INHS Reports July-August 1998

Ecological Numeracy

Ecological Numeracy is the ability to understand, interpret, and participate in the quantitative aspects of environmental debates and decision making. Acquiring ecological numeracy requires attentive thinking and dedicated practice, but not high-powered math. Skills that never go beyond one semester of calculus, and usually no more than advanced high school algebra, often allow one to slice through to the crux of a wide range of issues.

For several years, I have taught ecological numeracy to students of urban planning and natural resources and environmental sciences. Much of what we cover is summarized by four important equations, which I discuss below. I also provide an example of how we would use stock-flow thinking, related to equation 3, in the question of sequestering atmospheric CO2 by growing trees.

Four Important Equations:

1. I = PAT

The impact (I) of an anthropogenic activity can be considered the compounding of population (P), per capita consumption (A), and technology (T). Useful for reality checks: Can T be made small enough so that a doubled human population consuming at 4 times today's rate will produce only one half as much pollution? Equation 1 tells us that T must be reduced to 1/16 of today's value, i.e., that pollution control must become 16 times better.

2. 1+x+x²+x³+...x^n-1 = (1-xⁿ)/(1-x)

The sum of a geometric series, used to stress the impacts of exponential growth and depletion of resources, discounting in cost-benefit analysis, and new schemes to modify Gross National Product to reflect sustainability.

3. Fin - Fout = [[Delta]]M/[[Delta]]t

The dynamic relationship between the change of a stock (M) and the flows in and out (F) during a time period [[Delta]]t. Fundamental in understanding stocks and flows, pollution accumulation and dispersal, and population dynamics. Used in simulation modelling.

4.
The most complicated expression (it describes simultaneous linear equations) is a variation of input-output economics. Used to calculate indirect effects such as "sunlight in an eagle," the labor required to make a car, bioaccumulation of pesticides in food chains, trophic positions in food webs, and the pollution from (on-site) "clean" electric heat. A recent application is an "ecological footprint" --the size of area impacted by a nation's lifestyle. All but two of the industrialized nations (Canada, Australia) have a footprint larger than their actual area.

An example of stock-flow thinking:

Often one reads that an afforestation project will sequester X tons of atmospheric carbon dioxide per acre per year, with no mention of how long it can do this. The logistic curve (Fig. 1) approximates the buildup of fixed carbon (in the soil as well as in tree biomass) as a forest succeeds on previously bare ground. The graph shows the stock (in tons of carbon/acre), while the graph's slope is the uptake rate (a flow, in tons/acre-yr). For this example, uptake rate is maximum at around 50 years. Eventually the forest matures, its biomass levels out, and its net uptake rate goes to zero. At that point this forest has zero net effect on atmospheric CO2. To point to the value of X and not be explicit about its transient nature is irresponsible. A fruitful discussion requires that all participants agree where they are on the logistic curve.

Figure 1. Approximate biomass storage (expressed as carbon) in a successional temperate forest.

Three possible responses to this "saturation" problem are promoted, and each can be interpreted in terms of Figure 1.

1. Current research is asking whether increased CO2 concentrations will stimulate additional vegetation growth. In terms of Figure 1, we can ask if the eventual carbon storage (the asymptotic level) will increase, or just the rate of approach to it, that is, the slope.

2. To make a plot of ground useful in the long term for carbon storage, the trees could be cut and prevented from rotting or burning, and afforestation begun again. The average rate of carbon storage can be estimated from Figure 1, assuming a rotation time.

3. The trees could be burned in an application now using fossil fuel. A forest rotation for biomass fuel will have zero net long-term effect on atmospheric CO2 itself, but it will reduce overall emissions by displacing the original fossil fuel combustion. Again, calculations based on Figure 1 allow estimating this displacement.

Robert A. Herendeen, Center for Aquatic Ecology

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