and increasing degradation of environmental resources & services at the global spatial scale over time. This indicates and supports the notion that the environment-human relationship is anthropospherically-dominated. The validity of these indicated outcomes shall be determined in the dynamic analysis.

Analysis of Residuals: The plots of the predicted -v- actual residuals for the triumvirate categories are shown in Figure 7. In respect to the Population and GDP residual plots, both show a tightly compacted pattern with approximately equal points above and below the x-axis. In respect to the GHG residual plot, this is more evenly scattered above and below the x-axis. The residual plots for the individual triumvirate categories can therefore be considered as consistent with unbiased homoscedastic data.

In respect to the residual plot of the Triumvirate category, representing the cumulative category of the individual components, this indicates a different pattern trend. The pattern indicated is cone-shaped, increasing from the first residual point at approximately x = −0.165, and with approximately equal number of residual points above and below the x-axis, as the cone-shaped pattern increases in size towards the x-axis interval of −0.135. The triumvirate residual plot therefore would tend to indicate an unbiased heteroscedastic pattern. This is potentially because of the GHG residuals, as indicated in their residual plot, are unduly skewing the triumvirate residual plot. The reason for this may be due to the significant large numbers associated with the GHG category factors, as well as the degree of change (increase/decrease) which has occurred over the specified period. The Population and GDP categories tend not to have as dramatic changes occurring to their levels, unless there is an extreme event, such as a famine, pandemic, or the global financial crisis for example.

Therefore, the residual plots in Figure 7 have indicated that the individual triumvirate categories are unbiased and uniform in nature, whilst the cumulative triumvirate category is unbiased but skewed in nature due to the influence of the GHG category.

Dynamic analysis

Global dynamics

Based upon applying the SDF principles stated in Figures 2 and 4 to the results shown in Tables 5 to 12 and Figures 6 and 7, the indicated fundamental dynamics of sustainability occurring at the global spatial scale for the period 1995–2018, are as follows.

(i). The Environment & Societal Needs HNI (↔ $\stackrel{.}{A}$ ↔ H ↔ K_A) is throughout the specified period greater than E (↔ $\stackrel{.}{N}$ ↔ A,G,I ↔ K_N), resulting in unsustainability to occur consistent with equation (8) (Figure 2a). This indicates the significant and long-term transformation of environmental resources & services (i.e. natural capital) to anthropogenic capital (human, social, manufactured – re: Pearce and Atkinson, 1998; Phillips, 2020), which can in terms of the SDF be mathematically represented as K_N → K_A. This is due to the continuing increase in anthropospheric needs, that is occurring at significant increasing rate of change of the Earth System (dE/dt; re: Gaffney and Steffen, 2017), which is detrimental to environmental resources and services. Consequently, this has resulted in the environment (E ↔ $\stackrel{.}{N}$ ↔ A,G,I ↔ K_N) to be significantly depleted and rapidly degraded over time as the fulfilment of anthropospheric needs (HNI ↔ $\stackrel{.}{A}$ ↔ H ↔ K_A) increases. Therefore, as the available levels of the environment are utilized for their resources & services in fulfilling societal needs, then the lower the available level (and quality) of environmental resources and services available to the present and future generations.

(ii). Rate of Change (dE/dt) In Phillips (2020, 2022), the rate of change of the Earth System (dE/dt) (re: Gaffney and Steffen, 2017) in equations (2) and (3) has been discussed within the context of the stated principles in Section D of Figure 4b and equations (8) and (11) (Figure 4a). This has involved an objective judgement-based approach to evaluate the nature of dE/dt occurring and its impacts and effects, for example, lag effects, cause-effect mechanisms, feedbacks etc. However, in developing and discussing the results obtained in this study, it has been possible to make some significant advances in the evaluation of dE/dt by the determination of the net change of obtained S-values over time. As a result, this has provided the opportunity to begin to quantitatively determine the potential dynamic mechanisms operating at the global spatial scale. However, of more potential significance in respect to the Anthropocene Equation (re: Gaffney and Steffen, 2017; see: equations (2) and (3) and the SDF specifically, the net change in obtained S-values is representative of the rate of change dE/dt. As mathematically defined in Phillips (2020), and as shown in equations (5) and (8) (Figure 4a), the rate of change dE/dt is equivalent to: sustainability (S) (re: Phillips, 2010, 2015, 2016); the environment-human system and relationship ( $\stackrel{.}{N}$ , $\stackrel{.}{A}$ ) (Schellnhuber, 1998); and the rate of change in total capital dK/dt (natural and anthropospheric, re: Pearce and Atkinson, 1998; Phillips, 2020). Therefore, the net change in obtained S-values at any specified point in time (t) represents and indicates the rate of change in the Earth System (dE/dt).

To determine the nature of dE/dt occurring at the global spatial scale, it is necessary to determine and evaluate the dynamic trends and characteristics occurring in respect to net changes in E, HNI and S-values. By this, a fuller understanding of the role of dE/dt in determining the level and nature of sustainability occurring at the global spatial scale can be ascertained. The net change in obtained E, HNI and S-values over the specified period is shown in Figure 8.

Figure 8.

Net change in obtained E, HNI and S-values at the global spatial scale for the period 1995–2018.

As Figure 8 indicates, there are three potential dynamic trends occurring over the specified spatial-temporal scale. Firstly, there are indications of cyclic trends in HNI-values. This is represented by two periods where the change in HNI are reasonably similar in characteristics and longevity. The two potential cyclic periods indicated are 1997–2003 and 2004–2011. After these two periods, there is a pronounced flattening in the change in the HNI-values curve. This flattened curve potentially indicates a lag effect from the impacts of the 2008 global financial crisis and the post-crisis measures implemented by national governments, for example, austerity, reduced public services, tax rises etc., which suppressed social and economic activities and needs. In respect to the two cyclic periods, this would potentially indicate and reflect the lag effects of the economic cycle and improvements in quality of life. The cyclic trends indicated in Figure 8 would suggest it takes a period of time for the impacts of improved or declining social and economic conditions to occur.

Secondly, Figure 8 indicates that the change in obtained E and S-values are potentially dynamically coupled. This is based upon the fact that the curves of both are characteristically relatively similar to one another. Therefore, this is potentially suggests that a change in obtained E-values causes a reciprocal change in obtained S-values at the same point in time (t).

Thirdly and finally, the peak net change in obtained E and S-values occurs in the middle of the HNI-value cyclic period. Peak net change in obtained E and S-values occurred in 2001 and 2010, and peak HNI-value change occurred in 2003 and 2012, as shown in Table 7 and Figure 8. Therefore, this indicated that there is one of two potential dynamic mechanisms which may be operating: (1). The environment (E) and sustainability (S) are impacted by human needs and interests (HNI) immediately during the rapid increase of HNI, and both reach peak net change approximately 2 years before HNI; or (2). The impacts upon E and S lag considerably behind HNI by approximately 7–10 years, because of the time taken for environmental systemic changes to occur at all spatial-temporal scales.

These two distinct mechanisms reflect the potential dynamic operation of the environment-human system and relationship. Which of these is the primary or sole dynamic interaction mechanism in operation may require a further longer period than evaluated to determine with clarity and certainty. However, if the period of 2011–2018 is removed from consideration because it is indicative of the extreme event of the 2008 global financial crisis, then it is possible to tentatively consider the potential typical dynamic mechanism occurring in respect to the global environment-human system and relationship.

Consequently, based upon the normalized data spanning the period 1995–2010, it is indicated based upon Figure 8 that the environment (E) and sustainability (S) lags cyclically behind human needs and interests (HNI). As a result, the rate of change dE/dt [↔ S ↔ ( $\stackrel{.}{N}$ , $\stackrel{.}{A}$ ) ↔ dK/dt] reciprocally lags due to the cause-effect of HNI upon E and S. The reason for this is because as the rate of net change in HNI increases, the environment (E) is impacted upon. The impacts upon E occurs at a slower rate of change due to the complex dynamic interactions within the environmental system at all spatial-temporal scales. As a result, this causes the level and nature of sustainability (S) to also lag, but which occurs in a coupled dynamic relationship with E, due to the reasons stated earlier.

Therefore, the rate of change of HNI strongly influences the rate of change of E and S, but which occurs in a lag-cyclic process. This consequently indicates that HNI’s rate of change occurs more quickly than for E and S, due to the rapid conversion of natural capital (K_N) to anthropospheric capital K_A). The full impacts and consequences of the K_N → K_A transformation occurs over time in respect to the nature, severity and magnitude of impacts upon E, and the resultant changes over time to the environment-human system and relationship. Furthermore, the exploitation and utilization of environmental resources and services to fulfil anthropospheric needs occurs at a more rapid rate of change than the environment (E) can respond and adapt to.

Based upon the evaluation conducted of dE/dt, it has been potentially determined quantitatively and dynamically that a lag cause-effect mechanism operates in respect to HNI upon E and S, and thus determining the level and nature of the rate of change dE/dt. However, to determine whether the indicated dynamic processes are occurring, it is necessary to apply a more sophisticated quantitative analysis technique to the SDF methodology and quantitative analysis process originally developed in Phillips (2020, 2022). Therefore, using the obtained net change in S-values, Fourier analysis will be applied to determine whether any of the dynamic processes indicated are potentially present.

(iii). Determining the Dynamic Processes As just indicated, Fourier analysis was applied to determine the potential or actual presence of dynamic processes in the determination of the level and nature of sustainability at the global spatial scale, due to the level and nature of the environment-human relationship and system over time.

Fourier analysis is a mathematical analytical technique which attempts to identify patterns or cycles within a data series over a specified time period. It has multiple applications such as in physics, number theory, statistics and probability, optics, oceanography, image processing and cryptography to name a few. Fourier analysis has been applied in respect to environmental and sustainability-based case studies such as in Pata (2021), Wagenseil and Samimi (2006), Luan et al. (2018), Logan et al. (2010) and Bush et al. (2017), in order to determine cyclic processes. However, these and others have been case-specific to a defined area, issue, topic and/or environmental system, for example, agricultural sustainable development, weather/climate, tropics etc. Fourier analysis has not been applied however to determine the potential or actual dynamic processes occurring which influences or impacts upon the level(s) and nature of sustainability at any specified spatial-temporal scale. A potential reason for this may be due to insufficient data indicating the levels of the environment, the levels of impacts from humanity’s activities and needs, and the indicated level(s) of sustainability occurring at a specified spatial-temporal scale. This study however has generated sufficient data so that Fourier analysis is an appropriate and viable technique to determine potential fundamental dynamic processes occurring. Furthermore, Fourier analysis is a potentially valuable addition to the SDF methodology in the determination and evaluation of the dynamics of sustainability occurring at any specified spatial-temporal scale.

In respect to the use of Fourier analysis, the fundamental dynamic relationship that it determines, in respect to Figure 2 can be mathematically described as follows: $\mathrm{\Delta }\mathrm{S}\left(\mathrm{t}\right)\iff \frac{\mathrm{dE}}{\mathrm{dt}}\left[\iff \left(\frac{\mathrm{dN}}{\mathrm{dt}},\frac{\mathrm{dA}}{\mathrm{dt}}\right)\iff \frac{\mathrm{dK}}{\mathrm{dt}}\right]$ . In simple terms, the annual net change in S-values (ΔS(t)) fundamentally indicates the rate of change of the Earth System (dE/dt). This is because the net change of the S-value indicates the rate of change occurring between the environment and anthroposphere. Therefore, in terms of E, HNI and S, if sustainability occurs when [E(t) >HNI(t) ⇔ S(t) >0] and unsustainability occurs when [E(t) ⩽HNI(t) ⇔ S(t) ⩽0] (Phillips, 2010, 2015, 2016, 2020); then the degree of the rate of change between E and HNI will determine the level, nature and rate of change of sustainability (S), and thus indicate the level and nature of dE/dt occurring. As a consequence of this relationship, this infers the level and nature of the rate of change of N-A and K occurring. The mechanism controlling the development of cyclic forms indicated by the Fourier analysis is the annual variations of ΔS(t)-values, which consequently indicates and infers the nature of dE/dt occurring. Therefore, the degree of variations of ΔS(t) occurring on an annual basis will consequently indicate whether the environment-human system is relatively stable, fluctuating, or perturbed.

Therefore, using the obtained data of the net change of E, HNI and S-values, as shown in Table 7 and Figure 8, Fourier analysis was applied using the data analysis tool within Microsoft Excel 2019. Because of the limitation of the Fourier analysis tool within Excel that data series must be consistent with powers of two, the data was evaluated in 16-year period blocks over the period of 1996–2018. Therefore, a total of eight time periods were evaluated, which were labelled t = 0, t + 1, t + 2, t + 3, etc., reflecting the distance from the initial first value of net change obtained which was in 1996 (t = 0). Obtained Fourier transformation values of net change of E, HNI and S-values were converted from complex numbers to real numbers in Excel, and were then plotted on a smooth scatter graph for the time period. The eight time period plots, and a composite plot of all the time periods in respect to net change of S-values, are shown in Figures 9 and 10 respectively.

Figure 9.

Fourier transformation graphs of net change of E, HNI and S-values at the global spatial scale over the period 1995–2018, in 16-year time series. Initial time starting point (t) is 1996, and each starting point of a 16-year period time series is represented as t = 0, t + 1, t + 2, t + 3, etc. The dotted line represents Fourier transformation net change in E-values; the dashed line represents Fourier transformation net change in HNI-values; and the solid line represents Fourier transformation net change in S-values. The x-axis is calendar years (A.D.), and the y-axis is the Fourier real number values.

Figure 10.

Composite plot of Fourier transformation net change in S-values for time series t = 0 to t + 7, compared to original obtained net change of S-values at the global spatial scale for the period 1995–2018. The x-axis is calendar years (A.D.), and the y-axis is the Fourier real number values and S-values.

As Figure 9 shows, there are clear indications of a definitive pattern in all of the time series plots in respect to the Fourier transformation of net change in E, HNI and more importantly S-values. In specific respect to S-values, there are clear indications of a pattern consistent with a cyclic process. This is defined by a double peak in the middle of the time series plot, and two peaks either side 3–4 years before and after the double peak. The double peak is a potential significant indication of a cyclic process in operation, with the peaks either side potentially indicative of the previous and next double peak cycle. The indicated lag time between cycles, as indicated in Figure 9, as measured from the bottom of the previous potential double peak to the start of the next potential double peak is approximately 3 years. The S-value double peak pattern in Figure 9 is clearly reflective of the same pattern in relation to net change of E and HNI-values. The trend of net change of E and HNI-values in all of the plots in Figure 9, would seem to indicate a considerable lag period before a double peak occurs in relation to each of them.

Consequently, this potentially indicates, in conjunction with the S-values, that the double peak is indicative of the impact and influence of humanity upon the environment within a defined time period cycle, and which takes a period of approximately 3–4 years to cyclical perturbate and manifest throughout the Earth System. This reflects the period of time for the effects and impacts of anthropospheric activities and needs within a cycle to become significant in magnitude and intensity, to influence profoundly the level(s) and nature of the environment-human relationship (sustainability / unsustainability). This part of the cycle of impacts and feedbacks upon the environment-human relationship occurs over a period of 4 years, as indicated by the double peak in all of the time series plots in Figure 9. After this, the cycle begins again to the next lag-phase and double peak, which as indicated is approximately a total duration of 7 years. However, considering that all of the time series evaluated in Figure 9 indicate the same trends, this suggests some process of further potential significance.

Figure 10 indicatively shows the individual Fourier transformation time series plots on a single composite graph, along with the trendline of the original net change of S-values over the specified period. As Figure 10 potentially and significantly indicates, there are clear suggestions of a continuous cyclic process consisting of overlapping cycles of 7 years’ duration from the start of the lag tome to the end of the double peak. This process outlined of overlapping cycles is of further potential significance when examined in relation to net change in S-values.

As indicated in Figure 10, there are two potential clearly defined periods of significant net change in S-values, both of which have similar trend patterns to one another. The first period is from 2000 to 2006, and the second period is from 2009 to 2015. As indicated, both are 6 years in duration, with a lag time between them of 3 years. This potentially indicates the influence and impacts upon levels of sustainability from the continuous cyclic processes, which takes a relatively short period of time to trigger a definitive decline in the levels and nature of sustainability as described earlier. Figure 10 therefore potentially indicates that the continuous cycles represented by the Fourier time series plots (t = 0 to t + 7) reinforce each other within the time period cycle at the global spatial scale. This consequently results in a fundamental detrimental change in the obtained indicated net change in S-values, representing a deterioration of the Earth System and the environment-human relationship in a pronounced cyclic process as described.

Therefore, given that the obtained indicated values of E, HNI and S were based upon data from a vulnerability assessment, then the resultant implications are of greater potential significance. Specifically, it indicates that humanity’s influence and impact upon the environment at the global spatial scale is perturbated in significance, magnitude and intensity within the Earth System, due to the nature of humanity’s relationship with the environment. This relationship is indicatively reflected by the cycles in each of the time periods in Figure 9, and which when combined as indicated in Figure 10, results in a fundamental detrimental and continuous cyclic change in the Earth System and the environment-human relationship, consistent with an anthropospherically-dominant influence and impact. The nature of the cycles potentially indicated in Figures 9 and 10, in respect to their nature and duration, indicate a continuous and rapid rate of change of the Earth System (dE/dt) consistent with equation (3) – referred to as the Anthropocene Equation (re: Gaffney and Steffen, 2017). If the Earth System was not being significantly impacted upon or influenced by anthropogenic activities and needs, then it is reasonable to suggest that the indicated cycles would be of a far greater duration, much smaller in their peaks and troughs, and have a longer lag time between cycles. In essence, a very shallow cyclic process in respect to their nature and duration, would potential reflect a rate of change dE/dt consistent with equation (2) of the Anthropocene Equation theory (re: Gaffney and Steffen, 2017). This would consequently indicate a co-evolutionary relationship between the environment and humanity. Figures 9 and 10 are therefore potentially indicative and representative of the cyclical processes that produces the rate of change dE/dt consistent with equation (3), (the Anthropocene Equation – re: Gaffney and Steffen, 2017). This would potentially contribute towards the fundamental reasons for the obtained negative S-values indicating unsustainability, and the indicated vulnerability of the environment-human system and relationship at the global spatial scale. Figures 9 and 10 have consequently and potentially shown for the first time, a clear and direct indication of the influence and impact of humanity upon the environment globally. Therefore, Figures 9 and 10 are potentially humanity’s footprint signal upon the Earth System in the Anthropocene, resulting in unsustainability and vulnerability of the global environment-human system and relationship.

(iv). Role of the Triumvirate As indicated earlier, the triumvirate of influencing factors, both individually and cumulatively, have a significant role in influencing the level and nature of sustainability at the global spatial scale over time. Because of the dominance of these factors is negative in nature, as indicated earlier and in Table 8, this reciprocally means that their influence upon levels of sustainability is also negative. Consequently, this has meant they have a detrimental influence and impact over time upon the nature of the environment-human system and relationship.

In dynamic terms, the fundamental influence and impacts of Population, GDP and GHG (HNI ↔ $\stackrel{.}{N}$ ↔ H ↔ K_A) upon the environment (E ↔ $\stackrel{.}{A}$ ↔ A,G,I ↔ K_N) is detrimental in significance and magnitude over time. This is reflected by obtained HNI-values being greater than obtained E-values throughout the specified period. Consequently, this has resulted in obtained S-values consistent with unsustainability occurring. Therefore, increasing levels of population-based factors at the global spatial scale causes reciprocal increases in GDP as the means to fulfil anthropospheric activities and needs. This in turn increases GHG-based factors as the result of increasing economic growth and societal needs. The current literature supports the indicated relationship between the triumvirate factors in influencing the environment-human system and relationship at the national and international spatial scale – for example: (i). Increasing Population and GDP – Increasing CO₂ emissions (re: Begum et al., 2015; Chang et al., 2019; Dong et al., 2018; Wang and Li, 2016; Wang and Zhao, 2015; Zhao et al., 2014); (ii). Increasing GDP - Increasing CO₂. (re: Khan and Hou, 2021; Raihan and Tuspekova, 2022a, 2022b; Rahman et al. (2020), and (iii). Increasing Population – Increasing CO₂ (re: Alam et al., 2016; Rahman et al., 2020; Weber and Sciubba, 2019). As a consequence of this Population-GDP-GHG cycle, this impacts upon the environment due to the increasing exploitation and utilization of environmental resources and services. These outcomes of the cause-effect mechanisms of anthropospheric activities and needs upon the environment is potentially significant in advancing knowledge in respect to establishing the long-term nature and degree of influences upon sustainability, approximately equivalent to a human generation (>20 years).

The obtained S-values shown in Table 7 and Figure 6 would certainly tend to indicate that unsustainability at the global spatial scale has been present for a considerable period of time longer than the specified period. Therefore, this potentially indicates that the levels of population, GDP and GHG-based factors may have breached optimum system thresholds for a considerable period of time, causing the rapid exploitation and utilization of environmental resources and services to fulfil societal needs to occur (K_N → K_A). The triumvirate has consequently transformed the nature of the environment-human relationship from co-evolutionary to anthropocentric.

However, of greater potential significance is in relation to equations (2) and (3) of the Anthropocene Equation theory (re: Gaffney and Steffen, 2017), and in respect to Earth System Analysis (re: Schellnhuber, 1998) as shown in equation (1), both of which are component theories of the SDF.

In essence, the results obtained in Tables 5 to 12 and Figures 6 and 7 are consistent with equations (3) and (11) (Figure 2), and thus reflects the dominance of the anthroposphere upon the environment-human relationship. Furthermore, the results obtained indicate that current management and mitigation of humanity’s impacts upon the environment at the global spatial scale, in respect to M(t) in equation (1), are potentially and fundamentally inadequate. This is potentially because the fundamental influences and drivers which fosters unsustainability, such as and primarily the triumvirate, are not being adequately addressed or managed consistent with a co-evolutionary relationship.

In summary, the triumvirate has a definitive detrimental role and influence upon the level(s) and nature of sustainability occurring at the global spatial scale over time. The triumvirate has strongly influenced and even induced unsustainability to occur at the global spatial scale over a prolonged period of time. This is because of the long-term impact upon the environment from anthropogenic activities and needs. As a consequence, this indicates that unsustainability at the global spatial scale has potentially existed for considerably before the specified period, and that optimum system thresholds have been potentially breached to a significant degree.

(v). The Dynamics of Vulnerability-Sustainability The results obtained and shown in Tables 5 to 12 and Figures 6 and 7, as well as the analysis conducted thus far, have strongly indicated the continuing and increasing vulnerability of the global environment-human system and relationship. This has resulted in unsustainability occurring, which has persisted for a considerable period of time. As a consequence, this demonstrates that there is a quantitative and dynamic relationship between vulnerability and sustainability at all spatial-temporal scales. The reason for this is because of the degree of vulnerability occurring at any specified spatial-temporal scale, significantly influences the level and nature of sustainability at the same scale. Specifically, as the level(s) of societal needs are fulfilled by anthropogenic activities (HNI ↔ $\stackrel{.}{A}$ ↔ H ↔ K_A), this causes a reciprocal improvement(s) in human well-being and quality of life. The nature and rate of anthropogenic activities and needs determines the rate of exploitation and utilization of environmental resources and services (E ↔ $\stackrel{.}{N}$ ↔ A,G,I ↔ K_N). This in turn determines the rate of degradation and depletion of the available level of environmental resources and services. As a consequence, these feedbacks onto the fulfilment of anthropogenic activities and needs as the present generation or future generations have to manage existing or seek new or alternative environmental resources and services to fulfil the present-day needs. Therefore, by the process outlined, which are visually conceptually simplified in Figure 11, the environment-human system and relationship becomes more susceptible to the rate of change dE/dt, and thus resulting in vulnerability to occur. This consequently influences and determines the level and nature of sustainability [S ↔ ( $\stackrel{.}{N}$ , $\stackrel{.}{A}$ ) ↔ dE/dt ↔ dK/dt] occurring at the specified spatial-temporal scale.

Figure 11.

A simple conceptual visualization of the current indicated dynamic process of the environment-human system and relationship and its potential outcomes at the global spatial scale.

Therefore, the greater the level of dE/dt, the greater the deterioration of the environment-human system and relationship, and thus the greater the level of vulnerability and unsustainability occurring at any specified and/or all spatial-temporal scales. The presence of vulnerability and unsustainability within the global environment-human system and relationship is strongly indicative of an anthropospheric-dominant relationship as opposed to a co-evolutionary relationship. An anthropospheric-dominant relationship, and the resultant vulnerability and unsustainability, is consistent with equation (3) and equation (8) (Figure 2a). A co-evolutionary relationship, and the resulting resilience and sustainability occurring, is consistent with equations (2) and (5) (Figure 2a). Therefore, the continuation and increase in the vulnerability of the global environment-human system and relationship, and the resultant unsustainability occurring, presents significant and potential realistic risks to human society and activities. Specifically, through the further deterioration of the environmental system and the occurrence of species extinction and systemic failures, which increases the potential risks of the collapse of the global environment-human system. The results and analysis presented and discussed certainly indicates that these risks are potentially realistic, unless urgent measures are undertaken to correct the current dynamic global environment-human pathway of vulnerability-unsustainability.

Summary of outcomes and conclusions

The study has indicated that the levels and nature of sustainability at the global spatial scale over the specified period 1995–2018 was consistent with very weak unsustainability occurring. This is due to the inducement and entrenchment of vulnerability of the global environment-human system and relationship, and thus resulting in unsustainability occurring. As a consequence, this reflects and indicates the environment-human relationship at the global spatial scale is anthropospherically-dominant as opposed to co-evolutionary. The occurrence and entrenchment of vulnerability and unsustainability has been strongly indicated to due to the role and impact of the triumvirate of influencing factors on an individual and cumulative basis upon obtained S-values. Therefore, if resilience and thus sustainability is to occur at the global spatial scale, then the triumvirate must be at levels which operate within system thresholds, such as the Planetary Boundaries framework (re: Dearing et al., 2014; Häyhä et al., 2016; Rockström et al., 2009).

The results obtained have strongly indicated the presence of global unsustainability over the specified period, and that this has occurred considerably before the specified period. Global unsustainability has been significantly influenced by the triumvirate (Population, GDP and GHG) due to their rapid increase in fulfilling societal needs. This has consequently resulted in system thresholds being breached, and thus causing the vulnerability of the global environment-human system and relationship. This reflects the transformation of the environment-human system and relationship from co-evolutionary to anthropospheric-dominant, consistent with the stated mathematical definitions and principles of the SDF and its component theories. Furthermore, using Fourier analysis for the first time within the SDF application methodology, the cyclic dynamic processes of humanity’s impact and influence upon the environment at the global spatial scale have been indicatively determined.

Therefore, this study has highlighted the urgency of the potential fate that humanity finds itself facing. Humanity’s future has always been its own hands. The results and analyses of this study have indicated the consequences of humanity’s actions and inactions, which have resulted in increasing vulnerability and unsustainability of the global environment-human system and relationship. Unless humanity starts and continues to operate within the safe thresholds of the planetary system, then the results indicated in this study will continue to worsen until there is only one inevitable outcome for the environment and humanity - extinction. It is not too late to change humanity’s pathway and return it to a co-evolutionary relationship with the environment. It will require considerable long-term global co-operation, co-ordination, discipline and sacrifice at all societal and governance scales to achieve - However, the most important things always do.

The planetary alarm bells are ringing and have been for some considerable time - It is time for humanity to wake up and hear them, and to take the necessary and urgent actions to ensure humanity’s and the planet’s continuation and survival. Therefore, humanity now needs to truly and deeply ask itself: ‘What kind of a world do we want? and What must we do to get there’ (Blackburn, 1992; Clark, 1989).