FORAGING OPTIMAL THEORY
Lab 7
Reminder!
Bring a Calculator
I. FUNCTIONAL RESPONSE
In lecture, you studied the Lotka-Volterra predator-prey model:
different feeding behaviors in response to varying prey densities. Consequently, there are more complex functional response curves (Type II, and Type III) that are frequently used by ecologists to model predator-prey interactions (Figure 1). A Type I functional response assumes that the predator consumes a constant fraction of the prey. This translates into a linear increase in the number of prey consumed as the prey density increases. An example would be a mussel that filters plankton from a fixed volume of water each day. As prey density in the water increases, more prey are consumed. The linear nature of the Type I response makes it the simplest of the three response curves to model, accounting for why the Lotka-Volterra predation model assumes this type of functional response. A Type II functional response curve describes an initial increase in feeding rate with increasing prey density. However, as prey density increases further, the predator's response to the increase becomes less. Finally, a maximal feeding rate is approached wherein additional increases in prey density do not lead to higher predation rates. A Type II functional response is probably the most common of the three types among real predators. The declining rate of increase at higher prey densities can be explained by predator satiation-the predators are eating at their maximum rate per day. Under these circumstances, they have plenty of food and they won't consume prey at higher rates even if prey density increases more. Another way to think about the leveling of predation at high prey densities is that at very high prey densities, the predator is spending all of its time handling its prey (killing and eating), so the predator cannot possibly increase its rate of consumption. A Type III functional response is a sigmoidal (Sshaped) relationship between predation rate and prey density. At low prey densities, the capture rate is very low, but it increases dramatically as prey density increase (the middle part of the S),
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N = rN - cNP , t
P = acNP - dP t
where N is the prey population, P is the predator population, r is the intrinsic rate of prey increase, c is the encounter rate or efficiency of capture, a is the rate at which the predator converts victim biomass into predator offspring, and d is the predator death rate. As you probably remember, there were four major assumptions underlying the LV predation model. In this lab, one of the points we will be examining is the assumption that individual predators consume a constant fraction of the prey at any given time. The change in the predator's feeding rate with respect to its prey's density is termed the predator's functional response. In general, when prey are more abundant we expect a predator to consume more of the prey. In the Lotka-Volterra model, the functional response of the predator is given as cN, where c is a constant (efficiency of capture) and N is the prey population size or density. This relationship leads to a simple linear functional response for the predator (Type I, Figure 1, slope = c). Unlike our simple model, real predators are often expected to exhibit
I
Np/t
II
III
Prey Density
Prey Density
Prey Density
Figure 1. Type I, II, and III functional response curves. Np/t is the number of prey consumed per unit time.
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finally decreasing and leveling off at very high prey densities. A Type III functional response may occur when a predator forms a search image or increases its capture efficiency of the prey as the prey becomes more abundant. Another possible explanation is that a predator switches from one prey item to another, as it becomes more abundant. Eventually, as in the Type II response, the predator reaches a limit on the amount of prey it can eat.
II. OPTIMAL FORAGING THEORY
Over evolutionary time, natural selection favors behavioral traits that maximize survival and reproductive success (fitness). Many traits involve tradeoffs of fitness benefits and fitness costs. For example, longer leg bones may allow a predator to run faster (catch more prey) but longer leg bones may also increase the chances of breaking a leg. Optimality theory proposes that an organism's attributes or behaviors are the result of natural selection, which results in an optimal ratio of fitness benefits to fitness costs. The theory of optimality has been extensively applied to the foraging behavior of animals. Optimal foraging theory deals with the cost-benefit analysis of an organism's feeding behavior. There are major assumptions underlying optimal foraging theory. As detailed below, these assumptions are generally reasonable; however, if any of these assumptions do not hold, then an organism's behaviors might not be expected to match the predictions of optimal foraging theory.
Assumptions of Optimal Foraging Theory: Higher energy intake per unit effort leads to higher fitness. Foraging behavior is subject to natural selection. Animals are either foraging in the environments in which they evolved or have evolved the mental capacity to learn optimal behaviors (by trail and error) in new environments.
The principal determinant of an organism's success is its ability to locate and ingest food. For an organism to survive, it must obtain enough food to offset the costs of obtaining the food. If an organism consistently captures fewer caloriesworth of food than it burns while attempting to find the food, it will quickly perish because there is a net energy loss. An organism must secure significantly more calories than it burns if it is to reproduce, because much more additional energy must be invested to produce offspring. In species that rear their young, the parents that are best able to maximize their foraging yield will be able to provide the most resources, resulting in the fittest offspring. Time-efficient foraging behaviors may also allow a parent to spend more time defending their young from predators or teaching them important survival skills. Even in species with little or no parental investment, the individuals that feed optimally will be able to produce more, higher quality sperm or eggs or will be able to out-compete others for mating opportunities. Students of optimal foraging theory are interested in the following questions:
How long should an area be searched for food before moving to a new area? How far from a home site should an animal search for prey? Is a particular patch of food worth stopping at? Which prey should be pursued? In plants, how much energy should be allocated to roots (foraging for nutrients) versus leaves and stems (foraging for light)?
The "Functional Response" section of this lab discussed the response of predators to changing prey density. Even at high prey densities, it is beneficial for a predator to select prey types that maximize the ratio of fitness benefits to fitness costs. Often, the prey population is comprised of individuals of different ages and sizes. Larger prey obviously hold more nutritional worth than smaller prey, yet the larger prey may also require more energy to capture, subdue, and ingest (handling time). Likewise, predators are comprised of individuals of different sizes and ages. Younger, smaller predators are usually less
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experienced and expend more energy when capturing prey (handling time) or may even be harmed by large prey. It should be obvious from these different possibilities that understanding how and why predators consume their prey requires more than simply examining functional response curves for an individual predator and one of its prey items. Still, predictions from simple optimal foraging theory are surprisingly robust.
Factors that Affect Optimal Foraging Strategies
Often, it is not as easy to determine if a predator is feeding optimally. When a predator's foraging behavior does not seem to be optimal, key factors affecting the predator have probably been ignored. Although Pacific oystercatchers (a type of shore bird) were predicted to eat large mussels by an optimality model, they ate mussels that were mostly intermediate in size (Meire & Ervynck, 1986). It turned out that older mussels were more likely to be covered by barnacles, which made them very difficult to open. In other words, handling time was dramatically longer for older, larger mussels. Thus, optimally foraging oystercatchers would not be expected to consume the largest mussels because the costs outweigh the benefits. Competition with other predators may also affect the feeding habits of an organism. It may be optimal for a competitively inferior species to pursue less profitable prey in order to reduce competitive interactions (Milinsky, 1984). This illustrates how knowledge of detailed conditions may be needed to understand foraging behaviors. As another example of how conditions affect animal foraging, bluegills, a common pond fish on the mainland, sacrificed 50% of their potential feeding time to remain hidden when in the presence of largemouth bass, their predators (Gotceitas, 1990). The benefit of not foraging while largemouth bass were present was greater than the cost of the lost foraging time.
Evidence of Optimal Foraging
Do animals forage optimally? Often they do! There is a growing body of scientific evidence supporting the theory of optimal foraging. A specific example of optimal foraging has been shown in Northwestern Crows (Richardson & Verbeek, 1986). The crows were observed foraging for clams in the intertidal zone. The crows would dig up many different sizes of clams, and would often leave the smaller clams in favor of larger clams. The handling time was almost equal for all clam sizes, but larger clams contained more calories. Richardson & Verbeek (1986) created an optimality model based on searching costs to predict the percentage of clams that should be eaten, from each size class, after they were excavated. After creating the model, the crows feeding behavior was observed and recorded (Figure 2). The observational data fit the optimality model very well, indicating that the crows were foraging optimally for clams (maximizing caloric uptake per calorie expended).
120 100
Percentage Eaten
Optimal Choice
In determining whether an animal is foraging optimally, the fitness costs and benefits must be determined. An animal is considered to forage optimally when it obtains the maximum energy (E) per unit time (T). The time variable (T) can be split into two components: T s and T h. Search time (T s) represents the time spent looking for prey, handling time (T h) represents the time needed to subdue and eat the prey after it is found. Sometimes it's acceptable to assume that search time (T s) is equal for all available prey items. Fitness cost is then proportional to handling time
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80 60 40 20 0 20 25 30 Clam Length (mm) 35 40
Figure 2. Optimality model (solid line) and corresponding data on northwestern crows foraging for clams from Richardson & Verbeek (1986). As you can see, the model fits the actual data very well.
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(Th), and fitness benefit is proportional to the energy obtained per prey item (E p). The profitability (P) of a prey item is then defined as:
energy intake of a predator that chooses to eat only prey type 1 with the following equation:
P = Ep/Th .
Ep (caloric content of the prey) can be determined using an energy-measuring device such as a bomb calorimeter. Th can be measured by determining how long it takes the predator to consume a prey specimen after it has been located. Most predators do not feed exclusively on one prey type; therefore, it is desirable to calculate the profitability of feeding on multiple prey types. For example, the overall energy intake, E/T, for a predator feeding on three prey types is calculated as follows:
E Ei 1 = T 1 + Th1
These equations are useful for determining which predator feeding behavior should be followed to optimize E/T. Then you can observe a predator's actual behavior to determine if the behavior matches optimal foraging theory.
Marginal Value Theorem
According to the law of diminishing returns, remaining in an area (resource patch) until all prey in the patch have been eaten is not usually optimal (Charnov, 1976). This idea has been adopted as the marginal value theorem. As prey become rare in a patch, they become more difficult to find, i.e. search time becomes exceedingly high. The optimal time for a predator to leave a patch depends on the search time required to find a new patch, the expected prey density in the new patch, and the quality of the present patch. Predators should remain in a patch longer when it contains high quality prey, and when locating new resource patches is difficult. It is possible to determine the optimal time to leave a patch using graphical methods, but that will not be formally discussed in this class. Nevertheless, in today's lab you will need to make decisions about how long to forage in a patch. You should consider the factors described above in order to maximize your chances of survival.
E 1 Ei 1 + 2 Ei 2 + 3 Ei 3 = T 1 + 1Th1 + 2Th 2 + 3Th 3
where is the rate at which a prey item is encountered by the predator and Ei is the average net energy obtained per prey type (1, 2, or 3 as specified by the subscript). can be calculated as the number of prey encountered per total search time (T s) where T s can be calculated by subtracting the handling time (T h) from the total time (T):
Ts = T - Th
When a predator is equally skilled at locating all prey, (the encounter rate) for each prey item is proportional to the relative abundance of each prey item (i.e. more abundant prey items are encountered more often). Each prey type may have a different handling time and energy content. The big E/T equation above assumes that the predator's selection among the available prey items is random. If certain prey types have low profitability, it may be optimal for a predator to pass up certain prey when they are encountered, and spend time locating other prey rather than handling prey with low profitability. We can calculate the overall
Ideal Free Distribution
It is obvious that all food patches are not created equally. Some patches of food may hold more prey, higher quality prey, or both. As a result, it is expected that predators will gravitate towards the highest quality patches. As predators flood the high quality patches, it becomes more beneficial for predators to leave the high quality patches to forage in lower quality patches, due to the high level of competition in the high quality food patches. If allowed enough time, predators are
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expected to be distributed in such a manner that the number of predators in an area is proportional to the quality of the food patch. As a result, all of the predators will have relatively equal foraging success. This concept is called the ideal free distribution of predators (Fretwell & Lucas, 1970). A simple example of the ideal free distribution is the distribution of cats on campus. The prime food patch is on Maile Way, where people feed the cats most often. This also happens to be the place where the cats are the most abundant. Not all cats on campus inhabit this area, though. Some choose to hang around the cafeterias and dormitories. To test whether the distribution of campus cats matches the ideal free distribution, would we need to determine if all of the cats were receiving a nearly equal amount of nutrients over the long term, regardless of where they choose to forage on campus.
return the captured prey to the original feeding can. Recording the data for your functional response curve Prey only upon the barley (1 minute). The rice should be thought of as a non-food item (e.g. rocks or sand) that interferes with your ability to find prey. Repeat the feeding trail 2 - 3 times, each time returning the contents of your gut to the main patch can and shaking the patch can. Do not search for pray by stirring the content of the can with your forceps, but pick from whatever you can see/is on top!! The results for each student will be shared so that everybody has a complete data set. The data will be used to create a functional response curve for the predator.
B. Optimal Foraging In the Lab
We will conduct an optimal foraging experiment in the lab to see if we can predict the optimal foraging strategy for a juvenile bird predator and an adult bird predator. You will play the role of the bird predator. Patches of food (tin cans) will be set up on a table in the room. Each patch contains 50 kidney beans, 50 garbanzo beans, 50 black-eyed peas, and 50 barley grains. The seeds represent your prey, and each prey item contains a different amount of energy (Table I). The kidney beans are a non-food item (no caloric value).
Table I. Hypothetical net caloric worth of each prey item.
III. LAB EXERCISES
A. Functional Response
We will conduct an experiment to examine the shape of a predator's functional response curve. The predator in this experiment is YOU! Tin cans will contain both barley and rice (these are simulated prey) in varying proportions ranging from 1:0 to 0:1 (see Functional Response data sheet p. 7-11). An assumption we must make in generating the response curve is that these proportions do not change much during feeding. There are a large number of prey items in each can, so we do not expect a large change in prey proportions even if prey are selectively consumed. Each student has a can with a different ratio of barley to rice, as indicated on the side of the can. When a feeding period begins, pick up your forceps, remove your prey, and place prey in your gut cup or the lid of your can. To complete the handling process for each prey item, you must put down the forceps after successfully dropping a prey in your gut. When a feeding period is complete, stop removing prey and calculate your rate of prey consumption by counting the prey in your gut. Before beginning a new feeding period,
Prey Item Kidney beans (non prey) Garbanzo beans (Lrg) Black-eyed peas (Med) Barley grains (Sml)
Caloric Value (Ei) 0 1 5 10
Half of the class will play the role of an adult bird predator and half of the class will play the role of a juvenile bird predator of the same species. The adult predator, with its precision-shaped beak (consisting of a pair of forceps), is superior at handling all prey items. The juvenile predator is inexperienced and clumsier when it comes to food
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handling (especially small grains); the juvenile's beak consists of a pair of chopsticks. There are also other differences between adult and juvenile. Due to its larger body size and more intense feeding behavior, adults have higher energy requirements than juveniles (Table II).
Table II. Metabolic cost of foraging for 10 minutes.
Metabolic Cost
Adult 350 calories
Juvenile 150 calories
When you feel you have optimally depleted your patch (recall the marginal value theorem), you can get up and get a new patch from the table, but it will cost you 10 calories to get to the new patch. Make sure to first count and then return all prey items back into their original can before switching!!! In addition, there is interference competition between the adults and juveniles when moving between patches. When searching for a new patch, adults will always get first choice because of their more aggressive behavior. A juvenile's life is tough; as a juvenile, you must be gracious and allow the adults to pick a new patch first (let adults cut in line ahead of you). One day you'll be a big bird and you'll have your day. Handling Time Data For Theoretical Optimality Predictions (1 minute trials) First, we will collect handling time data to make predictions about the optimal foraging strategy for adults and juveniles. Each student will get a patch (can) of food and will be instructed to forage for one of the three types of prey. Half of the class will be juveniles (chopsticks), while the other half will be adults (forceps). You will have one minute in which to remove as many prey items as possible and place them in your gut cup (only one prey type per person). You will see plenty of prey items in your can, so during the one minute trials we will consider search time to be zero second. You are not spending time searching for prey. Instead, you are handling the prey (picking them
up and putting them into your gut cup). During the 1 minute trials, we will consider the handling time to be 1 minute (60 seconds). Again, to complete the handling process for each prey item, you must put down the forceps/chopsticks after successfully dropping a prey in your gut. Work as quickly as you can and record the number of seeds you were able to ingest in the allotted time. The class's handling time data for each prey type will be pooled. Return all seeds from your gut to the original cup and place it back on the table at the front of the room. Data from the 1-minute foraging trails will be used to make predictions about which foraging strategy is theoretically optimal over the long term. These predictions will then be compared to the optimal strategy when you are free to forage as you choose in the "Real World." Foraging Experiment to Test Which Strategy is Best in Your "Real Bird World" (10 minute trials) To determine which feeding strategy is best among real birds (your lab mates), you will run 10-minute "survival tests" with birds (lab mates) taking on different feeding strategies. You will then determine which strategies are viable (obtain enough calories to survive) and which is the overall optimal (maximum calories) in our real "life-and-death" Bird World. Each student should obtain 8 empty cups (your guts) and a pair of forceps or chopsticks, depending on whether you're an adult or a juvenile. Before beginning, each person will be assigned one of three specific feeding strategies: 1) only barley; 2) barley and black-eyed peas; or 3) barley, black-eyed peas, and garbanzo beans. Stick to your feeding strategy, and do not deviate from it. If your feeding strategy involves two or more prey items, you should try to consume both of those prey items as you encounter them rather than searching out one type of prey. When the experiment begins, you should go to the patches (cans of beans), and pick a number out of the box. The number will correspond to a patch on the table. Take that numbered piece of paper
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and the patch can back to your seat. Write the patch number on your gut cup (this way the patch can be replenished in the correct proportions when the lab is over). Your search time will be mostly the time you spend between patches and time needed to record your new patch numbers. We know from past labs that the average time spent changing cans (search time) is 2 minutes; therefore, each 10 minute trail will involve about 8 minutes of handling time. Initially, all cans contain equal proportions of the three prey items. Pick up your forceps or chopsticks and capture your prey. Ingest the prey by transferring them to your "gut". After you put the seed in your gut, you must put down the forceps/chopsticks. Recall that forceps/chopsticks on the table is required to complete the handling process of each prey item (that's just how our bird world works). It is your choice when you'd like to move on to a new patch (consider the marginal value theorem). To do this, place your present patch back on the table, put the numbered piece of paper back in the box, and get in line for a new patch. Do not look in your patch until you've reached your seat. You must keep track of your patch changes because each change costs an extra 10 caloric units. Remember, adults can cut in front of juveniles. The unforaged patches will disappear quickly. When you switch patches, your new patch may have already been substantially depleted, depending on the foraging strategy of the previous bird. When you return to your seat, use a new cup (gut), and write the patch number on the new gut cup. The exercise will last for 10 minutes. After 10 minutes are up, tally up your "score" (Table 1), subtract the metabolic cost of foraging for 10 minutes (Table 2), and the patch switching cost (10 calories per switch). The patches should be returned to their original state (replacing all prey in their original patch) by emptying your numbered gut cups into the appropriate patch. It is very important that you do this correctly
because all lab sections this week will be using the same patches. Adults/juveniles will switch roles for the second foraging period. At the conclusion of the second foraging period, tally your score; return the patches to their previous state, and the class's data will be compiled.
IV. ASSIGNMENTS
Functional Response
1. a. Plot the functional response curve (Figure 1) for the barley prey, based on the average prey captured from three trials at each prey density. Be sure to label the UNITS on the y-axis. b. Which standard functional response does it most closely match? c. What type of foraging behavior do you think you are following based on your functional response curve? (3 sentences max)
Optimal Foraging In the Lab
2. a. Use the E/T equations in the "optimal choice" section and the data on juvenile and adult handling times (the one-minute trials) to predict which feeding strategy should have been optimal for adults versus for juveniles. (You are coming up with theoretical predictions based on the 1minute trials)
Estimating (encounter rate): To calculate we will need to borrow some information from our 10 minute "real world" foraging trails: 1 = total number of prey1 encountered / Ts As was previously mentioned, for a particular prey item can be assumed to be proportional to that prey item's abundance. In our real world, each patch had an equal proportion of all 3 prey types so 1=2=3= (the encounter rate for all three prey items was the same) From past lab experience, we know that the average Ts (time spent moving between patches) is about 2 minutes during a 10 minute trial.
(Questions continue on next page)
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Finally, we will assume that each encountered prey was captured by adult birds during the 10 minute trial, so the number of prey items caught by adults represents the encounter rates (juveniles probably dropped some prey, so using the juvenile numbers would be less accurate). Using these assumptions, we can calculate a single , which is the same for each of the three prey items.
= (class average for number of total prey eaten per adult bird during the 10 minute trail X 0.33) / 2 minutes
Note: The value of should be calculated before you leave lab, since everyone in your lab should use the same value. Again, you will use the same for all prey items since they were initially equally abundant. The is assumed to be a constant that does not change with time.
"Real Bird World" Foraging Experiment 5. a. Calculate the E/T values for each of the 3 strategies based on your data from the "Real Bird World" b. Rank the "Real Bird World" results for the 3 feeding strategies from the best to the worst strategy. c. Do the rankings match those from the theoretical predictions (question 2)? d. How closely do you think your actual foraging behavior matched the theoretical foraging behavior? e. In the real bird world, did all strategies allow the birds to survive?
V. REFERENCES
Charnov, E.L. 1976. Optimal foraging: the marginal value theorem. Theoretical Population Biology 9:129-136. Fretwell, S. D. and Lucas, H. L. 1970. On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheor. 19: 16-36. Gotceitas, V. 1990. Foraging and predator avoidance: a test of a patch choice model with juvenile bluegill sunfish. Oecologia 83: 346-351. Krebs, J.R. and R.H. McCleary. 1984. Optimization and behavioral ecology, pp 91121. In: J.R. Krebs and N.B. Davies (eds.) Behavioral Ecology: An Evolutionary Synthesis. Sinauer. Sunderland. Meire, P.M. & A. Ervynck. 1986. Are oystercatchers (Haemoptopus ostralegus) selecting the most profitable mussels (Mytilus edulis)? Animal Behaviour 34: 1427-1435. Milinsky, M. 1984. Parasites determine a predator's optimal feeding strategy. Behavioral Ecology and Sociobiology 15: 3538. Richardson, H. & N.A.M. Verbeek. 1986. Diet selection and optimization by northwestern crows feeding on Japanese littleneck clams. Ecology 67: 1219-1226.
Theoretical Optimality Predictions 2. b. Is it theoretically optimal for adults to select only barley; barley and black-eyed peas; or barley, black-eyed peas and garbanzo beans? c. What are the predictions for juveniles? d. Present your results in a column graph with foraging strategy on the x axis and E/T on the y-axis. (See Lab 0, Column Graphs for instructions on making graph) (3 sentences max) 3. a. Based on your calculations of E/T in question 2, and knowing the metabolic costs of survival (Table 2), are the juveniles and adults theoretically predicted to survive when utilizing the feeding strategies employed in the 10 minute Foraging Experiment? Show calculations. b. Why or why not? (3 sentences max)
Remember: E/T has units of calories per minute. The metabolic costs given in Table 2 have units of calories per ten minutes.
4. If we increased the number of kidney beans in each cup by 10-fold, which of the following variables (Th, Ts, , Ei) would you expect would be affected and why? (4 sentences max)
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A. Functional Response Data Sheet (record # of barley eaten)
Barley%
100 90 80 75 70 67 60 50 40 33 30 25 20 10 0
Rice%
0 10 20 25 30 33 40 50 60 67 70 75 80 90 100
replicate replicate replicate 1 2 3 average
B. Handling Time Data for Optimality Predictions
Seeds/Min
replicate # 1 2 3 4 5 6 Garb Adults BEP Barley Garb Juveniles BEP Barley
Handling Time = Min/Seed
replicate # 1 2 3 4 5 6 AVG Garb Adults BEP Barley Garb Juveniles BEP Barley
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"Real World" Foraging Experiment Data Sheet
Caloric Intake
Number of Seeds Eaten
replicate 1 2 3 4 5 6 Avg Ei Avg Caloric Intake/item Avg Caloric Intake/strategy Barley Bar Adults Bar & BEP Bar, BEP, & Garb Bar BEP Bar BEP Garb Barley Bar Juveniles Bar & BEP Bar, BEP, & Garb Bar BEP Bar BEP Garb
X 10
X 10
X5
X 10
X5
X1
X 10
X 10
X5
X 10
X5
X1
E/T
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"Real World" Foraging Experiment Data Sheet
Caloric Output
Number of Patch Changes
Adults replicate 1 2 3 4 5 6 Avg Patch Change Caloric Expenditure STD Caloric Expenditure Total Average Caloric Expenditure 350 350 350 150 150 150 Barley Barley & BEP Barley, BEP, & Garb Barley Juveniles Barley & BEP Barley, BEP, & Garb
X 10
X 10
X 10
X 10
X 10
X 10
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