Exponential Growth in the Light of Potato Beetle Growth in Northern Vermont
About fifteen years ago my former partner Lisa and I decided to make a major life change. We wondered what it would be like to move to a parcel of land, build our house from the woods, and grow our own food without hooking up to the electrical lines. Our wonder became reality after we purchased land in Northern Vermont. By the second year we had gravitated into the system of division of labor. An example of our system of division of labor was evident in our garden. Lisa was in charge of nurturing the seedlings, planting the seeds, and in general planning where everything would go. I was in charge of plowing and tilling the garden, obtaining the manure and maintaining the tools, truck and machinery needed to do the job. We would help each other out and offer advice but for the most part we kept to our own division of labor.
Lisa had started teaching at a local Waldorf school and needed to go away for a week of training. She was nervous about leaving me alone with the garden, but I assured her that everything would be fine. Certainly, anyone who could build the house and fix the machinery could watch the plants grow, pull weeds, and water. The day Lisa left she gave me last minute instructions including to keep a watch out for the potato beetle.
For those of you that have never seen a potato beetle they are something to behold. They start out as an orange egg mass on the undersurface of the potato leaf. The eggs hatch into soft bodied orange larvae with black spots. These larvae grow rapidly until reaching lengths of one half to three quarters of an inch. In this stage their main purpose in life is to consume the leaves of the potato plants. After the larvae stage they metamorphose into the beetle stage. In this stage of development, the outer body is hardened, it acquires wings, and its appearance is all together different from the larvae. The wings allow the beetle to be somewhat mobile permitting the distribution of eggs after mating.
Since Lisa and I decided not to use insecticides we needed to deal with the potato beetles manually. Almost all of the potato beetles were in the larval form with an occasional adult and egg mass. The larvae and the adult beetle were both dropped into a jar of kerosene and the egg masses were squished in between two leaves to avoid mess.
The morning after Lisa left I headed out to the garden. I weeded first then strolled over to the potato patch. There were a good number of beetles so I began dropping the larvae and the adult beetles into the jar of kerosene and squishing the egg masses between the potato leaves, I spent approximately an hour and decided to get the rest the next day.
I worked on the house the rest of the day and the next morning went out to the garden. The number of larvae increased dramatically from the previous day. Dropping them into the jar of kerosene was much too slow so a new procedure had to be developed. The potato leaf was wrapped around the potato beetle and squished. The leaf prevented the orange goop from the larvae from flying outward. There is only one word that could describe this event…disgusting, Well, all I could do is do my best and finish the next day.
That night Lisa called home. She had food poisoning and had to come home. She said she could drive half way so my neighbor Barry and I drove down to meet her and drive her the rest of the way. She was quite sick that night but felt a little better the next day. She asked how the garden was and I replied, “Great.”. I assured her that I had everything under control. Since Lisa was sick there was no time to attend the garden, but what’s one day.
Lisa and I went to the garden together the next day. From a distance the potato plants seemed to have an orange tint. Lisa and I walked over and with a silent look Lisa gazed into my eyes. Finally she spoke. “I thought you had everything under control.” I reiterated a lesson on exponential growth but she didn’t want to hear it. We started squishing and squishing. The method had evolved once again. The larvae were squished between our thumb and finger with orange goop flying everywhere. We had so much to do we couldn’t spend all our time squishing potato beetles. In the days to come our battle with the beetles came to an end. The potato beetles had won. Our potato plants were nothing but stubs with ant least ten beetles on each stalk looking for something to eat. What a site, nature’s version of Agent Orange.
Surveying the barren potato patch we decided to dig a few up to determine the damage. Believe or not there were potatoes. They were small compare to the previous year’s crop, but at least we had potatoes for the winter. Next year we’ll be ready for the exponential growth of the potato beetle.
What is exponential growth anyway? Starting out with a straight definition may be helpful. Since the value of a increases considerably with an increase in x, the term exponential growth is used loosely in statistics to refer to the very rapid growth in number of a quantity over a period of time. What all that means in regular words is that exponential growth starts out slow, almost unnoticeable, but with time the numbers begin to accumulate excessively. As seen with the potato beetle, at first the population seemed manageable, but over time became overwhelming. There are other nice examples of exponential growth.
For example, if a paper is folded once the thickness of the paper is twice that of the unfolded piece. If folded again the thickness increases to four times that of the original unfolded piece. The next fold would be eight times, the next sixteen and so on, A piece of paper cannot be physically folded more than six or seven times, but if this limitation is disregarded how many folds would it take to have a thickness be greater than the distance from the Earth to the moon? Lisa and I read the answer in disbelief.
It was voting day and after exercising our civic duty we decided to have a pizza for dinner.
We ordered the pizza and I asked for a pencil and paper. I began to multiply.
1) 2 X 1 = 2
2) 2 X 2 = 4
3) 2 X 4 = 8
4) 2 X 8 = 16
and so on…
39) 2 X 1,380,203,186,944 =2,760,406,373,888
40) 2 X 2,760,406,373,888 = 5,520,812,747,776
Number 40 represented the number of single thickness of paper as a result of all
the folds. A ream of paper is 500 sheets and measures about two inches. After dividing the 500 into the extremely big number I converted the thickness into miles. The information had been right after all. In 40 folds the thickness was equal to 2/3 the distance of the moon and with one more fold (41) the thickness would surpass the moon’s distance from the Earth.
There is another great example. In a far distant cold land called Vermont was a king. Being a moral upright king he was loved by all that he ruled. His son, who loved to play chess, just turned 15 and his father was already thinking of his sixteenth birthday because it was such a special birthday. He thought about what he could give him and decided that he would love a chess set. He searched throughout his kingdom and finally came to the hamlet of Troy where he found a peasant who could build the finest chess set. The darker pieces would be carved out of serpentine while the light colored pieces would be carved from soapstone. The square of the board would be inlayed using walnut and birdseye maple. It would take almost a year to make such a chess set. The king asked the peasant how much he wanted for his labor. The peasant told the king that he was a simple man. All he wanted was one grain of rice for the first square, two grains for the second square, four grains for the third square, eight grains for the forth, sixteen for the fifth and so on. The king thanked the peasant and left with his men shaking his head. “No wonder that man is just a peasant, all he wants is some rice for a year’s labor.” The year passed by and the king set his mathematicians to work to figure out exactly how much rice the king owed the peasant. The king had to fill his kingdom of Vermont fifty-nine feet high with rice to pay the peasant for his labor. The king was unable to pay the peasant so being such a moral upright king he handed his kingdom over to the peasant because it was clear the peasant understood exponential growth and the king did not.
Both of these examples, however, only consider open systems. In other words exponential growth is allowed to grow to infinity without any boundaries. What happens when physical boundaries are set? Let’s consider a pond with a pondweed starting to grow on the surface of the water. Let’s say that the weed’s surface area (the amount of area covered by the weed) doubles each day and through calculations it has been determined that the pond will be completely covered in 30 days. On what day will the pond be half covered? The answer is day 29. On the last day the area of the weed doubles and covers the other half. Proceeding backwards on day 28 only ¼ is covered, day 27 only 1/8 and on day 26 only 1/16.2
Imagine now the proud owner of this beautiful pond. July 4th is approaching and only 1/16 of the pond is covered by pond weed. The family had planned a four day vacation to celebrate the 4th of July. His time away would be short and he could deal with the pondweed when he returned from his four day vacation. Thinking back to the potato beetle I can definitely relate.
Outside of the potato beetle fiasco there are many examples of exponential growth. By taking a good hard look at these numbers we maybe able to fill in part of the environmental picture. The increase in human population is an example of exponential growth. Technicological advances have allowed human population to increase past the natural carrying capacity. Carrying capacity is the maximum permanently supportable load.3 Back in the era of hunting and food gathering, the human population was thought to have reached a maximum of about five million. The natural environment would not allow an increase due to the limitation of a food source. Added technology of agriculture permitted human population to grow to approximately 250 million at the time of the birth of Christ. Labor-intensive agriculture, like that of hunting and gathering, had its limitations. Enough food could not be grown along with other goods and services provided through labor to increase population without undermining society. Immigration to the New World along with the Industrial Revolution extended the limit to population growth. Just before these two events (1650 AD) the population was beginning to level off at about 500 million. The population had doubled in the period between the birth of Christ and 1650. By 1850 the population had doubled again with one billion inhabiting the Earth.4 The second billion was added between 1850 and 1930, a period of 80 years. In the next 30 years (1960) the third billion appeared and by 1975 four billion humans, in 1992 there were five billion and now there are over six billion human beings walking this Earth.5
Thomas Malthus in 1830 published an essay, “A Summary View of the Principle of Population”. In this famous essay human exponential growth was introduced for the first time as a concept. He felt that both plants and animals would increase in a geometrical progression if there were no obstacles. Humans, according to Malthus, were no different. Man would not only increase, but even with superior intelligence would eventually reach the limiting factor of food requirement. If the population increased, the land needed to feed the populace would also have to increase. Since there is only a fixed amount of farmable land on Earth, as population increases arable land becomes the limiting factor.6
Back in my college years (1972) I enrolled in an ecology course. The class was very informal. New ideas, at least new to me, were introduced and discussed such as human growth. Another new idea was global warming. Green house gasses especially carbon dioxide were being investigated by the top environmental scientists. David Charles Keeling had been keeping track of atmospheric carbon dioxide concentration since 1950 and noticed an increase from year to year. Atmospheric carbon dioxide concentration is measured in parts per million (ppm) and in 1950 was about 310 ppm. As Keeling took his measurements he found it to increase to 315 ppm in 1957. The concentration kept right on climbing to 330. ppm in 1975, 345 ppm in 1985, 360 ppm in 1995, and 367 ppm in 1999. Methane another greenhouse gas holds about 25 times more heat than carbon dioxide and like carbon dioxide is steadily on the rise. Atmospheric methane measured in parts per billion and went from 1,147 ppb in 1950 to 1,725 ppb in 1998.7 Scientists have drill cylinders of ice called ice core samples. In these samples atmospheric gas concentrations from the past are trapped. Data from this process has shown that before the Industrial Revolution was in high gear (1850-1860) atmospheric carbon dioxide concentration was about 285ppm and methane concentration was 791 ppb.8 As one can see there is an increase in concentration of these gases from year to year.
When we compare the present rates of deforestation, desertification, number of fish caught per year, consumption of fossil fuels, amount the insurance companies are paying out for disasters, number of deaths due to terrorism with past rates we can observe an exponential growth in the increases from year to year. In the 1990’s these numbers increased even faster. In 1950 there were only 60 computers and along with all the other consumer products, computers increased in numbers exponentially. By 1959 there were 6000 computers, 1966-15000, and in 1970-80,000. According to Gartner Dataquest, a research firm, one billion personal computers were sold between 1975 and 2001. It is estimated that a second billion will be sold in the next five or six years.9 The 1990’s brought many technicalogical advances and the computers were connected world wide with the World Wide Web. This advancement has made it possible to find resources, extract resources, transport resources, transform resources into products, distribute and sell products, and dispose of waste at lightning speed.
What can we do with all this information? Like the king of Vermont, are we throwing our kingdom away because we lack the understanding of exponential growth? Alone exponential growth only paints part of the environmental picture. There are other environmental concepts that can fill out this painting. Could it be human destiny to find the right relationship to the material world? Could it be humanities future to penetrate these dynamic concepts? From today’s perspective this seems quite limiting, but our challenge is to view from other perspectives that may not only dissolve those limits away but provide unlimited growth.
 Donella H. Meadows, Dennis L. Meadows, and Jorgen Randers, Beyond the Limits, (Post Mills, Vt.: Chelsea Publishing Company, 1992), p. 15.
2 Paul R. Ehrlich and Anne H. Ehrlich, Population Explosion, (Simon and Schuster, 1990), p.15.
3 William R. Catton Jr., Overshoot: The Ecological Basis of Revolutionary Change, (Urbana and Chicago: University of Illinois Press, 1980), p. 4.
4Paul R. Ehrlich and Anne H. Ehrlich, The Population Explosion, (Simon and Schuster, 1990, p. 11-12.
5 Jeremy Rifkin with Ted Howard, Entropy: Into the Greenhouse World, (N.Y., Bantam Books, 1989), p. 118.
6 Three Essays On Population, (N.Y.: Mentor Books, 1964), Thomas Malthus, “A Summary View of the Principle of Population”, 1830, p. 15.
7 Natalie Goldstein, Earth Almanac: An Annual Geophysical Review of the State of the Planet, (Oryx Press, Westport, 2002), p. 122.
8 Natalie Goldstein, Earth Almanac: An Annual Geophysical Review of the State of the Planet, (Oryx Press, Westport, 2002), p. 151.
9 The World Almanac and Book of Facts 2003, World Almanac Books, 2001, p. 704.