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# Modeling Food Systems

As applied mathematicians and computational scientists, we don’t normally think of food systems as a potential research topic. Yet as explained in the accompanying article about Molly Jahn’s (University of Wisconsin) public lecture at the 2016 SIAM Conference on Mathematics of Planet Earth, many questions related to food systems and food security can challenge our mathematical and computational skills.1

Computational models of food systems are essentially economic models (i.e., input-output models, computable general equilibrium models). These are process models, accounting for as many actors and processes as possible to simulate actual food systems. They are carefully calibrated to match available data and designed to find equilibrium states subject to constraints, by minimizing costs or energy, for example.

From a mathematical perspective, food systems are complex systems. They have their own internal dynamics, subject to external forces and stochastic variations; in that sense, they are similar to Earth’s climate system. But there are several significant differences. While climate processes are governed by the laws of physics and chemistry, the processes that make up the food system don’t seem to follow any such laws, except possibly the law of supply and demand. Additionally, agents in the food system make choices based on cultural and societal norms, which are difficult to capture in mathematical terms. Lastly, conceptual models—highly simplified models that focus on a particular phenomenon, with just enough detail to identify critical parameters or highlight underlying mechanisms—are lacking for food systems.

Conceptual models play an important role in climate science and are ideal for mathematical analysis. For example, simple energy balance models allow one to easily show that the climate system can have multiple equilibria and transit from one equilibrium state to another, or that the release of increasing amounts of greenhouse gases into the atmosphere leads to global warming and possibly irreversible phenomena. In this article, we describe a recent proposal for a safe and just space for humanity, which could provide a blueprint for the development of conceptual models for food systems.

Food systems and food security are among topics of interest to the SIAM Activity Group on Mathematics of Planet Earth (SIAG/MPE). The American Institute for Mathematics hosted an exploratory workshop in San Jose, Calif., in April 2015 [5, 6], and a follow-up workshop occurred at the University of Oxford in July 2016. In addition to mathematicians, attendees of both workshops included food and nutrition specialists, social scientists, sustainability experts, food industry representatives, economists, and data specialists, thus emphasizing the subject’s multidisciplinary attributes. A tutorial on food systems and food security took place at the SIAG/MPE inaugural meeting in Philadelphia, Pa., in September 2016. Some of the material presented here originated at these events.

### A Safe and Just Space for Humanity

Food systems impact both natural and societal wellbeing. What does it take to achieve a balance between biodiversity and sustainability on the one hand, and fairness and social justice on the other?

Planetary Boundaries. In 2009, a group of Earth system and environmental scientists proposed measuring stress to the Earth system in terms of planetary boundaries [10]. They suggested nine such boundaries and presented measurable control variables (indicators) for seven of them. Their proposal led to the concept of a safe operating space for human existence on the planet. Exceeding a critical value of a control variable would risk triggering abrupt or irreversible environmental changes—a tipping point in the parlance of dynamical systems. Table 1 lists the boundaries and their indicators, while Figure 1 shows the current status of the indicators for seven of the planetary boundaries.

Table 1. Planetary boundaries and their control variables.

Given our limited understanding of the fundamental processes controlling each planetary boundary, one could argue that it is impossible to present reasonable numbers, or the borders are much more malleable than the boundaries suggest, or, with better or worse management, boundaries could be moved. The concept of planetary boundaries, however, is now generally accepted and has since been adopted. For example, the United Nations (UN) utilizes it for ecosystem management and environmental governance. Expert commentaries are available in [1, 2, 3, 7, 8, 11, 12], [13] provides an update to the original proposal, and [4] offers a scholarly discussion of boundaries and indicators.

Figure 1. Estimated status of the control variables for seven of the planetary boundaries. Image courtesy of [13].

Social Boundaries. Planetary boundaries represent the existence of biophysical and ecological constraints to the Earth system. They define an environmental ceiling, beyond which lie unacceptable degradation and potential tipping points. Similarly, there exist generally-accepted social priorities, which imply unacceptable human deprivation if unmet. A set of 11 priorities, together with their indicators, was proposed to guide discussion at the 2012 UN Conference on Sustainable Development (Rio+20). Table 2 lists the boundaries and their indicators. Together, the social boundaries define a “social foundation” for a just operating space for humanity.

Table 2. Social boundaries and their control variables.

Figure 2 presents estimates of the current status of indicators for eight of the social boundaries. The indicators are measured from the center, and social justice is achieved when all sectors reach the outer green boundary. The orange sectors indicate not only that we are falling short on social justice at the global level, but also that significant discrepancies exist among the various indicators.

Figure 2. Estimated status of the control variables for eight of the social boundaries. Image courtesy of [9].

Doughnut Economics. By placing the social boundaries inside the planetary boundaries, Kate Raworth [2] presented a visual representation of an environmentally safe and socially just space for humanity (see Figure 3). In this space, inclusive and sustainable economic development is possible. If we imagine this image in three dimensions, we have a torus, which is reminiscent of a doughnut; hence, the term doughnut economics.

Figure 3. A safe and just space for humanity. Image courtesy of [9].

The proposition of a safe and just space for humanity provides an organizing principle for a formal approach to sustainable development, and the concept of doughnut economics can serve as a blueprint for the creation of mathematical models. Though one must adapt the blueprint to the particular question under study, it provides a framework for both qualitative and quantitative analysis that includes the human value system. Here are some preliminary ideas.

Blueprint for Mathematical Modeling. Planetary and social boundaries are essentially degrees of freedom that determine the state of the socio-economic system in a multidimensional state space. If the number of degrees of freedom is $$n$$, then the state space is $$X = \mathbb{R}^n$$. A control variable quantifies each boundary, so there is a metric in $$X$$.

The state vector $$x(t) \in X$$ represents the state of the system at time $$t$$. In addition, there may be a vector $$\lambda$$ of parameters, which affect the state of the system but are not affected by the state. If the system is autonomous, the state vector’s rate of change is a (generally nonlinear) function $$f$$ of the state vector and the vector of parameters,  so a differential equation of the form $$\dot{x} = f (\lambda, x)$$ in $$X$$ governs the dynamics of the system. Assuming that not all relationships are equally important and not all variables change on the same time scale, it may be possible to apply dimension reduction techniques and thus design novel models that help us understand the dynamics of food systems.

While all this is standard fare for modelers, actually constructing the function $$f$$ is the major research challenge. This function incorporates the relationships between the state variables and the parameters. In the context of food systems, these relationships are not given as clean mathematical expressions; we must infer them from sources with which we as mathematicians are not necessarily familiar. With some luck, numerical data sets or statistical correlations may define these relationships. But more often they must be teased out of such ephemeral impressions as “variable $$x_1$$ is strongly influenced by variable $$x_2$$.” The challenge is then to translate these impressions into the language of mathematics – a messy process with many opportunities for misunderstandings and false starts. But it is a challenge we cannot ignore. The stakes are too high; we must reach out to all stakeholders and initiate the dialogue.

1 In "Assessing Risks to Global Food Security," Jim Case recaps Molly Jahn's lecture and discusses the vulnerability of globalized and regionalized food systems.

References
[1] Allen, M. (2009). Planetary boundaries: Tangible targets are critical. Nature Reports: Climate Change, 114-115.
[2] Bass, S. (2009). Planetary boundaries: Keep off the grass. Nature Reports: Climate Change, 113-114.
[3] Brewer, P. (2009). Planetary boundaries: Consider all consequences. Nature Reports: Climate Change, 117-118.
[4] Garver, J., & Goldberg, M.S. (2015). Boundaries and Indicators: Conceptualizing and Measuring Progress Toward an Economy of Right Relationship Constrained by Global Economic Limits. In P.G. Brown & P. Timmerman (Eds.), Ecological Economics for the Anthropocene: An Emerging Paradigm (pp. 149-190). New York, NY: Columbia University Press.
[5] Ingram, J., & Zeeman, M.L. (2015). Multiscale modeling of the food system: Workshop summary. In Multiscale Modeling of the Food System. Palo Alto, CA: American Institute of Mathematics.
[6] Kaper, H. (2015, October 1). Is There a Role for Mathematics in Food Security? SIAM News, 48(8), 1-2.
[7] Molden, D. (2009). Planetary boundaries: The devil is in the detail. Nature Reports: Climate Change, 116-117.
[8] Molina, M.J. (2009). Planetary boundaries: Identifying abrupt change. Nature Reports: Climate Change, 115-116.
[9] Raworth, K. (2012). A safe and just space for humanity: can we live within the doughnut? Oxfam Policy and Practice: Climate Change and Resilience (pp. 1-26).
[10] Rockström, J., Steffen, W., Noone, K., Persson, Å.,Chapin, F.S., Lambin, E.F.,…Foley, J.A. (2009). A safe operating space for humanity. Nature, 461, 472-475.
[11] Samper, C. (2009). Planetary boundaries: Rethinking biodiversity. Nature Reports: Climate Change, 118-119.
[12] Schlesinger, W.H. (2009). Planetary boundaries: Thresholds risk prolonged degradation. Nature Reports: Climate Change, 112-113.
[13] Steffen, W., Richardson, K., Rockström, J., Cornell, S.E., Fetzer, I., Bennett, E.M….Sörlin, S. (2015). Planetary boundaries: Guiding human development on a changing planet. Science, 347, 1259855.

Hans Kaper, founding chair of SIAG/MPE and editor-in-chief of SIAM News, is an adjunct professor of mathematics at Georgetown University. Mary Lou Zeeman, founding vice-chair of SIAG/MPE and co-director of the Mathematics and Climate Research Network, is the Wells Johnson Professor of Mathematics at Bowdoin College.