Mathematics Course: Reading Assignment and Field Project Instructions, Papers of Mathematics

Download Mathematics Course: Reading Assignment and Field Project Instructions and more Papers Mathematics in PDF only on Docsity! 1 COURSE SUPPLEMENT FOR Math 11A Calculus and Applications Professor M. Mangel This supplement has many objectives, but the main ones are: • To get you to think about the course each night: do not wait until the last minute to begin working on the problems. I have set this up so that you should work five nights per week, doing about four problems from the book per night. Start on the additional problems as early as possible in the week, because you will time need to think about them. • To get you to comfortable with computations. • To remind you that much (my partner says “all”) of life is a word problem. . Quantitative methods (modeling methods for prediction and understanding; statistical methods for inference about data) are an essential feature of modern biological science. If you do not understand these methods, you will be left behind. We focus on ecological and evolutionary applications for three reasons. First, many of these applications can be understood with a minimal amount of upper level biology. Second, this is my own field of research, so I know these applications well. Third, the animals and plants we’ll discuss are very cool. The quality of mathematical exposition (i.e. can the reader follow the steps, do you write full equations, etc.) matters. Graphs with numerical values must be done either on graph paper or using statistical software. NO CREDIT will be given for problems involving graphing that are done on regular notebook paper unless you are simply asked to sketch the graph. In problems involving verbal answers, the quality of your presentation (i.e. full sentences, sentence structure, choice of words) matters. In either case, when answering questions, be specific: points will be deducted for irrelevancies in your answer. In addition to problems and reading, you will do a field project that will help bring the mathematics that we discuss to life. Your reports of the field project should be typed double spaced in the following format: Introduction (Places the work in a broad conceptual context) Organize the introduction from general to the specific. Explain the rationale for the study (i.e., what are the questions you are trying to answer and why?). Methods (Include enough information so that the study can be repeated) Organize your material in a step by step fashion. Omit unnecessary information (e.g., "I wore a blue sweater while doing this experiment.") 2 Results (Summarize and illustrate (tables and graphs) your findings) Do not interpret the data or draw conclusions. Present the data in a logical order usually the same order that you use to describe the methods. Use the past tense. Discussion (Interpret the results, support your conclusions with evidence) Tell the reader what the findings (results) mean. Don't present every conceivable explanation. Recognize the importance of "negative" results. Convey confidence (Not "I think I found" but "I found"). What do these results mean in terms of the question you asked and how does it relate to the broader context of your research? You may work in groups of up to 4 people and turn in one group assignment on the field projects. If you do your project with a colleague who attends a different discussion section, you must put that person's name on the report and also include the name of their TA. You may also work together on the homework. If you do so, each of you must turn in a separate assignment, written in your own words. If we cannot tell who plagiarized whom, neither will receive credit. Homework is due at the start of the second discussion section; you may want to bring a copy for yourself to take notes on during the discussion section. The late homework policy is simple. ABSOLUTELY NO LATE HOMEWORK WILL BE ACCEPTED. In general, problems from the book will be graded by the reader on a 3 point scale: 3 – perfect, 2 – nearly right, 1 – tried, 0 – no serious effort. The additional problems will be graded by the TAs. 5 Due the week of 19 Jan 1) Read Section 2.2 and do the following problems on page 99ff: 2, 10, 15, 29, 71, 73, 74 2) Read Section 3.1 and do the following problems on page 127ff: 1, 2, 15, 18, 27, 32 3) Read Section 3.2 and do the following problems on page 137ff: 1, 4, 5, 10, 17, 19, 45 4) Field Project: You are to do one of these two field projects. When doing these field projects remember the following instructions: • You can work in groups of up to four students and turn in one report. • If you work in a group, all of your names and the names of your TAs must be on the title page of the report. • You must write the report in the format described at the start of this supplement: Introduction, Methods, Results and Discussion. I. DUCK FORAGING BEHAVIOR A. Introduction In lecture, we discussed several different concepts relating to foraging behavior (including group foraging). For your field trip project, your group will have an opportunity to test some of these ideas with ducks at Riverside Park (located along the San Lorenzo River next to the County building between Soquel and Water Streets). B. Equipment Needed Stopwatch or watch with a second hand, notebook paper, graph paper, 1 loaf of whole wheat bread with each slice cut into quarters. C. Procedure: You will need two groups of two people each (one bread thrower and one data collector in each group). Flip a coin to decide which group will be the "fast" group and which group will be the "slow" group. Space the two groups approximately 20 m apart. Starting at the same time, the slow group should throw out one fourth of a piece of bread every minute, while the fast group should throw out one fourth of a piece of bread every 15 seconds. The data collectors should record the number of ducks which are present at their site every minute. Also, the data collectors should record any interesting behaviors the ducks seem to be exhibiting (e.g. aggression towards one another, gender differences, etc.). If possible, try and count the number of "aggressive acts" by the group as a whole (per minute). After 15-20 minutes, switch the fast and slow groups and note the response of the ducks. D. Discussion What patterns do you see in the results (e.g. what differences were there, if any, between the two groups)? How do you explain your results? Are the individual ducks maximizing energy gained per time (energy unit = 1 chunk of bread)? 6 II. CALCULATING A GAIN CURVE A. Introduction A key component of the marginal value theorem (MVT) is the gain curve. In this project, you will calculate a gain curve using ducks and bits of bread. B. Equipment Needed Stopwatch or watch with a second hand, notebook paper, graph paper, two loaves whole wheat bread with each slice cut into eighths. C. Procedure Your group will need to have one person who observes an individual duck, and one person who records the data. If you have four people in your group, record data from two ducks at the same time (i.e., have two duck watchers and two recorders). Begin by throwing out a large number of pieces of bread. Starting as soon as the bread is thrown out, the data recorder will call out "time" every four seconds, and the duck watcher will report the number of pieces eaten by their duck in that interval. Continue until no more bread crumbs remain. Repeat the experiment a couple of times, switching your data collection roles if you wish. D. Analysis Calculate your gain curve(s) by plotting time on the x-axis and the total number of crumbs eaten (cumulative) on the y-axis. E. Discussion In what ways, if any, does your curve differ from what you expected? What do you think caused these differences? What other information would you need to have in order to make a prediction for how long an individual duck should stay in your bread crumb "patch"? Given the availability of other foraging patches, did your ducks behave as predicted by the MVT? 7 Due the week of 26 Jan 1) Read Section 3.3 and do the following problems on page 142: 1, 2,3,4,6,7, 12,13, 15,16,18,19, 23, 25, 26 2) Read Section 3.4 and do the following problems on page 148: 5, 9, 13, 14, 15 3) These questions pertain to the foraging problems that we discussed in lecture i) Show that E11 1 h11+ -------------------- E11 E22+ 1 h11 h22+ + ---------------------------------------> is the same as 1 E2 E1h2 E2h1– -----------------------------> . What does this mean if E2 increases, h1 increases, or h2 increases? ii) A forager encounters prey item type 1 one item every10 seconds, type 2 one item every 8 seconds, both items are worth 10 cal, but type 1 takes 5 seconds to handle, type 2 requires 10 seconds to handle. If the forager is maximizing energy, do you predict that it will specialize or generalize? iii) For a predator in a patchy environment, the rate of energy flow is R(t) = G(t)/(t+) where G(t) is the gain from a patch residence time of t and  is the travel time between patches. Suppose that the travel time is very large compared to patch residence time. What is an approximate formula for the rate of energy flow and what does this mean for predictions about how much of the patch is removed? 4) Sketch survivorship curves l(x) for organisms that live no more than 20 years for the following cases: i) 80% of the mortality occurs at ages >15. ii) a constant number of individuals die each year. iii) a constant fraction of individuals die each year through year 19. iv) 80% of the mortality occurs before age 1 Carefully label the axes and each curve. 10 Due the week of 16 Feb 1) Read Section 4.6 and do the following problems on page 221ff: 1,3,5, 8,10, 29, 40, 53, 60, 62 2) Read Section 4.7 and do the following problems on page 233ff: 3, 7, 17, 23,25, 29, 32, 37, 45, 62 3) A study of competition is described by the Lotka-Volterra competition equations: dN1 dt = 2N1    1 -    N1 + N2 125 dN2 dt = 3N2    1 -    N2 + 4 N1 200 Population size is measured in grams and time in days. i) What is the maximum per capita growth rate of the first species, including units? ii) In the absence of the second species, what is the carrying capacity for the first species, including units? iii) What are the units of the "4" in the second equation? iv) Sketch the isoclines for each species on a separate graph in the N1-N2 phase plane, showing where the species is predicted to increase, decrease or remain constant. 4) Peter Waser and Scott Creel published the following information on the life history of mongoose in the Serengeti: Age (a) l(a) m(a) 0 1 0 1 .41 0 2 .328 .21 3 .252 .39 4 .182 .95 5 .142 1.32 6 .085 1.48 7 .057 2.45 8 .031 3.78 9 .021 2.56 10 .014 4.07 11 .005 3.76 12 .005 3 13 .002 2 14 .002 0 i) Compute R0 from these data. 11 ii) What do you predict will happen R0 if the current survivorship for age 5 and beyond decreases by just 5%? Now compute the new value of R0. iii) Compute R0 if individuals delay reproduction from years 2-4 because of a food shortage. That is individuals are now 4 years old when they get the reproduction previously associated with a two year old, 5 years old when they get reproduction previously associated with a 3 year old, etc. iv) Interpret your results. 12 Due the week of 23 Feb 1) Read Section 4.8 and do the following problems on page 241ff: 1, 3, 4, 9, 10, 13, 22,33, 34, 37, 48 2) Read Section 5.1 and do the following problems on page259 ff: 19,20,23, 29, 33, 34, 38, 42, 49 3) In June 1994, I participated in the "Potential Biological Removal" (PBR) workshop held at the Southwest Fisheries Science Center, in response to the re-authorization of the Marine Mammal Protection Act. The amendments to the act determine that PBRs would be found from the formula: PBR = Nmin 1 2 Rmax Fr where Nmin is a minimum estimate of population size, Rmax is the maximum value of per capita reproduction and Fr is a safety feature defined as Fr = .9 for populations at or above 60% of carrying capacity, Fr= .5 for populations that were depleted or of unknown status and Fr =.1 for threatened or endangered populations. Scientists studied population dynamics using the "A logistic model": N(t+1) = N(t) + rN(t){1-(N(t)/K)A} - PBR(t) where N(t) is the size of the population in year t and PBR(t) is the PBR in year t. Generally, the value of A=2.4 is used. This is somewhat different from the logistic that we studied in class because of the focus on yearly differences, rather than rates of change over short intervals of time. Note, however, that N(t+1)-N(t) is the change in population size from one year to the next. i) In the absence of any removals, what is the carrying capacity in this model? ii) What is the maximum per capita reproduction in this model? iii) For harbor seals, K=25,000 and r=.12. Graph per capita reproduction vs. population size. iv) Graph total population reproduction vs. population size 4) This question deals with the Lotka Volterra predator prey equations: dV dt = 6V(1- V 100 ) - 0.5PV dP dt = 0.1PV - 0.07P where time is measured in days and biomass in kilograms. i) What is the maximum per capita birth rate of the prey, including units? ii) What is the per capita death rate of the predator, including units? iii) What is the total rate at which prey are removed by predators? iv) What is the total rate at which predators appear? v) Approximately how many prey does it take to "make" a predator? 15 Due the week of 8 Mar 1)Read Section 5.4 and do the following problems on page 296ff: 1, 2, 3,4,6, 9,12, 15 2) Read Section 5.5 and do the following problems on page 307ff: 1, 2, 3,5,12,13, 18, 21, 37, 42,51, 55 3) This problem involves analysis of optimal age and size at maturity, along the lines of thinking by the famous fishery scientist Ray Beverton. Assume that until it reaches maturity, a fish grows according to the von Bertalanfy equation L(t) = Linf(1- e -K(t-t0)) (1) where t is age measured in years and Linf is the asymptotic length. The parameter t0 is included to take account for the differences in the early life history (egg and larvae) growth patterns and later ones, to which the growth curve is usually fit. After maturity, at age tm, reproductive length is fixed. If mortality M is assumed to be constant across body sizes, survival to age t is e -Mt . Assuming that weight is proportional to length cubed and that reproductive success is proportional to weight, the definition of fitness as expected reproductive success is F(t) = e -Mt [Linf(1- e - K(t-t0))]3 (2) i ) For simplicity, we will set t0=0. Show that the optimal age at maturity, defined as the age that maximizes expected reproductive success is t* = 1 K log(M + 3K M ) (3) Why does asymptotic size does not appear in this relationship? ii) Define the size at maturity as Lm=L(t*) and show that the relative size at maturity is Lm Linf = 1 - M M+3K = 3K 3K+M (4) iii) On the next page are data collected by my colleague Kai Lorenzen on tilapia (Oreochromis spp.) 16 K M Relative Size at Maturiy 0.34000 2.4000 0.52163 0.26000 1.7000 0.44737 0.52000 1.6000 0.48055 0.45000 3.1000 0.43333 0.34000 1.9000 0.41284 0.43000 3.6000 0.38532 0.50000 1.0000 0.63934 0.67000 2.1000 0.58824 1.0800 5.8000 0.49342 0.82000 5.0000 0.37267 Make a plot of predicted relative size at maturity (from Eqn 4) on the abcissa and observed relative size at maturity on the y axis. Now add the 1:1 line and interpret your results. 4) No quiz or elaborate word problem this week. We will use the second discussion section as a review for the final.