Chapter 4 Problems
1. Using the chocolate isotopes I Chi model in workbook 4.1 (Chapter 4 folder on
this CD, I Chi Spreadsheets), what is the shape of the curves that result when
isotope “pickiness” or fractionation is set at 1 so that there is no preference for
heavy or light isotopes? Which part of the curves would you label “turnoverdominated” and which parts “close to equilibrium”? If you change the mixing
resupply or replacement rate to values of 75, 150, 500, 700, and 900, how do the
curves change, and why? (A good way to view the dynamics is to enter the
numbers in this list one-by-one, typing them in, then use the “undo” arrows to
replay the list forwards and backwards until you see what is going on). Is there
always a “turnover-dominated” portion of the curves, and one that is “close to
equilibrium”? What controls whether the system is at equilibrium or in the more
dynamic turnover stage?
2. Continuing problem #1 with Chocolate Isotopes, set the replacement rate to an
average value of 500 out of 1000 chocolates eaten, then manipulate the
“pickiness” factor or  fractionation. What curves result with  fractionation
values of 1,2, 4, 8,16 and 18? (Technical note: in theory, maximum possible
fractionations for the HCNOS isotopes are for hydrogen isotopes and approach
18; see Science 110:14-16,1949). What happens to the ratio eaten, light/darks, as
you change the fractionation factor? Why? (Hint: because the total number of
chocolates is constant in this system, steady state eventually occurs where inputs
must equal outputs in terms of both light and dark chocolates. Most models in this
book are not steady state models, but this chocolate isotope model is a steady state
model once amounts reach equilibrium).
3. Continuing problems 1 and 2 with Chocolate Isotopes, you may have the sense
now that fractionation and mixing sometimes oppose one another. See if this is
really true. First set the  fractionation to 1 (no selectivity) and resupply to 500.
Note that these values give a final result at the end of the year of about equal
amounts of light and dark chocolates. Now set the resupply to 750, and type in
fractionation values until you find a value that gives this same final result
approaching of an even mix of light and dark chocolates. What is this value for
fractionation, and how do you explain this effect that fractionation and mixing
(via resupply in this case) seem to cancel each other out? Give your answer in
both qualitative and quantitative terms. (Hint: as in problem #2, think about
equality between inputs and outputs in developing your quanitative answer).
4. Using the oxygen cycling I Chi model (see workbook 4.2 in Chapter 4 folder on
this CD, I Chi Spreadsheets), first locate the three master variables at the upper
left of the worksheet for: daytime photosynthesis, continuous respiration, and
continuous isotope fractionation during respiration. Change these variables one at
a time and note the effects to answer the following questions. What are the effects
of doubling each of these master variables on the a) oxygen concentrations with
time, b)  values with time and c) the trajectories of oxygen dynamics in the
(concentration, ) graph? What are the effects of halving the values? Finally,
using the original values for the three variables, what is the effect of changing the
fourth master variable, the isotope value of oxygen from photosynthesis, from 0 to
-10o/oo and then from 0 to +10o/oo? Considering the overall results from this
exercise, would you agree or disagree with the idea that isotopes can help
constrain ideas about how oxygen is produced and consumed in aquatic systems?
Explain your answer.
5. This problem addresses whether oxygen isotope measurements are really useful
for studying oxygen dynamics. Originally, oxygen isotope models were
developed for the ocean to study production/respiration ratios or P/R (Global
Biogeochemical Cycles 1:49-59, 1987). When P/R = 1, production and respiration
are balanced and there is no net change in oxygen concentrations. But when
photosynthesis due to algal growth is greater than respiration, oxygen
accumulates and P/R>1. Here we compare two growth scenarios where P/R>1,
considering only the first 12 hours when oxygen is increasing. Using the oxygen
cycling I Chi model in workbook 4.2 (see Chapter 4 folder on this CD, I Chi
Spreadsheets), change the fractionation factor to 15o/oo, the value used in the first
modeling study cited above in this problem. Using your cursor to click on the
graphs in the workbook, what are the values at 12 hours (0.5 days) for oxygen
concentrations and 18O values? What is P/R at 12 hours? Next change the P rate
from 4 to 6 and R rate from 2 to 4 in the workbook and repeat your observations
about oxygen concentrations, 18O, and P/R at 12 hours (0.5 days). Make a table
of your results for both scenarios showing initial and 12 hour values for oxygen
concentrations, 18O, and calculated P/R. Now that you have compiled the results,
look back at the experiment and answer these questions: Can you use the oxygen
concentrations alone to estimate the P/R ratio, the rates of photosynthesis and the
rates respiration? If not, does adding the 18O information help with these three
estimates?
6. Following the step-by-step approach I Chi modeling approach outlined in section
4.4 and in the Workbook 4.4 (Chapter 4 folder on this CD, I Chi Spreadsheets),
develop a spreadsheet model for CO2 dynamics in a closed greenhouse full of
large plants maintained 24 hours a day in well-lit conditions. The plants fix CO2
and remove it from the atmosphere, but also respire CO2 and return CO2 to the
atmosphere. Incorporate the following initial assumptions into your model: a) the
initial 13C value of CO2 is -8o/oo, the current 13C value of CO2 in the
atmosphere, b) the initial concentration of CO2 is 375 ppm, c) during
photosynthetic CO2 uptake, plants fractionate by 20o/oo and use a constant
amount of 5 ppm CO2 per hour, and d) during continuous respiration, plants do
not fractionate carbon isotopes but add back CO2 with a constant value of -28o/oo
at a rate of 4 ppm per hour. What are the predicted concentrations and isotope
values after 10 days, at 240 hours? Show your spreadsheet model and graphs of
your time-course prediction for CO2 concentrations and isotope values. Also,
compare your model to the oxygen model of problem 4 and sections 4.2-4.5.
What are the similarities and differences between the oxygen and carbon dioxide
models?
7. Use the gain-loss model for cows presented in section 4.7 and modify it to explore
carbon isotope dynamics for cows feeding in a pasture. Assume that the
fractionation involved in carbon loss, primarily as respired CO2, is 1o/oo, but that
there is no fractionation during assimilation of carbon from the diet. (This same
assumption of no fractionation during assimilation was discussed in section 4.7
for N assimilation, and also likely applies to carbon isotopes). Use the spreadsheet
cow model in Workbook 4.7a (Chapter 4 folder on this CD, I Chi Spreadsheets) to
work through various growth and starvation scenarios for carbon, parallel to those
given already for nitrogen. If animals in nature generally have carbon isotope
compositions about +0.5o/oo vs. their diets, what kind of gain-loss dynamics do
the models predict for these free-living animals? Why?