BIOL 301-003, Problem Set 1 KEY, Fall 2013 1. You are interested in studying populations of the purple coneflower, Echinacea purpurea, a plant that lives in prairie fragments. In your study area, you have identified several prairie fragments that have populations of E. purpurea. Describe the research study you would do to determine which populations were sources and sinks. Be explicit about your hypothesis, the data you would collect, your predictions and what you would conclude given the different possible results. Answers will vary, but the key (beyond describing a study that actually addresses the problem) is to distinguish the hypothesis, data, predictions and anticipated conclusions. There are a few studies that would address the problem. One approach is to focus on the intrinsic growth rates of each population (sinks are negative, sources are positive). Another approach is to use some sort of marking technique to track migration among populations. A third approach is to look at the demographic structure of the populations (e.g., sinks will have low per-individual reproduction rates, but in sources they will be high). 2. Draw graphs that illustrate exponential growth. In Graph A, show how population size changes over time, using a linear scale. In Graph B, show how population size changes over time using a logarithmic scale. In Graph C, show how population growth rate changes over population size using whatever scale is appropriate. Clearly label all axes. Graph A should look like Figure 11.4a, Graph B should look like Figure 11.4b. Graph C should have population growth rate (or “dN/dt”) on the vertical axis and population size on the horizontal axis. If you put “r” or “λ” on your vertical scale, your graph should be a horizontal straight line (note: these are per-capita rates, and depending on how a question is worded, they may or may not be acceptable). Here is one possible version of Graph C (I chose arbitrary values and did the calculations in Excel). In this case, r = 0.1 and I calculated dN/dt over population sizes from 100 to 2000. To completely answer Question 3 as it is written, you would not need to do calculations, just show the right shape of the relationship and the correct scale on the axes. You don’t need the numbers, but if you left the scales off entirely, that would not earn full credit): Increase in N0204060801001201401601802000 500 1000 1500 2000Population Size (N)Population Growth Rate (dN/dt)3. For one research project, I had to measure population growth of Daphnia pulicaria (a small crustacean living in lakes that reproduces continuously with overlapping generations) in the winter. On January 1st I found the population density was 1.24 individuals L-1, and on January 31st the population density was 1.31 individuals L-1. A. What was the population growth rate of D. pulicaria in January on a per-month basis? (in terms of change in the density of individuals L-1)? B. What was the exponential growth rate of D. pulicaria in January on a per-day basis, assuming it is constant throughout January? At the time I measured the population density on January 31st, what was the population’s exponential growth rate? (Daphnia population growth is reported per day.) A. Since the time change is only one month, you can easily figure out that dN/dt = 1.31 – 1.24 = 0.07 individuals L-1. B. The correct way to figure out the exponential growth rate per-day is to use the equation: Nt = N0ert and solve for r. In this case, t = 30 days. For simplicity, though, I’m going to solve for λ and then convert it to r (using r = ln λ). 1.31 = 1.24 λ30 30√(1.31/1.24) = λ 1.001832 = λ 0.0018 = r You might have been tempted to start with the change in population density in 30 days, and then divide by 30 to get 0.0023 = dN/dt (on a per-day basis). From there, since dN/dt = rN you might have thought you could figure out the rate of increase (r) at a particular time with the population size at that time (on Jan 31, N = 1.31) and r = 0.0023/1.31 = 0.0018. While this can be a useful approximation if you don’t have a calculator handy, it’s wrong. And that’s because if the exponential growth rate is constant from day-to-day, the population growth rate will NOT be the same – i.e., the 0.0023 number is not the same for every day. In this case, the difference between doing it the correct way, and doing it as an approximation leads to the same answer, but that is not always true. Another way to do this problem wrong, but end at an approximately right answer, is to figure out the r value on a per-month basis (here: dN/dt = rN --> 0.07 = r(1.24) --> r = 0.05645 per month) and then divide that by 30. Again, it will be wrong because if r is constant, it is not possible for daily population growth rate to be constant. Remember to show your work on exams, so I know you understand the right way to solve the problem.4. You own a consulting firm that was hired by the South Carolina Secretary of Agriculture to assess the risk of two agricultural pests that are invading South Carolina. He is trying to determine where spending taxpayers’ dollars on control measures is most important. One of the pests is the Orange Tiger Beetle, Clemsona idioticus, an insect with discrete reproduction and a per-capita growth rate of λ = 1.0170 yr-1. The other pest is the Phloem-Sucking Aphid, Chihuahua athensi, which reproduces continuously. C. athensi has a per-capita growth rate of r = 0.0090 yr-1. A. Which pest has a faster growth rate? How do you know? The Orange Tiger Beetle. You can convert between λ and r with the equation λ = er. If you do that, you find that the equivalent λ for the Phloem-Sucking Aphid is 1.0090, which is a lot less than the tiger beetle growth rate. (Alternatively, the tiger beetle’s equivalent r is 0.0169). B. Currently, the population density of C. idioticus is 17 ha-1, and the density of C. athensi is 102 ha-1. C. idioticus becomes a serious economic problem when its population reaches 50 ha-1, where as C. athensi isn’t a serious problem until its density is 115 ha-1. Which pest will become a serious economic problem first? When will that happen? C. athensi will become a problem first, in 13.3 years. You can use either the Nt = N0ert or the Nt = N0λt equation, depending on whether you want to work with r or λ. You need to solve for t in the equations. Your time zero population