Modeling the Effect of Acid Rain on a Fresh-Water Ecosystem

Transcription

1 Modeling the Effect of Acid Rain on a Fresh-Water Ecosystem Don Schwendeman Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York Third UK Graduate Modeling Camp Oxford University, March 29 - April 1, 2011.

2 Outline Plan for this talk: New York: Fun Facts Background: Acid Rain and the Adirondacks Model 1: Food-Web Model Model 2: ph-lake Dynamics Model Existing Models and Data Concluding Remarks

3 New York: Fun Facts Map: New York state consists of: New York City and vicinity (Long Island, Westchester county) Everything else which is collectively known as Upstate New York. The modeling problem concerns lakes in the Adirondack region of New York state.

4 New York: Fun Facts Upstate New York: Capital District Albany, New York: first European settlement (Dutch) in 1614, became the state capital in Schenectady, New York: home to Thomas Edison s Edison Machine Works company, 1887, and later headquarters of the General Electric Company. Troy, New York: home of Uncle Sam (Samuel Wilson, meat packer, War of 1812), and Rensselaer Polytechnic Institute (founded 1824). Voorhees Computing Center, RPI

5 New York: Fun Facts Upstate New York: Adirondack region Adirondack Wilderness: located in northeastern New York, covers about 16, 000 km 2 (approximately 3/4 area of Wales). Contains New York s high peaks: The 46 (46 peaks above 1200 km, including Whiteface Mt., site of 1932 and 1980 winter olympics alpine events). Heavily forested region with approximately 1,300 lakes and ponds. Many of the lakes in the region are strongly effected by acid rain. Whiteface Mt. Lake Placid

6 Background: ph review: The ph level of water in a lake is an important measure of the ability of the fresh-water ecosystem to support life. Recall that ph = log 10 H + ph levels: where H + = molar concentration of hydrogen ions (moles/liter). Also, recall that the hydrogen ion concentration in pure water at room temperature is about 10 7 moles/liter, thus neutral ph is 7.

7 Background: Lake acidification: some bad news... Studies suggest that a ph level equal to about 6 is a critical biological threshold. A lake ecosystem can support many biological species when the ph level is greater than 6. Below a ph equal to 6 the number of lake species decreases, and most species can not exist if the ph falls below 4.5. For example, a sampling of fish species in a set of lakes...

8 Background: Lake acidification: how it occurs... Our main focus will be... Acid deposition via wet precipitation: rain and snow. Acid transport via surface runoff, and groundwater and river flow.

9 Background: Acid rain: acid deposition in the northeast US is particularly severe Recall: ph levels

10 Background: ph effects:

11 Background: ph of 3 lakes in the Adirondacks:

12 Food-Web Model: Idea: Develop a prototype food-web model consisting of a predator, a herbivore, and its food source (algae). At the top of the food chain, the predator feeds on the herbivore, while the herbivore, in turn, feeds on the algae. The rates of growth, death and interaction depend, presumably, on the ph level of their environment. Cast of characters: Predator pumpkinseed sunfish Herbivore Daphnia Algae Chlorella

13 Food-Web Model: Pumpkinseed sunfish: Pumpkinseeds reproduce rapidly, and spawning occurs from May into August. The female will lay between 1,600 to 2,900 eggs. Growth is fairly slow; they are sexually mature at age two and have a life span of about six years. They are usually found in water less than 2.6 m deep. Mature fish are capable of surviving at low ph, but ph definitely affects the survival of young fish.

14 Food-Web Model: Daphnia: Daphnia are small water-born crustaceans, between 0.2 mm and 0.5 mm in length. The average life span is about 50 days. Typically, a clutch of 6 to 10 eggs is released into the brood chamber, which hatch in a couple of days. The time required to reach maturity is about 7 days. Larger Daphnia avoid fish predation by residing in the lower (and darker) portions of the lake during the day (where the fish are unable to see) and then rise up to the surface at night to feed. (This behavior is called diel migration.)

15 Food-Web Model: Chlorella: Chlorella is spherical in shape, about 2 to 10 µm in diameter. In culture, a population doubles in approximately 0.6 days. Rapidly growing algae removes carbon dioxide from the water during photosynthesis, which can result in a significant increase in ph levels.

16 Food-Web Model: A generalized predator-prey food-web model: df dt = af + bdf + F (0) = F 0 dd dt = cd ddf + D(0) = D 0 da dt = A(0) = A 0 where F, D and A are the population densities of fish, daphnia and algae, respectively, and t is time. Initial states are given by (F 0,D 0,A 0 ) and (a, b, c, d) are parameters (that depend on ph presumably).

17 Food-Web Model: Questions / issues: What are the model equations? Are simple predator-prey type equations sufficient to model the behavior? The evolution of the populations evolve on different time scales (algae is fast while fish is comparatively slow). Can this be exploited? How do the parameters of the model depend on ph? How does the ph of the lake environment vary with time? Are there existing models/data available to guide the development of the model?

18 Food-Web Model: Modeling the ph of a drainage lake: ph budget: Incoming ph: wet precipitation (direct and indirect via surface runoff) and inlet streams. Outgoing ph: evaporation and outlet streams.

19 Food-Web Model: Example: Big Moose lake surface area: m 2 max depth: 21.2 m type: thin tilled, drainage hydraulic retention time: 174 days Observation: significant seasonal variation in ph

20 Food-Web Model: Simple model of the ph behavior: A basic rate law: dh dt = (incoming rate) (outgoing rate) + (buffering) where H = moles of hydrogen ions Questions / issues: How to model the incoming rate (from streams, rain, snow, surface runoff) and outgoing rate (streams, evaporation) with some level of accuracy? How to model a healthy lake s ability to buffer spikes in ph? Also, longterm exposure to acid deposition reduces the lake s ability to buffer the ph level.

21 Existing Data and Models: References available on my website: Faculty Schwendeman OxfordGradCamp (under Quick Links)

22 Existing Data and Models: Lab study of daphnia/algae behavior: Observations: Graphs show concentrations of algae (green curves) and daphnia (blue curves). Behavior is similar to standard predator-prey dynamics.

23 Existing Data and Models: Field study of fish/daphnia behavior:

24 Existing Data and Models: Field study of ph in rain:!+*#,*"%&-*.&/("&$01#-*!#'$"!#'"!"#$%$&'&()**!#&$"!#&"!#%$"!#%"!#!$"!" (" (#&$" (#$" (#)$" $" $#&$" $#$"!+*#,*"%&-*!"#$%$&'&()**!#*"!#)"!#("!#'"!#&"!#%"!#$"!"!"#$%$&'&()*#2*"%&+*&+(,+-&()*!#!("!#$"!#$("!#%"!#%("!#&"!#&("!#'"!#'("!#("!#(("!#)"!#)("!#*"!#*("!#+"!#+("!#,"!#,(" $" "%&+*&+(,+-&()*.&+/0"1***

25 Existing Data and Models: Field study of rain vs. runoff: 6 5 R(I) = max( 0, 0.618*I ) Runoff Rate vs. Rainall Rate Runoff Rate (10^-2 inches/hr) Rainfall Rate (10^-2 inches/hr) Observation: The hydrogen concentration in runoff is empirically shown to be 1/5 of that in rainfall on the surface of the lake.

26 Existing Data and Models: Lab studies of the effects of ph on daphnia and fish: Survival rates for daphnia at various levels of ph

27 Concluding Remarks: Background required: Basic knowledge of dynamical systems (predator-prey, logistic equations; stability analysis) Only very basic knowledge of biology/chemistry. Modeling steps: Develop and analyze a basic food-web model. fixed ph. Perform simulations for Develop and analyze a basic lake-ph model. Perform simulations for various flow conditions. Develop and analyze a coupled food-web/lake-ph model. Perform simulations for lake death or recovery. Consider short-time and long-time behavior. Calibrate models with available data, as best possible.

28 Concluding Remarks: Advantages: Very little mathematical modeling has been done on this problem. This means that modeling paths/choices are wide open. Should be fun... Disadvantages: The faculty mentor presenting this problem doesn t know much about it. Perhaps this is an advantage... Thanks: Mark Holmes for doing the initial research on this problem and for presenting it at the RPI Graduate Student Modeling Camp in Chris Breward for organizing the grad camp here at Oxford.

29 Any questions???