Miniproject

Create and analyze a model of a hypothetical ecosystem. Your model should consist of 5 dynamic elements (at least), including detritus and nutrient (at least one) reservoirs. Include in the model a disturbance variable that varies with time due to an anthropogenic or natural disturbance. (The variation might be in the form of a step change from a steady reference value.) Your reported work should include a schematic diagram, a graph showing feasible steady states and their stability versus a selected parameter (e.g., the magnitude of a step disturbance) with all other parameter values fixed, and a simulation of the unsteady state for that same fixed set of parameters. Submit your work on Mathcad and Excel sheets.

Course Projects

For the latter part of the course each group will undertake a project. The list below gives a brief description of a number of possible themes. Aside from these, groups may propose a topic on their own, so long as it is commensurate with the subject matter of the course.

Whatever the theme of the project, the principal aim is to develop a mini-lecture that complements or extends a subject covered in class meetings or introduces a new subject. The groups will submit a report and present their lectures during regular class hours from April 19 through April 28. A specific schedule of presentations will be announced later. Each group will be given a 20-minute time slot.

In all cases the projects should include a mathematical model of the system or process being studied and the results of computer simulations. Insofar as possible, models should incorporate both biotic and abiotic components, but they need not be developed de novo. Models taken directly from the literature, or modified versions of those found in the literature, are acceptable.

(1) Greenhouse gases and global warming [3,7,8]. The increase in "greenhouse gases", mainly due to the increase in CO2 from fossil fuel burning, has led to concerns regarding the potential for global warming and climate change. While there is no question the concentrations of CO2 and other important "greenhouses" gases are increasing rapidly in the atmosphere, there is disagreement over whether significant global warming has or will occur as a consequence. Mathematical models that attempt to predict the effects track the greenhouse gases through the Earth's major reservoirs in response to increasing anthropogenic inputs. A global energy balance then can be applied to estimate the future changes in the average global temperature.

(2) Ozone dynamics in the troposphere and stratosphere [3,7,8]. Ozone occurs naturally in the stratosphere where it protects the Earth from harmful ultraviolet radiation. Its level is being depleted rapidly, however, owing mainly to the chemical effects of increasing nitrous oxide and chlorofluorocarbon (CFC) concentrations in the atmosphere. Increases in the former are due mainly to the greater fertilizer applications worldwide. Increases in the latter are due to industrial production and consumer use, until recently, of CFCs. Ozone in the troposphere, on the other hand, is a harmful pollutant, and its level is increasing due to increased levels of the oxides of nitrogen formed in combustion processes. Concentration profiles of ozone and other chemicals in the atmosphere can be studied and predicted by way of mathematical models based on transport and chemical reactions in the atmosphere -- i.e., by treating the atmosphere as a chemical reactor.

(3) Lake dynamics and eutrophication [7,8]. Lakes are generally not permanent features of landscapes. They slowly become filled with sediment, devoid of life, and eventually disappear. This process is accelerated by human/industrial activities that lead to eutrophication of lake waters. Eutrophication is defined as the enrichment of waters by inorganic plant nutrients (commonly nitrogen and phosphorus) that increase the production of algae. The sources include fertilizers entering the lakes in ever-greater quantities, due to agricultural activities, and animal and human wastes, due to increasing population levels. The consequence is a marked increase in organic detritus, caused by the seasonal death of algal populations, which leads to a major disturbance in the food web and the local extinction of some or all fish and invertebrates. Mathematical models, used to study these dynamics, account for nutrient input, plant and animal population dynamics, and other major components of lake ecosystems.

(4) Water acidification by acid rain [3,7,8]. The influence of acid rain has received considerable attention, not only because of changes in potable water quality and damage to exposed materials, but also because of changes in ecosystems owing to a reduction in the pH of aquatic environments. In addition, the problem is international in scope; the sources of the problem may be in one location (country), and the deleterious effects felt in another. The principal components responsible for acid rain are sulfur dioxide and the oxides of nitrogen, pollutants emitted to the atmosphere largely from fossil fuel combustion, automobile exhausts and metal-ore smelting. These components are oxidized in the atmosphere and deposited in acidic form on the earth's surface with rain and snow. The lowering of the pH of surface waters affects aquatic life directly and indirectly -- indirectly due to the release of toxic metal ions, such as aluminum, which otherwise exist in harmless form in lake sediments. A mathematical model would describe the dynamics of biotic and abiotic components of an aquatic ecosystem, including the dissolution reactions of metal compounds in sediments. Analysis and simulations would inquire into the effects of changes in the pH of incoming streams.

(5) The transport, fate and impact of toxic organic pollutants and trace metals-- e.g., in the Great Lakes system [8]. There are many examples for which the fundamentals of transport phenomena and chemical kinetics can be applied to construct mathematical models to track the fate of pollutants released into the environment. One example is that of polychlorinated biphenyls (PCBs), a persistent organic pollutant that has contaminated fresh water systems in recent decades.PCBs, produced by the chlorination of biphenyls, found widespread use for about 60 years before their impact on the environment was recognized. Their production is now severely restricted, but many of their applications resulted in direct or indirect release of PCBs into the environment where they have been slow to degrade and dissipate. A primary example is PCB persistence in the Great Lakes. PCBs bioaccumulate in fish, and are harmful (carcinogenic) to humans who consume the fish. A mathematical model in the literature describes the spread of this pollutant through the Great Lake system and the various chemical, physical and biological mechanisms by which it enters and leaves the environment. The model could be modified to describe the level of fish contamination and the dynamics by which that contamination is eventually reduced to safe levels.

(6) Contamination from landfills; methane production [7,8]. Methane is an important and relatively long-lived greenhouse gas. Its atmospheric concentration has tripled since pre-industrial times. Municipal landfills, where methane is generated by anaerobic bacterial decomposition of refuse, are believed to be among the major factors in accounting for the increase. A mathematical model, that would treat the landfill as a large biochemical reactor with transport to and from the atmosphere through a porous cover soil, would describe the rates of production of methane and other chemicals and the effect of conditions on those rates. An added consideration is the possible flaring or even commercial use of the methane to prevent its discharge to the atmosphere.

(7) Dynamics of the biogeochemical cycles of sulfur and phosphorus [3,7,8]. As with the cases described in class for carbon and nitrogen, the natural cycles of the nutrient elements sulfur and phosphorus are strongly impacted by human/industrial activities. These include the increased mining of phosphorus for the production of fertilizers and detergents and the production of sulfur dioxide by fossil fuel combustion. The ultimate consequences are the diminishing reservoir of phosphorus, acid rain, and the pollution of the Earth's water systems. Box (compartment) models can be used to simulate these cycles, to examine trends in the distribution of these essential nutrients through the Earth's reservoirs, and to predict future consequences.

(8) Extensions of our coverage of biota dynamics

(a) Populations with age and/or stage structure [1]. Mathematical models would be constructed to inquire into the effects of age or stage (egg, larva, pupa, adult) structure on the dynamics of single populations and of a few types of interacting populations. With such effects on birth and death rates incorporated into the governing equations, questions could be raised about their consequence in terms of population stability and general dynamic behavior -- including the outcomes of age- or stage-dependent disturbances (e.g., a temporary disturbance that selectively destroys elderly members, or newborn members, or eggs.)

(b) Spatial distribution of species populations [1]. None of the study models used in class accounted for the fact that species do not exist in spatially uniform environments in nature. The most common models for studying spatial effects are "patch" models, which represent spatial areas as boxes. Species may migrate among the boxes -- similar in mathematical form to the transport of heat between connected catalyst particles in a fixed bed reactor. (Spatial uniformity is approached if the migration rate is very high.) Such models for a few types of interacting populations might be used to examine the dynamics of distributed populations and to compare them with those predicted by spatially uniform models.

(c) Continuation and extension of the mini-project [6]. Models would be constructed of larger more complex ecosystems. For example, the schematic diagram of the ecosystem might include decomposers, nutrient cycling and some abiotic dynamics in a "complete" simulated world. Simulations involving such a model could then be studied for better insight into the impacts of anthropogenic disturbances. An alternate approach would be to build an ecosystem by starting with basal species and allowing other species, randomly selected one at a time from a pool of candidates, to attempt an invasion. The community is thus built of those species that can invade and survive. (A Matlab program showing a construction of this type is presented in the book by Roughgarden [6].)

(9) Dynamics of a chemostat [2,4]. The chemostat is a chemical reactor, often of the CSTR-type, designed to carry out biochemical reactions. Nutrients are continuously fed and products, produced by living organisms within the reactor, are continuously withdrawn. A model of the reactor dynamics, therefore, accounts for the growth and death of organisms and the biochemical reactions occurring within them. As such, they resemble models of open ecosystems, but with simpler food webs and specific purposes. Chemostats may exhibit all of the dynamics of interacting populations, including various types of instabilities and multiple steady states.

(10) Design of Biosphere 2 [5,9]. The subject "Biospherics" deals with ecosystems that are closed to material flows but open to energy inputs. Obviously, the Earth is a natural system of this type. An artificial one was constructed for the Biosphere 2 project. Biosphere 2, started in 1991, was an ecological experiment conducted in a sealed structure on 3.15 acres in Arizona. During a two-year initial experiment, all air, water and nutrient cycles were completely closed and recycled within the system while several thousand species (including humans) inhabited the structure. The main purpose was to provide data to help scientists better understand how our world works -- with the possible eventual application to the design of self-sustained habitats on other planets. Overall the project has had some failures and some successes. What went into the design? -- mathematical models and simulations?

References

1. Among many books that can be found through a library subject search on "population ecology" -- Gotelli, N.J. (1995) "A Primer of Ecology." (Life Sciences Library)

2. Bazin, M. J. (Editor) (1983), "Mathematics in Microbiology." Chapter by A. Cunningham and R. M. Nisbet, Transients and Oscillations in Continuous Culture, pages 77-103. (Life Sciences Library)

3. Butcher, S. S., R. J. Charlson, G. H. Orians and G. V. Wolf (editors) (1992), "Global Biogeochemical Cycles," Academic Press, New York. (Life Sciences Library)

4. Butler, G. J., S. B. Hsu and P. Waltman (1983) "Coexistence of Competing Predators in a Chemostat," Journal of Mathematical Biology, 17, 133-151. (Hesburgh Library)

5. International Journal of Life Support & Biosphere Science (1997) 4, 3/4. (RAS collection)

6. Roughgarden, J. (1998), "Primer of Ecological Theory," Prentice Hall, New Jersey. (RAS collection)

7. Schlesinger, W. H. (1997), "Biogeochemistry, An Analysis of Global Change," Academic Press, New York. (Life Sciences Library)

8. Schnoor, J. L. (1996), "Environmental Modeling, Fate and Transport of Pollutants in Water, Air, and Soil," John Wiley & Sons, New York (Engineering Library)

9. Web site: http://www.biospherics.org