The PelicanWeb's Journal of Sustainable Development

Research Digest on Integral Human Development,
Solidarity, Sustainability, and Related Global Issues

Vol. 6, No. 5, May 2010
Luis T. Gutierrez, Editor

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A Mathematical Model
of Sustainable Development
Using Ideas of
Coupled Environment-Human Systems

Jason Phillips
Dept. of Geography & Camborne School of Mines
University of Exeter, United Kingdom

The following is based on, and extracted from, the PhD thesis and paper in Sustainability Science, Vol.5(1), p127-142 of the author (see: Phillips, 2009, 2010).


Earth System Analysis was postulated as a theory by Hans-Joachim Schellnhuber in 1998 as a way to characterise the Earth System - the coupled relationship between the environment and humans. Within this theory, is the notion of Geocybernetics – the management of the Earth System in order to achieve strategies and mechanisms of co-evolution between the environment and humans. This is regarded as the concept and application of sustainable development. However, whilst fundamental definitions in Earth System Analysis are presented for the coupled relationship between the environment and humans, no such definitions exist for sustainable development within the Earth System context.

Consequently, the paper presents a mathematical model of sustainable development which provides for the fundamental abstraction of the key concepts and parameters necessary for sustainable development to occur within the context of the Earth System – the coupled system of relationships and dynamics between the environment and humans. The model utilises basic mathematics to detail these concepts and parameters, as well as the conditions required for sustainable development to occur.

Keywords: Sustainable Development; Earth System Analysis; Geocybernetics; Earth System Science; Sustainability Science.

1. Introduction

The crux of sustainable development is gained by the understanding of fundamental dynamic relationships between the environment and humans, and how to apply such knowledge. What is the role the environment plays upon the development of the anthroposphere and vice versa? In particular, part of the fundamental weakness within the concept and development of sustainable development is the failure is ask the most basic question of all: what is the environment? There can be no sustainable development without an effective answer to this question, as everything follows from this question as to the role of, and relationship with, humans. This consequently determines how sustainable development will be achieved and maintained. It is the conceptual understanding of these fundamental concepts that this paper will provide through the support of Earth System Analysis (Schellnhuber, 1998, 1999, 2001; Schellnhuber and Kropp, 1998). Earth System Analysis, contrary to the predominantly word-based arguments that have been dominated within the subject, uses mathematics to define the nature of the dynamics between the environment and humans.

The development in understanding of the systematic interconnectivity and interactions between the environment and humans in Earth System Analysis, has provided coherent framework for understanding the feedbacks and synergisms between the ecosphere (N) (the Environment) and the anthroposphere (A) (the Human World & Society). However, whilst Earth System Analysis does provide for a fundamental grounding in the dynamics of the Earth System in the historical past and present day, the question of defining the fundamental dynamics of the co-evolutionary state that implies sustainable development remains unresolved. Therefore, it is necessary to determine the fundamental dynamics of co-evolution between N and A, which in turn can determine the nature and level of sustainable development.

Therefore, by using Earth System Analysis as a fundamental basis for supporting the conceptual development demonstrated in the paper and undertaken through independent research, this paper will propose and present a mathematical model in order to attempt to contribute to the debate: ‘What is sustainable development?’

2. Research Background & Context

2.1 Sustainability Science

There has been increasing attention in the recent development of the concept of sustainable science, which seeks to understand the interactions between nature and society (Omer, 2008). Sustainable science focuses on the role and use of science in the prudent management of the environment, and for future human development (Afgan et al., 1998). Consequently, it recognises that knowledge should be applied in supporting the goals of sustainable development through the use of scientific assessment of the current conditions and future prospects for the Earth system (Afgan et al., 1998).

“Sustainability science has emerged over the last two decades as a vibrant field of research and innovation” (Clark, 2007). The specific questions that sustainability science attempt to address are:

  1. How can those dynamic interactions be better incorporated into emerging models and conceptualizations that integrate the Earth system, social development, and sustainability?
  2. How are long-term trends in environment and development reshaping nature-society interactions?
  3. What factors determine the limits of resilience and sources of vulnerability for such interactive systems?
  4. What systems of incentive structures can most effectively improve social capacity to guide interactions between nature and society toward more sustainable trajectories?
  5. How can science and technology be more effectively harnessed to address sustainability goals?
  6. (adapted from Kates et al., 2001)

Therefore, as Clark (2007) goes onto to state: “from its core focus on advancing understanding of coupled human-environment systems, sustainability science has reached out with focused problem-solving efforts targeted to urgent human needs”. However, understanding such coupled systems are “made all the more unpredictable by both our interest in what goes on in particular places and by our active, reflective engagement in the system whose behaviour we are trying to predict” (Clark et al., 2005).

A key part in the development of sustainability science has been Earth System Analysis (Schellnhuber 1998, 1999, 2001) which characterises the dynamic relationship between the ecosphere (the environment) and the anthroposphere (the human world & society). Therefore, given its pivotal role in the development of the model and its application, it is worth briefly outlining this important work.

2.2 Earth System Analysis

Earth System Analysis (Schellnhuber, 1998) is concerned with the coupled dynamic relationship between the environment and humans. There are three principal components to Earth System Analysis – Global Change; Global Environmental Management; and Geocybernetics.

2.2.1 Global Change

The notion of ‘Global Change’ is concerned with the effect upon nature in respect of the magnitude of such impacts. Therefore, it requires an understanding of the historical and present behaviour of humans upon the environment adopting a global perspective.

In the historical past (the pre-industrial age - the period before the beginnings of the Industrial Revolution of the late 18th Century), the impacts of humans upon the environment tended being local in magnitude (Schellnhuber, 1998). This meant that such impacts could be almost compensated for over time by the environment, and consequently, there was no long-lasting impact caused. Therefore, this dynamic relationship is defined by Schellnhuber (1998) as follows:

Equation 1
The Dynamic Relationship between the Ecosphere and the Anthroposphere in the Historical Past as stated by Schellnhuber (1998).

Equation 1 states that the ecosphere (N) was independent of any influence by the anthroposphere (A) over time. On the other hand, the development of humans was influenced by its own actions, and those of the environment. However, as Schellnhuber (1998) admits, this is an extremely compacted version of the true dynamics of the Earth System. Therefore, the question is how the relationship changes in the present day.

The influence and actions of humans upon the environment in the present day are uncomparable in respect to the historical past, (Schellnhuber, 1998). This is due to human actions and influences which are significantly greater in respect of intensity and geographical extent (Schellnhuber 1998). This refers to Schellnhuber’s (1998, 1999, 2001) observation that since the Industrial Revolution, humans are intervening on “the scale of the system”. Consequently, humans are affecting the operation of the Earth System by impacting upon the processes of the environment.

By using the same approach as for the historical past, the coupled dynamic relationship between the N and A for the present day can be determined. Schellnhuber (1998) defines this relationship as follows:

Equation 2
The Current Dynamic Relationship between the Ecosphere and the Anthroposphere, as stated by Schellnhuber (1998).

Equation 2 indicates that in the present day scenario, the relationship has changed so that N and A are now strongly coupled in the evolutionary development of planet. This is because human actions and by-products (i.e. pollution, infrastructure, industry, urbanisation, waste etc.) are now directly influencing the systems and processes of the environment. This means impacts upon one component of the environmental system will have knock-on effects upon the other components at the same geographical level and further up the system.

2.2.2 Global Environmental Management

Humans are changing its habitat on a global scale (Schellnhuber, 1998, 1999, 2001). However, science and technology, by the same token, is capable of channelling human development along acceptable environmentally-sound pathways. This would be considered as sustainable development within the Earth System (Schellnhuber, 2001). For this to occur, humans, as a global actor in the Earth System, would need to obtain a new identity – the “Global Subject” (S). This is a self-conscience role to “conceive the planetary system in its entirety” (Schellnhuber, 1998, 1999, 2001). This means that the design of nature is shaped and influenced by the characteristics of the perception system of humans – the construction and maintenance of a set of laws/rules for an image of the outside world from various and specific impressions. However, an individual person’s senses are not sufficient to perceive, for example, any global changes to the environment. Instead, the Global Subject perceives the Earth System, as Schellnhuber (1998) observes, through the senses of the global “scientific-medial complex”, i.e. monitoring devices, computers and data storage, electronic networks (e.g. remote sensing). Then humanity, through scientists, translates the results to the best of their ability (e.g. the 4th IPCC Climate Change report), consequently undertakes appropriate measures to modify its course of action, either on its own volition or as part of an organised approach (i.e. recycling household waste, car-sharing; the Kyoto protocols).

With all this in mind, Schellnhuber (1998, 1999, 2001) states it is possible to conceive the Earth System at the most basic level as follows:




Equation 3
The Composition of the Earth System, as stated by Schellnhuber (1998).

This conceptualises the Earth System to have two primary components, namely the N and the Human Factor H (Schellnhuber, 1998, 1999, 2001). N represents the various planetary sub-spheres that can be “spelled” alphabetically: a - atmosphere; b - biosphere; c - cryosphere etc. The Human Factor H consists of two key sub-components – the anthroposphere (A), which is the total sum of all individual lives, actions and products of humanity; and the Global Subject (S).

Due to existence of the Global Subject, the dynamic relationship between the ecosphere and anthroposphere in the present day era changes to include a new factor – M(t). M(t) represents the chosen management strategy to steer sustainable development along a desired path over time. The new dynamics associated with the management aspects of the Global Subject (Schellnhuber, 1998, 1999, 2001) are as follows:

Equation 4
The Dynamic Relationship between the Ecosphere and the Anthroposphere controlled by the selection of strategy M(t), as stated by Schellnhuber (1998).

Therefore, the choice of the management strategy will determine the path and nature of sustainable development. This is this issue that Geocybernetics addresses.

2.2.3 Geocybernetics and Sustainable Development

Geocybernetics is “the art of controlling the complex dynamic earth system under uncertainties of all kinds” (Schellnhuber and Kropp, 1998). This suggests that if humans are to continue to live on this planet, then there needs to be prudent and effective use of resources - sustainable development. In order to achieve this, there must be co-evolution between the environment and humans. This is achieved by utilising strategies to achieve co-evolution paths over a specified period of time. This can be categorised into one or more of five fundamental geocybernetic paradigms of sustainable development which are:

  • Standardization - This paradigm provides for the setting of clear values, targets and indicators to control the N-H system. Therefore, it sets thresholds for the safe conduct of the system over the long-term (Gallopin, 2003).
  • Optimization - This paradigm is concerned with choosing the “best” design for N-H co-evolution by choosing the optimal path / strategy over a fixed period of time (Gallopin, 2003; Schellnhuber, 1999).
  • Pessimization - This paradigm is concerned with the minimum amount of damage for the maximum amount of potential benefit. Thus, avoiding bad management is crucial (Schellnhuber, 1999; Gallopin, 2003).
  • Equitization - This paradigm is about the preservation of options for future generations (Gallopin, 2003). Therefore, the paradigm’s notion of “equity” is associated with equality of the environmental and development options for future generations (Gallopin, 2003).
  • Stabilization - This paradigm brings the N-H complex into the desired state of sustainable development, and then maintains by utilising good management (Gallopin, 2003).
From this, humanity can select the appropriate master principle for its strategy to control and achieve sustainable development; or it can develop a strategy that utilises a suitable combination of paradigms to achieve the same purpose (Schellnhuber, 2001). Where a combination of fundamental geocybernetic paradigms are used, this is defined by as a complex paradigm (Schellnhuber, 1998).

With the research context outlined, it is appropriate to now discuss the mathematical model of sustainable development detailed in Phillips (2009, 2010). This model was developed and supported through utilising and expanding upon ideas and concepts within Earth System Analysis (Schellnhuber, 1998, 1999, 2001) and the ideas and concepts of weak and strong sustainability (see: Pearce and Turner 1990; Daly 1994; Costanza and Daly 1992; Goodland and Daly 1996; Gowdy and O’Hara 1997; Comolli 2006; Victor et al. 1995; Bowers 1997).

3. Mathematical Model of Sustainable Development

3.1 Defining Sustainable Development

Utilising the concept of co-evolution of the N-A system, as noted by Schellnhuber (1998, 1999, 2001), then it is possible to begin to infer what is sustainable development. Sustainable development, for any specific point in time, can be described as follows:

Equation 5
S = Sustainable Development
E = (the) Environment
HNI = Human Needs and Interests
t = time

This suggests that for determined or attributed values of E and HNI, the level and nature of sustainable development is dependent upon the nature and level of impact which humans inflict upon the environment and which consequently results in the depletion and degradation of resources and services offered by E. Determined values of E and/or HNI refer to real-time data collected through experiments, observations or quantitative measures such as indicators or quantitatively-based Environmental Impact Assessment (EIA); and attributed values of E and HNI are data obtained using a value judgement approach, such as in the case of a qualitative or semi-quantitative EIA.

If Equation 5 is related and compared to Equation 2, then this supports the notion that humans tend to have a negative role to the operation of the environment. Therefore, the need to “quantify” E and HNI through determined or attributed valuation becomes paramount. This can be achieved in respect of the notion of weak and strong sustainable development, which describes the relationship between natural and human capital.

Natural capital is typically considered to be the stock of natural resources or environmental assets that yields a flow of useful goods and services, now and in the future (Pearce and Turner, 1990; Daly, 1994). Such resources or assets can be differentiated into two broad categories: 1) renewable and 2) non-renewable (Costanza and Daly, 1992; Goodland and Daly, 1996; Gowdy and O’Hara, 1997; Comolli, 2006). Gowdy and O’Hara (1997) provide examples of each, such as economically valuable biological species for renewable resources, and minerals and fossil fuels for non-renewable resources. Human (or manufactured) capital is all human-made machines, tools and buildings that are used in economic production (Daly 1994; Victor et al., 1995; Goodland and Daly, 1996). Hence, this requires the physical transformation of natural capital and the use of human labour to produce (Daly, 1994; Victor et al., 1995).

Consequently, the notion of weak and strong sustainability is based upon the concept of natural capital, and how much of such capital should be preserved in perpetuity or set aside for use for present and future generations (Bowers, 1997). Figure 1 shows an illustrative example of the differences in the use of capital in respect to weak and strong sustainability along the lines discussed.

Figure 1
A diagram based on and adapted from Roberts (2004, p82) which illustrates the differences in the conversion of natural capital to human capital in the weak and strong sustainability approaches.

So, if E (i.e. natural capital) in respect to its substitution to HNI (i.e. human capital) is necessary in the determination and evaluation of the level and nature of sustainable development, then the lower the impact of HNI upon E, the greater the value of S. However, this does require the highest possible value of E. The components of E and HNI both contain separate and integral relationship which define each in its nature and extent, which shall be explored.

3.2 Defining the Environment (E)

The environment (E) can be defined as the four planetary and integral sub-spheric systems which is necessary for all planetary operation at all spatial-temporal scales. These are: Atmosphere (A), Biosphere (B), Hydrosphere (H), and Lithosphere (L). These work as an integrated dynamic system, which contain numerous smaller sub-systems, e.g. the Gulf Stream, the Tropical Rainforest biome, the Hydrological Cycle, and Plate Tectonics. Such sub-systems have an impact or role to play in influencing the development and characterisation of the whole system (E). Further, such sub-spheres take time to evolve to a perceived end result, or continuously adapts due to changes within the sub-spheric system or an event(s) occurring sub-sphere(s). Therefore, the Environment (E) can be characterised mathematically, at any point in time, as follows:

Equation 6

This definition tends to be more direct in its determination of the Environment, and is comparable to Schellnhuber’s definition of N = (a,b,c...) (Schellnhuber, 1998, 1999, 2001). This however offers the potential opportunity for application at any specified spatial-temporal scale to be more reasonable approached. However, any system, irrespective if it is natural or anthropogenic, has pre-determined maximum threshold for its safe operation. The Environment (E) is no different in this respect as there are clear limits and tolerances of operation, which if exceeded, would begin to cause the step-by-step failure of the sub-spheric system, similar to a ‘cascade effect’ which Schellnhuber (1998, 1999, 2001) and Lovelock (2000) support. This is due to the interdependence of the operations within and between the sub-spheres. Therefore, given the clear operational parameters and constraints of E, in respect to spatial-temporal considerations, then this can be expressed as follows:

Equation 7

Consequently, E is dependent upon the space needed or taken by the sub-spheres to operate in; and the time required for evolution, adaption, mitigation and repair of the system(s) in respect to the other sub-spheres. This means for any determined or attributed values of E obtained, they must reflect the significance and magnitude of the impact / effect (potential or actual) in respect to the present state of the condition of E.

3.3 Defining Human Needs and Interests (HNI)

From a Darwinian and even an anthropogenic viewpoint, humans have developed further as a species than any other. This is due to the fact that humans not only satisfy their own basic needs, but have gone beyond to satisfy needs and interests that are tangible and intangible in nature. This is comparable to the definition of the Human Factor (H) in Equation 3 earlier, and suggests that each generation continually desires more than the past generation. This could result in one of two potential outcomes: 1). To succeed and improve the human condition through acts of inspiration; or 2). To deteriorate the human condition through acts of self-destruction. Therefore, the appearance is that HNI is infinite, and would vary according to the level of human societal hierarchy it occurs at. However, is this actually the case?

HNI is dependent upon the resources and services available and produced by E which ensures tolerable conditions for human to live and survive. If HNI increased at a rate that is at the increasing detriment of E, then this infers that there is a maximum limit for HNI based on the resources and services of the E left available. This means as a consequence that when E eventually is degraded beyond a point of no return, at whatever spatial scale, then humans would be required to live somewhere else. This would therefore suggest that there is a limit to the potential determined or attributed value of HNI obtained for any specific point of time, and can be characterised as follows:

Equation 8

The parameters and factors that determine the degree and/or value of HNI, and which informs the use of E by HNI, can be described as follows:

Equation 9


Equation 10


Equation 11

Equation 9 defines HNI in respect to the human hierarchy, by which at each level, it adopts Needs and Interests (NI) relevant to that level. NI itself is defined in Equation 10, where at each level of human hierarchy (based on the ideas of Maslow, 1943), there will be different requirements of NI components, based on their status and development at the time. As the human hierarchy changes due to environmental and human pressures of evolution and development, then NI changes accordingly in order to meet the aspirations and challenges faced at each level of the hierarchy. Each successive generation of the hierarchy would seek improvements in their condition from the previous generation. However, each hierarchical level would still fundamentally require to meet Basic Needs (BN). BN (Equation 11) ensures that the absolute minimum necessary conditions for human survival and development are met at any level of the hierarchy. As humans evolve to become more sophisticated in respect to NI over time, then their wants and desires becomes more intangible in nature in conjunction with advancing social development. Consequently, BN becomes by the Society level of the human hierarchy, almost automatically achieved. This is because the Society level will ensure that everyone within it has a satisfactory level of subsistence in meeting the parameters of NI (e.g. the welfare state). However, where human conditions only meet the requirements of BN, then this is due to environmental and/or human factors that hinder further development along the human hierarchy. Such factors would include: war; changing or poor environmental conditions; natural disasters; famine; poor political structures; overpopulation; poor education; lack of social cohesion; insufficient technological development or capacity etc.

Human development and the needs and interests that fuel it, is therefore dependent upon the level and extent available of three factors at the time – Social Development (SD), Technology (T), and Knowledge (K), and which can be suggested by the following:

Equation 12

These factors’ boundaries are potentially infinite, as indicated in Equation 12, as well as the human potential to meet its needs and interests.

3.4 Determining the Level and Nature of Sustainable Development

As E and HNI have been defined, it is now possible to determine what this mean for sustainable development. For a level of S to occur at any point of time and for a specified spatial scale, then the determined or attributed value of E must be greater than the determined or attributed value of HNI, and which can be expressed as follows:

Equation 13

If on the other hand, the determined or attributed value of E is less than or equal to the determined or attributed value of HNI, then S would not occur as described in Equation 14. This is because there must be a continuous source of E and HNI to utilise, and that it must not endanger the safe operation of E.

Equation 14

If considering the totality of the equations described thus far, then it is possible to suggest the dynamic relationship of E and HNI over time, as can be seen in Figure 2, and which is based on the following:

  1. Assumes an unmitigated situation to the scenarios described in Equation 1 and 2;
  2. Utilising the relationships and rules described in Equations 5, 13, and 14; and
  3. Adopting a conceptual approach in the development of the dynamic relationships, through a process similar to a Newtonian thought experiment.

Key: IR - Industrial Revolution
Figure 2
A simple conceptualisation of the potential unmitigated and unmanaged relationship between the Environment (E) and Human Needs and Interests (HNI) in respect to the governing dynamics of the mathematical model of sustainable development, related to the coupled dynamics of the Earth System as outlined by Schellnhuber (1998, 1999, 2001) and using ideas of Thomas Malthus. The E / HNI label refers to values for both represented on the diagram.

Figure 2 would certainly suggest that a mode of assessing, implementing and managing sustainable development is required to correct the detrimental situation inferred. This would be achieved by implementing ‘a pool of management options’ (Schellnhuber, 1998, 1999, 2001) as proposed in Equation 4. This would include measures such as: Environmental Impact Assessment (EIA), Environmental Management System (EMS), Strategic Environmental Assessment (SEA), and Global Reporting Initiative (GRI). If this approach is adopted using a suitable quantitative approach, then the potential determination of E and HNI within a range of, which would lead to a determination of S using Equation 5. This would consequently suggest that sustainable development would be deemed to occur at a value greater than or equal to 0.001, using an accuracy of 3 decimal places.

The determination of the value of S is dependent upon the unit of measurement of the mode of assessment used to determine E, HNI and thus S. Consequently, the values of E, HNI and S are representative indicators of the original data used to allow the ability to determine mathematically the potential or actual level of sustainability that will or could be attained.

For example, in the application of the model to Environmental Impact Assessment (EIA) in the work of Phillips (2009, 2010), the model would use the ‘units’ prescribed by the EIA methodology used in order to prescribe and calculate values of E and HNI. The values of E and HNI in Phillips (2009), were derived from the obtained scores of weighted values of the parameters (the Battelle Environmental Evaluation System), and semi-quantitative totals of categories, based on values derived on data collected and professional judgement (the Rapid Impact Assessment Matrix).

Therefore the purpose of a S-value of 0.001 or greater, is that if an overall determined value of E was 0.500, and HNI was 0.499, then in respect to Equation 13, E is greater that HNI - then S is occur (just!!) because there enough E left to perhaps allow HNI to occur afterwards, if E was given time to recover. However, if E and HNI were both 0.500 - then in theory, nothing would be left for humans to use afterwards as S would 0.000 - in essence, take with one hand and give out with the other, and under the conditions detailed in Equation 14, this would constitute no sustainable development whatsoever. This is a very extreme example, however, Phillips (2009) has obtained values of S in respect to EIAs with very low S-values, indicated high environmental and human impact. Further, Phillips (2009) has also obtained values of HNI which have been greater than the obtained values for E, which with reference to Equation 14, no sustainable development occurring.

Therefore, the less impact that HNI does onto E, the greater the level and nature of sustainable development occurring, and reflects the notion of maintaining the highest possible natural capital in order to achieve sustainable development. In order to achieve this, the choice and implementation of the desired assessment and management option will be of importance, and requires careful judgement and expertise in balancing the delicate needs of the Earth System. Using the same approach as in the construction of Figure 2, it is possible to suggest a new dynamic relationship that exists to implement sustainable development as shown in Figure 3.

Key: IR - Industrial Revolution, EM - Environmental Movement (from late 1960s)
Figure 3
A simple conceptualisation of the revised potential relationship between the Environment (E) and Human Needs and Interests (HNI) in respect to the governing dynamics of the mathematical model of sustainable development, and related to the coupled dynamics of the Earth System as outlined by Schellnhuber (1998). The E / HNI label refers to values for both represented on the diagram. The emergence of the environmental consciousness and awareness, starting from the Environmental Movement, has developed the political, economic and social necessity and will to develop alternative paths for environment-human relationships through policy, law, assessment and new scientific understanding.

In the context of Earth System Analysis, Figure 3 suggests that the introduction of a pool of management options (M) in controlling and mitigation of the anthroposphere (A) results in numerous new potential pathways for sustainable development available following the emergence of the Global Subject (S). From the perspective of the model, the number of paths for E and HNI could be significantly greater or less those shown in Figure 3, dependent upon the potential options available at the time. The degree of success or failure of the chosen path for sustainable development is dependent upon numerous factors such as: strategy undertaken, the personnel employed, the available time and resources, progress in knowledge and technology, the social development in environmental awareness etc.

The management options will include: the obtaining and interpretation of data concerning the project or issue being assessed (e.g. the use of an EIA or indicators); developing appropriate strategies for achieving sustainable conduct (e.g. through a SEA); implementing the chosen strategy; the maintenance of the chosen strategy (e.g. through the use of an Environmental Management System such as ISO 14001); or change the strategy in light of assessment and performance.

4. Conclusion

It is fair and acceptable to state the fact that the mathematical model of sustainable development presented is still a somewhat work in progress.

Nevertheless, the primary issues concerning what sustainable development is, are more than adequately addressed within the model construct suggested. Primarily - the nature of the environment-human relationship, in relation to the use of and impact upon our planet at any spatial scale at any specified point in time. The potential quantification of sustainable development provides for a great possibility to use the ‘scientific medial complex’ (Schellnhuber, 1998,) and to utilise the pool of management options (M) to calculate and implement sustainable development.

However, the model does represents a contribution to the debate concerning sustainable development, as well as providing the opportunity to utilise the model in order to make an assessment of the level and nature sustainable development of a project / development. Such a model may be viewed with some scepticism. One potential criticism of the model could be that it is merely a re-invention of Earth System Analysis. This is not the case, as when the model was first developed in 2000/2001, the author was not aware at all of the existence of Earth System Analysis, and derived the model through independent research and considerable thought and reflection. The fact that the model has some strong correlations with Earth System Analysis provides validity for the integrity of the model and the ideas conveyed. Further, the model goes further somewhat than Schellnhuber, in the definition and role of humans within the Earth System, by defining and simplifying the human hierarchy and motivations for actions that may and do impact upon the environment.

It has been a long held view within sustainable development theory that it is not only about the use of resources, but also the impact that humans have upon the environment in terms of production, consumption and waste. Therefore, the model attempts to address these issues, and the factors behind them. This provides for the opportunity to re-evaluate the way humans interact with the environment at all spatial scales over time.


Glossary of Mathematical Terms:

1) Earth System Analysis

N - ‘Macro-state’ of the Ecosphere (N)
A - ‘Macro-state’ of the Anthroposphere (A)
t - Time variable (t = 0)
F0, G0, F1, G1, F2, G2 - Function
M - coherent voluntary strategies of management which are available to the “global subject” (S)
E - the Earth System contains the ecosphere N, and the human factor H.
N - represents the various sub-spheres that can be “spelled” alphabetically: a - atmosphere; b - biosphere; c - cryosphere etc.
H - represents the Human Factor comprised of:
A - The physical sub-components of anthroposphere: all of the individual lives, actions and products (e.g. pollution, waste, overpopulation, famine etc.); and
S - the “metaphysical” sub-component, reflecting the emergence of a ‘global subject’ (e.g. the Environmental Movement, environmental law, environmental politics, the ‘fear’ of climate change and its consequences)

2) Mathematical Model of Sustainable Development

S - Sustainable Development
E - Environment
HNI - Human Needs and Interests
t – Time
A - Atmosphere
B - Biosphere
H - Hydrosphere
L - Lithosphere
I - Individual
Comm - Community
Soc - Society
Sp - Species
NI - Needs and Interests
QL - Quality of Life
Ec - Economic
So - Social
BN - Basic Needs
Sh - Shelter
F - Food/ water
En - Energy
Rep - Reproduction of species
SD – Social Development
T – Technology
K – Knowledge
f – function of (mathematical notation)


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Phillips, J. (2009). The development and application of a geocybernetic model of sustainability. Unpublished PhD Thesis. Department of Geography (Streatham Campus) and Camborne School of Mines (Cornwall Campus), University of Exeter, United Kingdom.

Phillips, J. (2010). The advancement of a mathematical model of sustainability. Sustainability Science, 5(1):127-142.

Roberts, J. (2004). Environmental Policy. Routledge, London.

Schellnhuber, H-J. (1998) Part 1: Earth System Analysis - The Concept. In Schellnhuber, H-J. and Wenzel, V. (eds), Earth System Analysis: Integrating Science for Sustainable development, Springer-Verlag, Berlin Heidelberg, pp 3-195.

Schellnhuber, H-J. (1999). ‘Earth system’ analysis and the second Copernican revolution. Nature 402, Millennium Supplement, 2nd December 1999, pp.C19-23.

Schellnhuber, H-J. (2001). Earth System Analysis and Management. In Ehlers E and Kraft T (eds), Understanding the Earth System: Compartments, Processes and Interactions Springer-Verlag, Berlin Heidelberg, pp.17-55.

Schellnhuber, H-J., and Kropp, J. (1998) Geocybernetics: Controlling a Complex Dynamic System Under Uncertainty. Naturwissenschaften 85:411-425. Victor, P.A, Hanna J.E., and Kubursi, A. (1995) How strong is weak sustainability? Economic Applique 48:75-94.

© 2010 Jason Phillips

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Jason Phillips is a recent PhD graduate of the University of Exeter where he was under the supervision of Dr Stewart Barr (Geography) and Dr Patrick Foster (Camborne School of Mines). Jason’s PhD was titled: The Development and Application of a Geocybernetic Model of Sustainability, which was of his own conception and design. Jason’s past academic background is predominantly within the environmental area, holding a HND, BSc (Hons) and MSc in the fields of environmental monitoring and management, and is also a Master of the Camborne School of Mines. Jason’s main research interests are mainly in the fields of: Sustainable Development; Environmental Management & Assessment; Earth System Science; Sustainability Science; and Sustainable Mining. Jason is currently pursuing and actively seeking the development of a career in academia or industry in which he can develop the model further and its applications. He is presently submitting papers concerning and based on the PhD, which the first was published in Sustainability Science this year (see: Vol.5 (1), pp.127-142), and focused on the model developed during the PhD, and further extended the conceptual aspects of model in respect to the developments over time.

Jason Phillips can be contacted at jp1@tiscali.co.uk.


Integrated Research System for Sustainability Science (IR3S), Tokyo, Japan

Research Institute for Sustainability Science, Osaka University, Osaka, Japan

Sustainability Science Program, Center for International Development, Harvard University, USA

Potsdam Institute for Climate Impact Research, Potsman, Germany (Hans-Joachim Schellnhuber is Director of the Institute)

Quantifying and Understanding the Earth System (QUEST), University of Bristol, United Kingdom

Earth System Analysis for Sustainability (book)

Understanding the Earth System – Compartments, Processess and Interactions (book)

Sustainability or Collapse? - An Integrated History and Future of People on Earth (book)

"I saw God in myself
And I loved her ...
I loved her fiercely."

Ntozake Shange, USA (b. 1948)


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