STEP@USF

The bends or decompression sickness in SCUBA divers is linked to fast ascents from depth. When ascending, the ambient pressure is reduced (for approximately every 33ft underwater another atmosphere of pressure is added), which causes absorbed gasses to come back out of solution and form micro-bubbles in the blood. The air consumed by SCUBA divers is pressured and consumed at depth, so as divers ascend the micro-bubbles in the blood will also start to grow larger in volume (i.e. gasses expand when pressure is reduced). If the ascent is slow enough the volume of the bubbles should not rise too high and they can safely leave the body through the lungs, as divers continuously exhale on ascent. Large bubbles of gas can impede the flow of blood to the brain, central nervous system and other vital organs. Create a model that describes the movement of gasses out of solution and back into the blood and then ultimately out of the body as a SCUBA diver ascends. Using your model, determine the maximum ascent rate from a depth of your choosing that would ensure a SCUBA diver would not get the bends. What would your model look like if the diver did not exhale on ascent? Why do deep diving mammals, like seals, dolphins, and whales, not suffer from the bends?

Theory:

• Compartment models (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 587; Stephen P. and John Guckenheimer. Dynamic Models in Biology. New Jersey: Princeton University Press, 2006. 7)
• Systems of differential equations (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 560)

References:

Compare decay rates in bacterial populations when more than one population is present.

Theory:

• Tilman’s model for resource competition (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 98)
• Derivates of exponential functions (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 131)
• Exponential population growth (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 358)

References:

• Kayser, H. 1979. Growth Interactions between Marine Dinoflagellates in Multispecies Culture Experiments. Marine Biology. 52(4): 357-369.
• Bokhamy, M. et al. 1997. Survival and activity of Comamonas testosteroni in mixed population. Water Research. 31(11): 2802-2810.
• Palmai, M. and G. Kisko 2003. Studies on the growth of Listeria monocytogenes and Lactobacillus casei in mixed cultures. Acta Alimentaria. 32(1): 103-111.

At Superfund sites there is a concern that chemical toxins, for example, mercury, chromium, PCB and carcinogenic hydrocarbons, will cause contamination within the aquifers (groundwater bearing soils). Model the rate of import and export of the toxin of your choice into and out of the aquifer. How long would it take for the toxin to be removed from aquifer in a situation where there is a high rate import of the toxin into the aquifer (e.g. a rainy year causes excess runoff)? Would the water be drinkable in a single human lifetime?

Theory:

• Compartment models (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 587; Stephen P. and John Guckenheimer. Dynamic Models in Biology. New Jersey: Princeton University Press, 2006. 7)
• Systems of differential equations (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 560)

References:

In most countries alternative fuels, such as ethanol, are actually a mixture of ethanol and gasoline. These ethanol fuel mixtures have “E” numbers which describe the percentage of ethanol in the mixture by volume. For example, in the US many ethanol fuel mixtures are designated E10, which means 10% of the fuel is composed of Ethanol and the remaining 90% is composed of gasoline. Determine how efficient, in terms of gallons of gasoline consumed per mile, E10 is when compared to gasoline. As the ratio of ethanol to gasoline in an ethanol fuel mixture is increased determine how the efficiency of the fuel changes.

Theory:

• Derivative as an instantaneous rate of change (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 95)

References:

• Shapouri, H., J.A. Duffield, and M. Wang. 2002. The energy balance of corn ethanol: an update. USDA Agricultural Economic Report Number 814.
• Pimentel, D. 2003. Ethanol Fuels: Energy Balance, Economics, and Environmental Impacts are Negative. Natural Resources Research. 12(2): 127-134.
• De-gang, L., H. Zhen, L. Xingcai, Z. Wu-gao, and Y. Jian-guang. 2005. Physico-chemical properties of ethanol–diesel blend fuel and its effect on performance and emissions of diesel engines. Renewable Energy. 30: 967–976.
• Niven, R.K. 2005. Ethanol in gasoline: environmental impacts and sustainability review article. Renewable and Sustainable Energy Reviews. 9: 535–555.

Skull size in human adults is relatively small when compared to their body size. The same cannot be said for babies, whose skull size is quiet large in comparison to their body size. Use this information to determine at what developmental stage humans have larger skulls relative to their body length.

Theory:

• Allometric growth (Neuhauser, Claudia. Calculus for Biology and Medicine. New Jersey: Prentice Hall, 2000. 366)

References: