Australian judges’ commentary for 2019


The International Mathematical Modeling Challenge continues to grow strongly in Australia, and the Australian IM2C judging panel congratulates all those who engaged with the IM2C this year.

The judging panel received 142 team submissions this year, from teams in 49 schools in the ACT, New South Wales, Queensland, Victoria and Western Australia. An additional 130 teams had registered to participate in the IM2C, but for a variety of reasons did not complete and submit a report.

This Judges’ Commentary aims to provide feedback to teams and teachers on their efforts with the 2019 IM2C problem, and provide advice to others who may wish to use the problem as part of their mathematics teaching and learning. We recognise that for many teachers and schools, mathematical modelling has not played a large part in mathematics instruction, but that there is a keen and growing interest among teachers in learning more about modelling. The resources available through the IM2C Australia website, including the consolidated Guide to Mathematical Modelling for Teachers and Teams as well as the growing collection of past IM2C problems and commentary from the national and international judging panels, may contribute to filling this gap.


The 2019 IM2C problem

Problem: What is the Earth’s carrying capacity for human life?

1. Identify and analyze the major factors that you consider crucial to limiting the Earth’s carrying capacity for human life under current conditions.

2. Use mathematical modeling to determine the current carrying capacity of the Earth for human life under today’s conditions and technology.

3. What can mankind realistically do to raise the carrying capacity of the Earth for human life in perceived or anticipated future conditions? What would those conditions be?

Note that IM2C is aware of available resources and references that address and discuss this question. It is not sufficient to simply re-present any of these models or discussions, even if properly cited. Any successful paper must include development and analysis of your model.

Your submission should consist of:

  • One-page Summary Sheet.
  • Your solution of no more than 20 pages, for a maximum of 21 pages with your summary.
  • A complete list of references with in-text citations.
  • Reference list and any appendices do not count toward the 21-page limit and should appear after your completed solution.


Carrying Capacity: The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water, and other necessities available in the environment.  

The 2019 IM2C problem asked teams to determine the current and possible future sustainable population of the Earth.


The following commentary is based on approaches observed in the Australian team submissions, and aspects of particular interest to the judging panel members.

Common approaches

A common approach was to consider a range of factors (eg, water availability, land availability, air quality and availability, food production, sometimes also energy and resource usage) one at a time, identify the environmental or resource factor that constrains population growth the most (ie, leads to the lowest maximum population possible, among the factors considered), and use the maximum population due to that factor in responding to the second part.

The panel looked for evidence that teams also considered treatment (amelioration) of phenomena of human behaviours that are not sustainable (eg, evidence of thought about replacing non-renewable energy sources – indeed this tended to be regarded as a threshold issue: population cannot be sustainable if it depends on unsustainable practices or resources, such as continued reliance on non-renewable fuels).


Two different approaches were observed to the notion of sustainable life. Some teams did attempt to define this and to look for a solution that maintains some semblance of current lifestyles; others went for the ’Spartan approach’ (adopting harsh, meagre living conditions) and estimated the minimum biological requirements for humans to stay alive. The panel generally took the view that the Spartan approach did not fit with the ‘current conditions’ part of the question statement.

However, this does raise the question as to how ‘current conditions’ should be defined and interpreted, and many different responses to that aspect of the question were noted. Most team responses assumed currently available technologies and building, transport and agricultural practices should be the bases of calculations.

Other observations related to the assessment criteria

1. Problem specification and model formulation

Mathematical modelling begins with a process that transforms a problem often expressed in terms of ‘real-world’ factors and information into a mathematical form. The real world problem has to be examined carefully in order to identify the type of solution sought (ie, what kind of answer would be acceptable for this question). The problem must then be specified in a mathematical form that can be analysed.

The ‘carrying capacity’ problem was expressed as a real-world challenge, and too few teams were able to translate this into one or more mathematical problems. The most common mathematical formulation was some version of an expression that represented the lowest maximum population levels calculated from among selected factors. Each constraining factor would set a maximum limit on the global human population (eg, the supply of food).

The main modelling opportunities were presented by the third part of the question, and virtually no team found a way to translate that part of the question into a mathematical form amenable to analysis and solution.

2. Variables considered and assumptions required by approaches taken

The variables chosen and the assumptions made by teams are critical features of a modelling task, particularly one as complex as the 2019 IM2C problem. In general the judging panel was happy to consider whatever assumptions teams specified, as long as these were explained and justified. In addition, teams really needed to consider how their solution and findings might have changed if some of their assumptions were to change.

Teams considered a wide range of variables: oxygen levels, global warming, water availability (for a variety of purposes including drinking, domestic, agricultural and industrial uses) and water replenishment, agricultural production, food and dietary needs, land usage (for a variety of purposes including housing, industry, agriculture, roads and other infrastructure), population density of different locations, energy sources and usage, and treatment of waste products.

A relatively straight-forward approach observed was for teams to identify several different variables, and make a choice about which ones to consider, giving reasons for their choice.

However, to model sustainable capacity by assuming all people can live in apartment towers constructed on a scale that is comparable to the world’s currently tallest building seemed far-fetched. Assumptions like that really do need to be explained and justified.

One example of an often unjustified statement related to a link between atmospheric oxygen levels and the global number of trees. Some teams recognised there are other significant sources of oxygen (such as from ocean-based organisms) and teams making big assumptions and statements about this without proper justification, and without considering other related variables, did not impress the judging panel.

The judging panel looked for evidence of a clear sight line from the biological/environmental constraints identified through to the policy implications. For example, consideration of a population policy or a resource management policy. Responses that assumed nations can make an overnight shift to sustainable practices were generally not well regarded. Panel members looked for at least some evidence of thought about not only what to achieve, but also how the goals might be reached.

Approaches to food consumption varied – ranging from assuming that a single food (eg, sweet potatoes) could be considered a satisfactory human diet; through to copious detail on dietary recommendations from a variety of seemingly reputable sources. The Australian teams tended to be very western in their food orientation, and did not display awareness of such issues from a global perspective. But the ‘sweet potatoes only’ approach is also not particularly acceptable, unless clearly identified as a kind of ‘average food’ that represents a complete diet. Consideration of some representative food or diet also opened an opportunity to consider how sensitive a team’s solution might be to different assumptions, or to variation in crop-growing or other environmental conditions that may suit the representative food but not necessarily other foods. For example, how might productivity and yield vary under much hotter or much wetter conditions? Very few responses considered water-based foods such as fish and other seafood or sea-based plants.

Many teams took the approach of defining some kind of Nirvana on Earth (ie, how much space, food, air, etc each person would need under the ideal conditions described) then scaling this up under an assumption that this Nirvana could be more or less replicated right across every suitable or available space on the planet. This was regarded as an over-simplistic and unrealistic approach, and so most instances of that approach were rejected in the initial stages of the judging process. However some elements of such an approach were observed in reports of some of the national finalists. That approach might have been a useful way to think about future sustainability, but this opportunity was not seriously taken up by any teams.

3. Mathematical treatments

In general, teams applied relatively simple, low-level mathematical treatments which suggested that teams at different levels could participate meaningfully. However, much of the work done was not really mathematical. It was typically descriptive and qualitative, rather than quantitative.

Some teams included a large number of graphs in the body of their report. Too often these were included without proper labels, and without adequate explanation or commentary. In general, graphs might be included to show a point, but multiple graphs of a similar kind would be better placed in an appendix. The same can be said about tables of data – these should have a clear purpose in relation to the models being developed and the arguments being advanced, and multiple tables of a similar kind are usually better placed in appendices rather than the body of the report.

A common observation related to the way significant figures were treated. It seems inappropriate to include large strings of decimal places in the large numbers used in work on this problem, implying a degree of precision that is just not realistic.

Similarly, the problem this year invited teams to consider a large range of factors that were often reported in source materials in a wide range of units. Teams frequently did not use consistent units to handle commonly understood variables, for example mixing imperial and metric units for no good reason. It was particularly problematic when teams included different units in the same calculation.

Some teams had an interesting way of treating the concept of ‘average’. Too often the average was not treated as a representative concept that included a range of states, which could be used to scale an approach up to a global context; rather it was seen as a fixed state that every individual human could adopt or had already achieved.

The concept of ‘equilibrium’ was raised in a number of reports, but its treatment was often rather confused. One approach assumed that equilibrium would be (or has been) achieved via some natural process, that would see global population oscillating around the sustainable figure. This also seemed a bit unrealistic without some indication of the mechanisms needed for achieving that state (including for example discontinuation of unsustainable practices).

4. Model evaluation

Evaluation of models developed was virtually non-existent in the 2019 team submissions. For example, there was virtually no attempt by most teams to consider how their findings might have changed were some of their assumptions to be modified, or if some of the input values they assumed changed. This is a factor that could have been a really important way to identify higher quality reports.

5. Report preparation and structure

The organisation of the report, and the quality of the summary are two of the areas looked at closely by the IM2C judges. Many teams decided that a fancy cover page, and a detailed table of contents were an essential part of their report. This was often seen by the judges as a way of using space, rather than contributing usefully to a report.

It was also frequently observed that teams had trouble including equations in their reports – sometimes these were very difficult to read and interpret, and some contained obvious errors.

Another point was the way references were used and cited. Whenever some external information of data was referred to in the report, the judges looked for proper citing of each information sources at the point in the report where the information was mentioned, together with the detailed references given at the end.

Some difficulties with the question statement

The problem statement was very open, and could be taken up quite productively at different levels (for example, by younger students). However it seems that something about the wording of the problem did not steer teams enough towards a mathematical modelling approach, even though the second part of the question specifically mentioned using mathematical modelling.

This was particularly evident in approaches taken to the third part. Most team responses used an essentially qualitative consideration of the possibilities for the future, with no real attempt to consider mathematical models for future population change.