## Problems

14.1. Decay of radioactive elements deep within the Earth produces a flux of heat energy toward the surface in the form of geothermal energy. Visible results include hot springs, geysers, and volcanoes. Humans can extract this energy to run machines. Could plants or microorganisms use this energy directly for primary productivity in nature? Explain.

14.2. The net primary productivity of a tall form of cord grass (Spartina alterniflora) has been measured to be 5800 g/m2- yr in a Georgia salt marsh. What is the efficiency of energy capture assuming an average solar energy input for the southeastern United States of 3888 kcal/m2 - day? Comment on the result.

14.3. The United States as a whole receives an energy subsidy in the form of fuels in the amount of about 1.8 x 103 kcal/m2- yr. The world receives about 100 kcal/m2- yr. Compare these figures to the gross and net primary productivity figures given in the text for the United States and for particular ecosystems.

14.4. Tommy's Pond averages 2.0 f deep and has an area of 0.40 acres (1 acre = 43,560 ft2). The stormwater entering the pond was measured to have an average phosphorus concentration of 0.300 mg/L. The average flow rate of the stormwater may be assumed to be 5000 ft3/day (1 ft3 = 28.3 L). Thirty ducks live on the pond. Each duck is estimated to contribute phosphorus at the rate of 0.50 kg/y.

(a) What will be the steady state phosphorous concentration in the pond?

(b) What is the turnover rate for phosphorus in Tommy's Pond at steady state?

14.5. It can be difficult to visualize the huge quantities in global biogeochemical cycles. Earth has a land area of 148,429,000 km2 and a water area 361,637,000 km2. Normalize the fluxes and reservoirs in the carbon cycle as shown in Figure 14.5 on a per square meter basis. Separate the atmosphere into the portions over land (29.1%) and over water (70.9%). How does productivity on land compare to that on water?

14.6. Look at the magnitude of abiotic fixation of nitrogen. How long would it take to convert all the atmospheric nitrogen to nitrate if life suddenly disappeared from Earth? Use information from Figures 14.7 and 14.8, plus Earth's area from Problem 14.4.

14.7. Write a the balanced chemical equation for the reaction of water, oxygen, and nitrogen to nitric acid. How many moles of oxygen are needed for each mole of nitrogen? Would there be an excess left over of either, based on the current atmosphere of 79 mol% N2 and 21 mol% O2? Are there any other sources of oxygen?

14.8. Write a spreadsheet program to solve matrix (14.8). To do this in Microsoft Excel, first place the coefficients in a 4 x 4 array of cells, then the right-hand-side (RHS) vector in an adjacent column of cells. Say that the matrix is in cells A1 to D4 and the RHS vector is in cells E1 to E4. Use the cursor to highlight another empty 4 x 4 array: say, cells A6 to D9. Then type the following: "=minverse(A1:D4)." Now, hit "Cntrl-Alt-Enter." Cells E1 to E4 will display the inverse of the matrix. To get the solution, we must multiply the inverse by the RHS vector. To do this, highlight four empty cells in a vertical column. Then type '' =mmult(A6:D9,E1:E4)'' and hit "Cntrl-Alt-Enter." The cells will now hold the solution of the matrix system in order.

Notice that you can now change entries in the original matrix without reentering the functions. The solution changes automatically. The coefficients in Table 14.4 represent summer conditions. In winter the situation can be represented by the following values: k\ = 1.0; k2 = 8.0; k3 = 4.4; k4 = 2.5; k5 = 1.7. Use your program to compute the resulting distribution of nitrogen forms in this ecosystem. Compare and explain the differences between summer and winter.

14.9. What type of species interaction describes the human-mosquito pair?

14.10. To see how doubling can rapidly cause populations to outstrip their resources, try the following math experiment. Suppose that you were to take a piece of paper 0.1 mm thick and fold it in half twice. This would quadruple the thickness to 0.4 mm. Suppose, instead, that you had folded it 50 times, if you could. Before computing the thickness, take a guess (a big guess). Then do the calculation. To what physical distance can you compare this?

14.11. A population of insects lives for two years. The first year they are a juvenile form that does not reproduce; the second year is an adult form that reproduces and then dies. Sampling finds 1000 juveniles and 10 adults per square meter. Create a life table for this insect. Compute the natality and mortality for each age group, and use the results to compute the net reproductive rate, the mean length of a generation, and the growth rate. Is the growth rate what you would expect?

14.12. For a carrying capacity K, what population, N, would result in a specific growth rate that is one-half of the maximum?

14.13. Both the logistic equation and the Monod model produce sigmoidal growth curves. (a) Why do we not use the logistic model for microorganism growth in wastewater treatment plants? (b) Conversely, why not use the Monod model for ecological population modeling?

14.14. (a) How many possible pairwise interactions are there among nine species? (b) If each species has, on average, only two interactions, how many interactions will there be in this case?

14.15. As of the year 2005, the human population exceeds 6 billion people vs. 3.5 billion in 1950 and 1.6 billion at the start of the twentieth century. Find the values of r and K that make the logistic equation fit these data. According to this, what is the carrying capacity of Earth? How does this compare with the estimates given in the text? What are possible explanations for the discrepancy?

14.16. In the Lotka-Volterra predator-prey model, the period of the oscillations depends on the initial conditions. But in the vicinity of the steady-state solution given in Section 14.4.1, it approaches 2n(ad)~1=2. Thus, a decrease in the host growth rate or the predator death rate would increase the period. Can you explain in words why this is so?

14.17. Suppose that someone were to enter a large temperate-zone hardwood forest and harvest all of the trees in a 100-ft circle. Describe the changes that might occur at the site if it were left undisturbed for the next 100 years.

14.18. Compute the Simpson and Shannon-Weaver diversity indices and Pielou's evenness index for a community consisting of four species that each comprises one-eighth of the organisms and a fifth species that comprises the other 50%. Which species distribution in Table 14.9 does this community most resemble?