Bi1: The Great Ideas of Biology Homework 2 Due Date: Thursday, April 20, 2017

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1 Bi1: The Great Ideas of Biology Homework 2 Due Date: Thursday, April 20, 2017 But I will not stop to explain this in more detail, because I would deprive you of the pleasure of learning it yourself, and the utility of cultivating your spirit by the exercise, which in my opinion is the principal benefit one can draw from this science. - Descartes, La Géométrie 1. Decimation of otter populations in the Aleutians. In class, we gave a detailed discussion of trophic cascades in the Aleutian Islands culminating in conceptual diagrams like those shown in Figure 1. In this problem, you are going to do some simple estimates to see if the disappearance of the otters as a result of killer-whale predation actually makes sense. Here, we repeat some of the same estimates that appeared in the primary literature as the heated debates that surrounded this topic took place. Doing simple order-of-magnitude estimates is our vehicle to developing intuition. My claim is that the numbers that emerge from this analysis were quite surprising to those who originally advanced the killer-whale predation hypothesis. Our first exercise will be to try and figure out the dietary requirements of killer whales. One of the arguments made in the Williams et al. paper that reported many of the key results we consider here is that killer whales eat sea otters whole, a proposition that is obviously physiologically reasonable given the relative dimensions of these animals as shown in Figure 2. Note that one of the reasons that Estes and his team favored the killer-whale hypothesis was that they hardly ever saw otter carcasses. Otters that die from malnutrition are usually seen on the beach as our own Caltech Galapagos field trip sadly observed during the 2016 El Niño with baby sea lions. Williams et al. made ghoulish measurements in which they homogenized whole sea otter remains and then performed calorimetry measurements to learn their caloric value. This allowed them in turn to see what kind of pressure the killer whales could put on the otter populations. To make progress on the estimate you are asked to consider below, one of 1

2 Figure 1: Trophic cascade interrupted. The left side shows the trophic cascade in which otters control the number of sea urchins, which in turn eat the kelp. When killer whales switched their diet to include sea otters, the feedback control imposed by the otters was removed from the ecosystem, resulting in an explosion of sea urchins and the resulting sea urchin barrens. 2

3 the key issues that you will need to think about is how to scale the metabolic rate of animals with their size. There is a long history of thinking about this complicated question. For our purposes here, you may invoke the empirical Kleiber law which asserts that the metabolic rate of animals scales with the 3/4 power of their mass. Note that the mass of a typical adult male killer whale is 4700 kg and 2800 kg for a female (see Figure 2). Question 1a) Using street-fighting estimates, work out the daily caloric requirements of male and female killer whales. Express your answer in units of kcal-day 1 and also in watts. Please don t look anything up. Just make simple estimates based upon things you know. Explain your logic, do not provide estimates involving long strings of significant figures and make sure you explain how you used your everyday knowledge to produce the estimate. As an aside, what does this tell us about the number of pounds of fish that Sea World must be feeding its captive killer whales such as the famous Shamu each day? It is interesting to consider how our estimates fare relative to the measured values presented in Fig. 1 of the paper by Williams et al. included with the homework, and reproduced as Figure 3 here. What you will see is that there is a difference between the so-called basal metabolic rate (BMR) and the field-metabolic rate (FMR). To get some intuition for this, note that riders in the Tour de France need to eat 8000 kcal/day in order to avoid losing weight during the three-week long race. Similarly, Michael Phelps famously ate 10,000-12,000 kcal/day training for the Beijing Olympics. Together these facts imply field metabolic rates are a factor of 3-6 larger than basal metabolic rates. Given the extremely large dietary requirements for adult killer whales, we now explore the potential impact of small pods of killer whales on the sea otter population. The idea of the sequential megafaunal collapse hypothesis is that some small subset of killer whales changed their diets as a result of the unavailability of sufficient large whale prey, first switching to Steller sea lions and later to sea otters. Adult male sea otters have a mass of roughly 35 kg, while adult female sea otters have a mass of roughly 23 kg. 3

4 Figure 2: Relative size of killer whale and otter. Ecology, 85(12), 2004, pp ) (From Williams et al. Question 1b) Given that an adult sea otter yields roughly 7 kj/g of wet mass, how many sea otters would adult male and female killer whales need to eat each day if this were their sole diet? In this case, use the data from Figure 3 on the field metabolic rate of a large mammal such as a killer whale as the basis of your estimate. Use Python to make a plot of the number of otters lost per year from the population as a function of the number of killer whales that have switched to this kind of predatory behavior, assuming now that the only source of caloric intake for this subset of killer whales is otters. See Tutorial 0c for a refresher on plotting in Python. Question 1c) Given the population size data provided in Figure 4 and knowing that the total population was 60,000 sea otters at its maximum, what is your best estimate for how many orcas are implicated in the massive population losses observed, if in fact this is the mechanism of additional otter loss beyond the usual deaths that take place in the absence of killer-whale predation? As noted in class, the sequential megafauna collapse hypothesis (see Springer et al., Proc. Nat. Acad. Sci., 100, (2003)) considered here has been controversial and challenged on multiple fronts. The key ar- 4

5 Figure 3: Scaling of metabolic power with animal body size. FMR refers to Field Metabolic Rate and BMR refers to Basal Metabolic Rate. (From Williams et al. Ecology, 85(12), 2004, pp ) guments set forth against this hypothesis included: 1) killer whales rarely attack or eat sea otters, 2) the multispecies collapse was not sequential, 3) the declines were caused by nutritional limitation, 4) the timing of the multispecies collapse is inconsistent with the timing of the great whale depletions, and 5) the geography does not work. For more details see Springer et al., Marine Mammal Science, 24(2), (2008). We now explore some of these objections in more detail. One of the arguments made against the idea that some killer whales had shifted their diet from great whales to smaller marine mammals is the infrequency of observations of killer whales attacking otters. In six years of observation over the 2000 km of coast along the Aleutian Island chain, there were only 6 observed attacks of this kind. In this part of the problem, we will follow the authors of the original study and work out a number of simple quantitative estimates to develop some intuition about how frequently we would expect such attacks to be observed. Interestingly, James Estes notes that despite the presence of 60,000 otters at the peak of their population, there were similarly no reported observations of even a single live birth of an 5

6 Figure 4: Sequential collapse of megafauna. You will use the otter loss from its maximum population size of roughly 60,000 otters to figure out the number of extra deaths due to predation each year. Adapted from A. M. Springer et al., Proc. Nat. Acad. Sci. 100, 12223=-12228,

7 otter pup, though demonstrably such births do indeed occur! To make the estimates called for below, you will need to exploit the streetfighting divide and conquer approach already discussed in class and in some of our readings. Some of the facts that you will need to consider include that the typical density of killer whales in the Aleutians and elsewhere are 4 orcas/1000 km 2 (resulting from a 3000 km transect during a 1994 survey). Another key fact is that out of the entire killer whale population, only roughly 10% of those killer whales depend upon mammalian prey, while a larger fraction eats fish and squid. Question 1d) Given the transect results noted above for killer whale densities, make an estimate of the number of killer whales within 200 nautical miles of the Aleutian Island chain. Killer whales have different foraging preferences that classify them into different ecotypes: i) transients - they feed on marine mammals, ii) residents - they feed on fish and iii) offshores. If roughly 10% of killer whales are in the transient category, figure out how many of the killer whales within this 200 nautical mile region are potential hunters of marine mammals. When trying to estimate the total number of hours humans observed the island coasts, note that when scientists such as Estes and his team are at work in the Aleutians, it is during the summer months only, meaning that each summer there are roughly 10 weeks of observation, with the long daylight hours of northern latitudes taken advantage of. My own experience in these same Alaskan archipelagos tells me that multiple people will be on the bridge of the boat at a time, alternating between looking by naked eye and using binoculars. Question 1e) As part of your divide and conquer strategy, make an estimate of the total number of human hours over the 6 years of the expeditions in question that the coast was being watched, thus making it possible to spot a killer whale attack if it occurred. Though killer whales live over a vast area, our next estimate will be predicated on the idea that all attacks on sea otters occur near the shore and 7

8 hence rather than having to consider a survey of the entire area of the ocean surrounding the Aleutian Islands, we will focus on a one-dimensional survey of the several thousand kilometers of island coastline. Question 1f) Use your result from 1c to estimate the total number of otter attacks that occurred during the 6 years of expeditions. Then, estimate what fraction of those attacks you expect would have actually been seen, given the vast extent of the coastline over which such attacks could take place and the fraction of the coast under actual surveillance by the expedition. Finally, how many total attacks would you therefore expect to see? Explorations in Population Dynamics Modeling In this part of the problem, we consider a second facet of the study reported in the work on predation of otters by killer whales, namely, the modeling of how the population would change over time as a result of predation. Our strategy is to consider several models of increasing sophistication that make appearances in population dynamics. We make a series of toy models of the population dynamics of otters to see the kinds of effects that are taken into account, again with an eye to seeing whether the numbers make sense. If we first consider the number of otters at time t + t after sea otter hunting was banned, we can write an equation for the growth in the population of the form N(t + t) = N(t) + (k t)n(t). (1) This equation states that the total number of otters at the next time step (N(t + t)) is the current number of otters plus the net growth rate (births - deaths) that will occur during the time step (k tn(t)) (note that in the early stages of population growth, the net rate of births is higher than deaths, meaning the population has not achieved steady state). We can simplify by rewriting our result as N(t + t) N(t) t = kn(t). (2) 8

9 We notice that the left side of the equation gives us the discrete approximation to the derivative and use this definition to write the more standard form dn = kn, dt (3) with the simple solution N(t) = N 0 e kt, (4) as can be verified simply by taking its derivative. This is the classic differential equation for exponential growth. Note that in our numerical treatments of population dynamics introduced below we will be openly naive in our handling of the numerics using the simplest methods to integrate our equations (known as the Euler method first presented by Euler in his lessons on calculus and presented here in eqn. 2), not because they are the best numerical methods, but rather simply because they preserve the intuition of what we are actually doing when we integrate a dynamical equation. As your mathematical education advances, you will learn more sophisticated methods for handling differential equations numerically. Question 1g) Use Python to write the relevant code to perform a numerical integration of eqn. 1 using the Euler method. You can see tutorial 2 for an example of how to write such an integration routine. For the purposes of that equation, use the plausible approximate rate constant of k = 0.5 yr 1. The reason we choose this street-fighter s rate constant is because we make the simplifying assumption that every female has one offspring each year and that half of the population is female. Make a plot of your solution by considering the recovery of the population for a time period of 40 years after hunting stopped. Your plot should show number of otters as a function of time. Use the convenient but oversimplified initial condition that you initially start out with only 2 otters. Of course, no population can grow indefinitely. To account for this, population growth models include a term that reflects the carrying capacity of the population. The carrying capacity is implemented as a penalty to population growth that grows quadratically with increasing population size and 9

10 resulting in the modified growth equation N(t + t) = N(t) + (k t)n(t)(1 N(t) ), (5) K where we introduced the carrying capacity of the population denoted as K. We can also use the definition of the derivative to rewrite this in the more familiar form dn dt = kn(1 N K ) (6) an important result known as the logistic equation. The essence of this simplified model of population dynamics is that it acknowledges that populations cannot grow without bound. Question 1h) Use Python to perform a numerical integration of the logistic equation (eqn. 5). Choose a value of the carrying capacity so that the population reaches a dynamical steady state at the observed maximum otter population size of 60,000. Make a plot of the population size as a function of time over a 40 year period, starting with an initial condition in which you assume there are only 2 otters left. We are finally ready to make a simplified analysis of the reduction of the sea otter population over time as a result of predation by killer whales. Question 1i) Using your results from earlier for the number of sea otters eaten per year by one killer whale, now figure out how to amend the logistic equation to account for a steady predation rate for two different cases, one in which 5 killer whales have taken to the sea otter diet and a second case in which 10 killer whales have taken to the sea otter diet. Specifically, put in a term that simply accounts for the bleeding off of otters at a constant rate by killer whales. Make a plot of the number of otters over time for these two different cases, starting with the steady state number of otters in the absence of predation as your initial population number. The models we developed above still fall short of those used to analyze real world problems such as the loss of otters in the Aleutian Islands. For 10

11 those of you interested in learning more, the next step in sophistication is to acknowledge that the population has some age structure with the recognition that otters of different ages have different probabilities of giving birth to young as well as different probabilities of dying in a given year. This leads to the interesting idea of an age-structured population described by a concept called the Leslie matrix. A nice book on the topic is An Illustrated Guide to Theoretical Ecology by Ted Case. One of the key conclusions of the Aleutian Island study is that it was another example of the kind of trophic cascade resulting from a keystone species discovered by Robert Paine. 11