SustainabilityCan mathematics help us understand issues of sustainability and make it possible to realize enjoyable lives for all the people who share this planet, now and far into the future?...
Joseph Malkevitch
Introduction
Life gives birth to new lifeplants generate new plants, fish new fish, and animals their progeny. When these life forms have economic value to mankind as the food we eat or as biofuels, it is tempting to try to arrange the management of, say, fish, so that we always have an adequate renewable supply. Why catch halibut to the point where the remaining stock of fish will die out? This is one notion of sustainability. Harvest some halibut but leave enough so that these remaining halibut can reproduce in great enough numbers so that forever into the future one can catch halibut. Similar issues arise for cutting down trees but reforesting so that there will be new trees for the future uses that mankind may put them to (wood for construction and paper). Some of these potentially renewable resources reproduce and grow relatively quickly, but others are relatively slow growing, like trees. Population growth
The philosopher and economist Thomas Robert Malthus (17661834) famously raised the issue of the relationship between the growth of human populations and the ability of the earth and mankind's skill in pursuit of agriculture to provide enough food for the earth's human population. Portrait of Malthus (Courtesy of Wikipedia)
While there have been some noteworthy periods of famine where large groups of people died, on the whole farmers and others who support feeding mankind have met the challenge of keeping up with the world's growing population. This does not mean that there are not many people who die regularly from malnutrition and its consequences, and that many, including children, don't get enough food regularly and perceive themselves as hungry regularly. Most of the world's population eat relatively little meat and using the earth's resources to produce meat is a relatively inefficient way of feeding the world. Courtesy of Wikipedia While Malthus and others question the ability of the earth to provide food for the world's growing population, through the use of technology, biology and traditional knowledge of agriculture, so far no "catastrophe" has occurred. Yet, despite the dramatic gains for increased productivity for specific crops, many people still see what is happening as a cat and mouse game between demand for specific crops and the ability of the planet to meet those demands. The chart below shows the wheat yield in developing countries for a 55year period.
(Courtesy of the Food and Agriculture Organization of the United Nations)
A somewhat different look at the situation for wheat is given by this table of data from the United States which shows a generally growing yield of bushels of wheat per acre but also surprisingly large downturns and upturns over short periods of time.
(Courtesy of the Economic Research Service of the United States Department of Agriculture)
Exploitable yield gaps for wheat: actual versus obtainable yield:
(Courtesy of the Food and Agriculture Organization of the United Nations)
Offsetting some of this encouraging data is that the price of grains and food in world markets has shown a generally upward trend, putting strain on people in all countries to feed themselves without otherwise endangering their standard of living. (Courtesy of the Food and Agriculture Organization (UN))
Mathematical models of population
A mathematical model involves finding ways to simplify the complex realities one might see and representing these simplified issues in mathematical terms. For example, what affects the number of people on the earth? It is affected by birth rates, death rates, weather or volcanic catastrophes that might cause a large number of people to die, deaths due to wars, or largescale epidemics such as the flu epidemic of 1918, etc. However, one might simplify the situation by saying that "the more people there are now, the more people there will be in the near future." Using discrete mathematical methods, if we know that the number of people (population) at time t is N_{t} we can express the population in the next time period N_{t+1} using the following recursion equation. Here I will use either N(t) or N_{t} to represent the number of people at time t. Time for the moment will be thought of as taking on discrete time values rather than continuous ones. In a recursion equation one will not always get integers but we can still work with the "predictions" made by thinking of numbers which are not integers as being rounded up to the next largest integer. So here is an equation which expresses the N(t+1) in terms of prior values of N(t).
In words this equation says that the population in the next time period (next year, say) is the current population added to the growth (which could be negative) in population for one year. This growth is given by computing the growth rate r (a constant) times the population. So if r = 3% = .03 and the current population (N_{0}) is 100,000, the growth in a year is equal to .03(100,000) or 3000 people. So, in the next time step the population would 103,000 people. It is worth noting that from a mathematical point of view this is exactly the same equation that would be used to model the growth of a bank account that pays r percent interest per time period, and where there were no withdrawals or additions(besdies interest) to the account.
where f(0) = 0 and f(1) = 1.
While to fully understand what is going on with (**) one needs to study calculus, this is not the case for (*). For (**), if the initial population at time 0 is N_{0} then the value of the population at any time will be N_{0} times an exponential function. Both the differential equation and difference equation models predict that as time goes on, the population will get bigger and bigger and never stop. This is unrealistic, so what a mathematical modeler (someone who is using mathematics to get insight into the way the world behaves) does is to simplify the situation at hand in a different way. One way to do this is to assume that at some point the growth rate of population will not be constant but will reflect the size of the population itself. After all, in nature one sees some situations where the growth of a population will be "cyclical" and others where the population grows but stays in the vicinity of some value. Fisheries
Since ancient times the oceans have been a major source of food for mankind. Boats were built and nets sewn for the purpose of catching fish, and some cultures have been heavily dependent on the harvesting of what for a long time was thought to be the inexhaustible supply of sea life that the oceans could provide. However, the ever growing demand for fresh fish and the efficiency of modern fishing boats soon caused certain fisheries that seemed to have an unending supply of fish to collapse. An example of what seemed initially to be "let the good times roll," is shown in what happened to the stock of Atlantic cod in the fishery off Newfoundland. Courtesy of Wikipedia
A similar situation arose with regard to the harvesting of Atlantic herring. (Courtesy of the National Oceanic and Atmospheric Administration)
Does mathematics offer insight? The answer is yes, but we must cope with the complexities of using models that are often developed for understanding one phenomenon in situations where many phenomena are at work. First, mature fish, unlike mature people, come in a very broad range of sizes. For example, Atlantic tuna can be as large as 4.6 meters with a weight of 680 kg while an Atlantic herring maxs out at 45 cm and a weight of 1 kg. It would be helpful to have a model for relating length and weight in fish, and mathematics obliges:
where W is weight, L is length, and a and b are constants. For many species b is about 3 but the constant a varies a lot from species to species. Second, along with the size issue come issues about length of life, age to reach reproductive maturity, and public taste in what fish they most want to see on their plates. Third, there are traditions of fishing particular types of fish in different countries and in different fishing grounds. The fact that there are so many variables to consider has challenged modelers to find ways to "manage" fisheries so that the data needed to construct the model remain valid by the time the model's conclusions can be implemented in the "real world" so as to obtain the best results for all concerned. This is another big issue for fisheries models. The interests of a specific group of fishermen may not coincide with the interests of the whole nation they are a part of, and even less likely to the interests of fishermen in other countries. Many countries have no coastlines, no fisheries fleets, and yet they have an interest in the dynamism of the fisheries that provide catch for their citizens. When a group of people who are accustomed to making their living in a particular way have that pattern altered or when people are unable to feed themselves in the way that they prefer, it often will set off chains of events that can have far reaching, sometimes negative, consequences. Courtesy of Wikipedia
The same approach used by Verhulst was later rediscovered by Raymond Pearl and Lowell Reed, both with ties to biology and Johns Hopkins University.
In words it says that the instantaneous rate of change of the number of fish N is given by the product of two firstdegree terms. The function that solves this differential equation is known as the logistic equation. Its solution function is known as the logistic function. Its graph is shown in Figure 1. In the equation above, when the initial number of fish N_{0} starts close to the carrying capacity K, the exponential term in the denominator is being multiplied by a number close to 0 so the value of N does not grow much. Also, when t = 0 the value of N(t) is N_{0} as it should be.
Figure 1 (Logistic curve) (Image courtesy of Wikipedia)
In order to understand what is going on here it is also helpful if we plot the righthand side of the equation (***) along the horizontal axis, and the rate of change of the population along the vertical axis. We get the diagram below (Figure 2), which is a parabola.
Figure 2 (Image courtesy of Wikipedia)
The point in Figure 2 at the top of the parabola is labeled MSY, for maximum sustainable yield. Clearly, K/2 gives the maximum value for the population growth. If the fish population is at half the carrying capacity K, this model predicts that this is when the largest rate of growth is occurring. If one could remove this exact amount of "growth" of fish, then the population would still be the same size and continue to grow at this maximum rate of growth. Harvesting
The step of adding the harvesting of fish whose population growth is being governed by a logistic growth model is usually credited to H. Scott Gordon and Milner Baily Schaefer. Schaefer was born in 1912. He died at 57, having worked for the Washington State Fisheries (19371942) and later for the United States Fish and Wildlife Service. H. Scott Gordon appears to have developed his work on fisheries independently of Schaefer. Today their work is often referred to as the GordonSchaefer model. The appearance of the book Mathematical Bioeconomics: The Optimal Management of Renewable Resources by Colin W. Clark helped bring broader interest in this model, as well as bring the issue of the management of renewable resources to a much larger audience.
Figure 3 (Courtesy of Wikiedia)
Let H (for harvest) represent the amount of catch that one wants to remove from the fishery. H can be thought of as a horizontal line. The situation shown in Figure 2 has been redrawn in Figure 3 where several different levels of harvesting are indicated by different horizontal lines, showing three cases of interest: when the amount of fish removed is exactly at the height of the curve when the population is K/2 (the maximum sustainable yield, MSY), a line below that level, and a line above that level. If one could succeed in harvesting at exactly the situation where a horizontal line meets the growth curve at a single point, one could be harvesting the largest potential number of fish possible. However, the chance that one is operating at exactly this ideal level is not that likely; it is likely one will have slipped into one of the other two cases. In one of these cases one has a line which intersects the growth curve in two points. Neither of these two levels of harvesting are "stable" or "an equilibrium" in the sense that one can continue to harvest at these rates forever without any change in the population level of the fishery. The only equilibrium value is the maximum sustainable yield level, when H = K/2. However, merely in practical terms, harvesting at exactly this level is not very likely. If one harvests above this level, as indicated by the fact that the horizontal line does not intersect the population growth curve, the harvesting cannot be sustained. Eventually if this level of harvesting were to continue, though it might not be initially apparent, so many fish will be removed from the system that the ability of the fish stock to recover might be destroyed. There is probably a level of "overfishing" which reduces the population of fish to a level where recovery might actually be impossible for some species, causing species extinction.
One can draw a similar diagram to Figure 3 to analyze what happens by drawing a line through the origin of slope E (E for effort) instead of a horizontal line. The shape of the growth curve used in the differential equation above and shown in Figure 3 is very symmetrical. One can do the analysis here in a more qualitative way, by using nonparabolic shapes for the population growth curves. While the combining of economics and biological growth considerations are suggestive of issues that fisheries management experts must take into account, and that make a very interesting "playground" for people who are learning about mathematical modeling issues, they only hint at the complexities of the issues involved. The reason for this involves other ways of thinking about the whole system, before simplification, of managing a fishery. When a fisherman or group of fishermen invest in a very expensive new boat that meets the requirements that government may place on them, they have a strong incentive to use their boat to maximize their income. There is also the fact that when total income from fishing does not meet the needs of the individual fisherman whose life's work is fishing, boats may be used to increase income in some aspect of fishing that is not "regulated." Sometimes when governments see fishermen hurting and are nervous that people with expertise will leave the profession because in the short run they cannot make a living, the fishermen will be subsidized in a way that makes for additional complexities. Courtesy of Wikipedia (Vertical scale in millions of metric tons)
When modeling natural fisheries, especially their economics, one increasingly has to take into account the effects of aquaculture on what happens with regard to wild capture. Garett Hardin (Courtesy of Wikipedia)
Hardin's worked resonated with people who were interested in game theory. His ideas related to situations where seemingly "rational" behavior resulted in results that were bad for society (which often did not show up explicitly in the model of the game at hand) and for the individual participants in the game. With regard to ocean fisheries his work set in motion a more nuanced look at the reasons why there were fishery collapses and how to prevent future collapses. One approach to doing this is to limit access to "common resource" fisheries, to charge taxes of various kinds for being able to fish in these "common resource" fisheries, and buyback programs where governments limit catch by purchasing "excess capacity" that exists in the fisheries industry. The intrinsic complexities and game theory considerations have to some extent stymied experts in designing systems that meet the needs of both fishermen and different societies in both the developed and developing world. References
Andelson, R. (Ed.), Commons Without Tragedy, ShepheardWalwyn, London, 1991.
Joseph Malkevitch The AMS encourages your comments, and hopes you will join the discussions. We review comments before they're posted, and those that are offensive, abusive, offtopic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

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