Holling Disc Lab - St. Olaf Pages...Holling “disc equation.” It was given this moniker because Holling generated it by having a blindfolded assistant pick up (“prey upon”)

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HollingDiscLabThislabisslightlymodifiedbyEmilyMohlfromalabfoundattheBioMathLabwebsiteprovidedbyDr.JamesHaefneratUtahStateUniversity:http://www.indiana.edu/~oso/lessons/BioMath/BioMathLab.html

Introduction

There are many examples in nature that demonstrate that predators can control the numbers of their prey. For example, predatory lampreys almost eliminated the lake trout in the Great Lakes between 1950 and 1960. Wolves kept the elk numbers in check in the Yellowstone area before humans eliminated the predator. The goal of this lab is to take a closer look at the nature of predator/prey interactions. When predators are faced with increasing local density of their prey, they often respond by changing their consumption rate. This relationship of an individual predators’ rate of food consumption to prey density was termed the functional response by C.S. Holling in 1959. In this lab, we will repeat the method Holling used to establish the basis for the analysis of the functional response. He developed the conceptual model using blindfolded human subjects (the “predator”) and 4-cm sandpaper discs (the “prey”).

As you complete this lab, think about how herbivores might be similar to or different from the predators typically used to study functional responses.

Objectives In addition to learning about functional responses, this lab is designed to

1. Introduce you to data analysis using the statistical computer program R.

2. Help you think about ways to efficiently collect, enter, and analyze data.

Background: Three Types of Functional

2

Responses

Figure 1 illustrates the three general types of curves expected in various predator-prey situations.

Type I is a linear relationship, where the predator is able to keep up with increasing density of prey by eating them in direct proportion to their abundance in the environment. If they eat 10% of the prey at low density, they continue to eat 10% of them at high densities. The dotted line indicates a maximum consumption rate that some authors attach to Type I foraging.

Type II describes a situation in which the number of prey consumed per predator initially rises quickly as the density of prey increases but then levels off with further increase in prey density.

Type III resembles Type II in having an upper limit to prey consumption, but differs in that the response of predators to prey is depressed at low prey density.

Figure 1. Three functional responses, showing the relationship between prey density and the number of prey eaten by a predator in a given amount of time.

Spring 2001 Holling Disc Equation: Student Manual Biology 1240

The Holling Disc EquationIntroduction

There are many examples in nature which demonstrate that predators can control the numbersof their prey. For example, predatory lampreys almost eliminated the lake trout in the GreatLakes between 1950 and 1960. Wolves kept the elk numbers in check in the Yellowstone areabefore humans eliminated the predator. The goal of this lab is to take a closer look at the natureof predator/prey interactions. When predators are faced with increasing local density of theirprey, they often respond by changing their consumption rate. This relationship of an individualpredators’ rate of food consumption to prey density was termed the functional response by C.S.Holling in 1959. In this lab, we will repeat the method Holling used to establish the basis for theanalysis of the functional response. He developed the conceptual model using blindfolded humansubjects (the “predator”) and 4-cm sandpaper discs (the “prey”).

The Three Types of Functional Responses

Figure 1 illustrates the three general types of curves expected in various predator-prey situations.

Type I is a linear relationship, where the predator is able to keep up with increasing density of preyby eating them in direct proportion to their abundance in the environment. If they eat 10% of theprey at low density, they continue to eat 10% of them at high densities. The dotted line indicatesa maximum consumption rate that some authors attach to Type I foraging.

Type II describes a situation in which the number of prey consumed per predator initially risesquickly as the density of prey increases but then levels o↵ with further increase in prey density.

Type III resembles Type II in having an upper limit to prey consumption, but di↵ers in that theresponse of predators to prey is depressed at low prey density.

Type II

Type III

Type I

Num

ber

Prey

Eat

en /

Tim

e

(Pe)

Prey Density (N)

Figure 1: Three types of functional response relating prey density (N) and the number of preyeaten by one predator (Pe).

–1–

3

The Math and Logic of the Holling Disc Equation

Two factors dictate that the functional response should reach a plateau. First, the predators may become satiated (i.e., their stomach completely filled), at which point their rate of feeding is limited by the rate at which they can digest and assimilate food. Second, as the predator captures more prey, the time spent handling and eating the prey lowers the searching time. Eventually, the predator reaches the minimum time it takes to search, capture and consume. The predator cannot find prey and eat it any faster. The consumption rate cannot increase when the ability of the predator to catch and eat is at a maximum.

C.S. Holling described this relationship between search time, handling time, and consumption rate by a simple expression known as the “disc equation.” The equation was developed using blindfolded human subjects trying to find and pickup small discs of sandpaper on a flat surface. Any such task, including subduing and eating a prey item (whether a model of sandpaper disc “prey” or a real prey ) requires the following measurable elements that we will incorporate into a mathematical equation following the logic of Holling:

N = density of prey

a′ = attack rate or searching efficiency

Ttot = total time spent

Ts = total search time for all prey

Th = handling time per prey item

Pe = number of prey eaten during a period of time searching �

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We make the following assumptions. Pe increases with the time available for searching (Ts), the prey density (N), and with the searching efficiency or attack rate of the predator (a′). This is summarized as:

Pe = a′TsN (1)

As you will see when you take the Holling Disc data for yourself, search time (Ts) decreases as prey numbers (N) increase. As a result, Ts is not a constant. Thus, this equation as written will change for each different circumstance which a predator experiences. To achieve generality, we need to continue to reduce the predation process to more fundamental terms. We begin by removing the Ts term in the following way.

The time available for searching will be less than the total time, Ttot, because of time spent handling prey. Hence, if Th is the handling time of each prey item, then the product ThPe is the total time spent handling all prey consumed during the foraging bout:

Ts = Ttot − ThPe � (2)

Substituting this into equation 1 we have:

Pe = a′(Ttot − ThPe)N (3)

Rearranging, the equation gives us the Holling Disc Equation itself:

(4)

Equation 4 describes a Type II functional response, and is known as the Holling “disc equation.” It was given this moniker because Holling generated it by having a blindfolded assistant pick up (“prey upon”) small sandpaper discs placed on a table. Note that the equation describes the amount eaten during a specified period of time: Ttot. The


Rearranging, the equation gives us the Holling Disc Equation itself:

Pe =a

0NTtot

1 + a

0ThN

(4)

Mini-quiz: Test your algebra skills and show how to get from equation 3 to 4. It will take three steps.

Equation 4 describes a Type II functional response, and is known as the Holling “disc equation.” Itwas given this moniker because Holling generated it by having a blindfolded assistant pick up (“preyupon”) small sandpaper discs placed on a table. Note that the equation describes the amount eatenduring a specified period of time: Ttot. The density of the prey is assumed to remain constantthroughout that period. In experiments, this can be guaranteed by replacing any prey that areeaten.

Holling was not content to merely show that this was a basic relation for a blind human searchingfor sandpaper discs. He also demonstrated that Type II curves were generated when three speciesof small mammals preyed upon cocoons of the sawfly in a controlled laboratory setting.

Questions

1. Equation 4 describes the curves in Figure 1a (II). Look at equation 4 above and describe whatwould happen to the shape of the graph Th, got smaller and smaller and approached zero.

2. Using simple algebraic manipulation, derive equation 4 from equation 3

3. How would you estimate a

0, the attack rate or searching e�ciency?

Today’s Lab

You will now simulate several bouts of predation using one of your group as a predator and theothers to keep track of handling time, total elapse time, and number of prey “consumed.” We haveset up the details exactly as Holling did in his 1959 study. Work in groups of 4 or 5. At each labbench there is a predation arena with a di↵erent density:

–3–

5

density of the prey is assumed to remain constant throughout that period. In experiments, this can be guaranteed by replacing any prey that are eaten.

Holling was not content to merely show that this was a basic relation for a blind human searching for sandpaper discs. He also demonstrated that Type II curves were generated when three species of small mammals preyed upon cocoons of the sawfly in a controlled laboratory setting.

Stop and Think: 1. Can you rearrange eq. 3 to get eq. 4 in three steps? 2. Equation 4 describes the curves in Figure 1a (II). Look at equation 4 above and describe what would happen to the shape of the graph Th, got smaller and smaller and approached zero.

3. How might you estimate a′, the attack rate or searching efficiency?

4. If herbivores showed a functional response like predators, how might you modify this logic to be relevant to the process of herbivore foraging?

6

The Lab

You will now simulate several bouts of predation using one member of your group as a predator and the others to keep track of handling time, total elapse time, and number of prey “consumed.” We have set up the lab in a similar way as Holling did in his 1959 study.

Work in groups of 3 or 4. At each lab bench there is a predation arena with a different density:

Equipment for Each Lab Bench

• 1 predation board with discs � • 1 stop watch � • 1 small plastic tub to put eaten prey into � • 1 small plastic tub with new replacement prey • 1 blindfold Assign members of your team to the following chores:

• 1 predator (to be blindfolded) [person a] � • 1 person to measure handling time on a stopwatch [person b] � • 1 person to replace “eaten” prey with a new sandpaper disc [person c] Note: Additional group members can monitor time for the 1-min predation sessions.

Once you are organized, follow the instructions on the next several pages to explore the effects of alternative predator behavior.


No. of prey discsBench per 9 square feet

1 2562 1733 904 305 106 4

Equipment for Each Lab Bench

• 1 predation board with sandpaper discs

• 1 count up/down timer

• 1 small plastic tub to put eaten prey into

• 1 small plastic tub with new replacement prey

• 1 blindfold

Assign members of your team to the following chores:

• 1 predator (to be blindfolded) [person a]

• 1 person to measure handling time on a stopwatch [person b]

• 1 person to monitor the 1 minute predation bout duration with a timer [person c]

• 1 person to replace “eaten” prey with a new sandpaper disc [person d]

Note: The last two assignments can be a single person.

Once you have organized yourselves with the above materials and assignments, follow the instruc-tions on the next several pages to explore the e↵ects of alternative predator behavior.

Type II Response

You are now ready to investigate the predation rates of a Type II forager.

Blindfold the predator and instruct him/her to search for prey by tapping a finger tip randomlyaround on the board. No sliding allowed, only tapping. The predator must be hungry and in ahurry. No leisurely tapping allowed. When the predator detects a piece of sandpaper, he picks itup with the thumb tack and locates the prey tub (stomach). When the predation technique hasbeen approved by the whole group, you are ready to begin.

Here is what should happen in your group:

1. Person c [total time] start your stopwatch while shouting “go”.

2. When the predator encounters a disc, the handling time person [b] starts his or her watch.

–4–

7

Type II Response

You are now ready to investigate the predation rates of a Type II forager.

Blindfold the predator and instruct him/her to search for prey by tapping a finger tip randomly around on the board. No sliding allowed, only tapping. The predator must be hungry and in a hurry. No leisurely tapping allowed. When the predator detects a disc, he picks it up and locates the prey tub (stomach). When the predation technique has been approved by the whole group, you are ready to begin.

Here is what should happen in your group:

1. The predator [person a] should first put on the blindfold. 2. Set up with the correct number of discs distributed randomly. 3. When the 1-minute foraging period begins, start time and shout

“go”. � The predator can begin tapping for discs. 4. When the predator encounters a disc, the handling time person

[b] starts his or her watch 5. The predator removes the disc and places it in a tub to the side.

As soon as the predator’s finger is back tapping on the board, the handling time person [b] stops the handling time stopwatch. � For handling time, you are recording the accumulated or total time spent in each activity. You do not have time to write down how much time each handling event takes. The value you record will be total handling time (

Th, “sum of all Th”). �

6. The prey replacement person [c] puts a new disc out. It must be placed in a different place than the location of the preyed upon item. (or the predator will know to go right back to that spot for another snack) �

7. Continue with this procedure until 60 seconds have elapsed and the timer yells, “stop.” �

8. Repeat these 60-second predation bouts two more times and record the results below on the table titled “Your Group Data Type II”. �

9. Record what the predation density is at your lab bench: _______ discs per 9 square �feet. �

8

10. Calculate the mean of all your trials and enter them into your groups data table. �

(Note: Total Handling Time (

Th) is the cumulative time recorded by person b.) �

11. Use the mean values from your group data table to enter your

data into the class data table on google sheets. The “Class Data Type II” table below shows a model of what the table will look like. Be sure to enter your data and the prey density correctly. For Type_FxnlResponse, enter “2”.

12. We will work together to use these data to make some plots using the statistical program R.

13. Use the space below to sketch the relationship between

prey density and prey eaten.


3. The predator removes the disc and places it in a tub to the side. As soon as the predator’sfinger is back tapping on the board, the handling time person [b] stops the handling timestopwatch.For handling time, you are recording the accumulated or total time spent in each activity.You do not have time to write down how much time each handling event takes. The valueyou record will be total handling time (

PTh, “sum of all Th”).

4. The prey replacement person [d] puts a new disc out and tacks it securely to the board. Itmust be placed in a di↵erent place than the location of the preyed upon item. (or the predatorwill know to go right back to that spot for another snack)

5. Continue with this procedure until 60 seconds have elapsed and the time minder [person c]yells, “stop.”

6. Repeat these 60-second predation bouts two more times and record the results on the suppliedcharts.First record what the predation density is at your lab bench: per 9 squarefeet.

7. Calculate the mean of all your trials and use these numbers for the class report.

Your Group Data Type IITrial Prey Eaten Total Handling Time Handling Time/Prey

(Pe) (P

Th) (Th = (P

Th)/Pe)

1

2

3

Mean

(Note: Total Handling Time (P

Th) is the cumulative time recorded by person b.)

8. Place your mean values in the chart on the front board or overhead and then copy otherstudent values to fill in your own chart of the complete Type II response:

CLASS DATA TYPE IIPrey Total Prey Consumed Total Handling Time Handling Time/Prey

Density (N) (Pe) (P

Th) (Th = (P

Th)/Pe)

4

10

30

90

173

256

–5–


3. The predator removes the disc and places it in a tub to the side. As soon as the predator’sfinger is back tapping on the board, the handling time person [b] stops the handling timestopwatch.For handling time, you are recording the accumulated or total time spent in each activity.You do not have time to write down how much time each handling event takes. The valueyou record will be total handling time (

PTh, “sum of all Th”).

4. The prey replacement person [d] puts a new disc out and tacks it securely to the board. Itmust be placed in a di↵erent place than the location of the preyed upon item. (or the predatorwill know to go right back to that spot for another snack)

5. Continue with this procedure until 60 seconds have elapsed and the time minder [person c]yells, “stop.”

6. Repeat these 60-second predation bouts two more times and record the results on the suppliedcharts.First record what the predation density is at your lab bench: per 9 squarefeet.

7. Calculate the mean of all your trials and use these numbers for the class report.

Your Group Data Type IITrial Prey Eaten Total Handling Time Handling Time/Prey

(Pe) (P

Th) (Th = (P

Th)/Pe)

1

2

3

Mean



8. Place your mean values in the chart on the front board or overhead and then copy otherstudent values to fill in your own chart of the complete Type II response:

CLASS DATA TYPE IIPrey Total Prey Consumed Total Handling Time Handling Time/Prey

Density (N) (Pe) (P

Th) (Th = (P

Th)/Pe)

4

10

30

90

173

256

–5–

9

Type I Response

Next, we consider a predator whose handling time is extremely short. Recall our basic disc equation:

As handling time gets shorter and shorter (Th → 0), the denominator in the above equation approaches 1.0. The graph of this function (prey eaten (y axis ) vs prey density (N)) becomes more like a straight line with slope a′. This is the Type I response described in Fig. 1 on an earlier page.

To simulate this predator/prey situation we try to shorten Th as much as possible. This time, run three bouts of predation lasting one minute (as before) with the following difference in steps 4 and 5.

4. As soon as the predator locates a disc, the predator indicates it by stopping the tapping. At this point, the prey replacement person whisks the prey away from the spot for the predator and places it in the tub to the side. The predator continues to forage without interruption. The prey replacement person then takes a prey item from the side tub of extras and places it in a new location as before. The density of prey stays the same, as before.

5. The handling time person still times the short time between location of the disc and the time the predator starts tapping again. Th should be shorter, but not zero, compared to the previous Type II series.

Complete the following table for your group.


Figure 2: Class data for functional type II.

(Note: Total Handling Time is the cumulative time recorded by person b.)

9. Next, in Fig. 2 plot this curve: Prey eaten as a function of prey density.

When all the groups are ready, proceed to the next experiment.

Type I Response

Next, we consider a predator whose handling time is extremely short.

Recall our basic disc equation:

Pe =a

0NTtot

1 + a

0ThN

As handling time gets shorter and shorter (Th ! 0), the denominator in the above equationapproaches 1.0. The graph of this function (prey eaten (y axis ) vs prey density (N)) becomes morelike a straight line with slope a

0. This is the Type I response described in Fig. 1 on an earlier page.

To simulate this predator/prey situation we try to shorten Th as much as possible. This time, runthree bouts of predation lasting one minute (as before) with the following di↵erence in step 4.

–6–

10

(Note: Total Handling Time ( Th) is the cumulative time recorded by person b.)�

When all of the data and means have been calculated, enter your data into the group Google Sheet. Be sure to enter “1” for the type of functional response, and to enter the correct prey density.

In your group, use R to plot the relationship between pretty eaten and prey density. You may need to subset the data.

Question: Can you think of a predator/prey situation in which a Type I type of functional response would be expected?

Type III Response

If the prey has a refuge in the predation arena, or the predator learns while foraging (thus, making him/her a better forager with time), the Type III curve would be expected (see Fig. 1). In this functional response, the predator does less well at low densities, but increases its foraging rate at intermediate densities. At high prey density, the predator’s handling time limits its predation rate. This process results in the “S-shaped” curve in Fig. 1 above.


4. As soon as the predator locates a sandpaper disc prey, the predator indicates it by stopping thetapping. At this point, the prey replacement person whisks the prey away from the spot for thepredator and places it in the tub to the side. The predator continues to forage without interruption.The prey replacement person then takes a prey item from the side tub of extras and places it in anew location as before. The density of prey stays the same, as before.

NOTE: The handling time person still times the short time between location of the disc and thetime the predator starts tapping again. Th should be shorter, but not zero, compared to the previousType II series.

Complete the following table.

Your Group Data Type ITrial Prey Eaten Total Handling Time Handling Time/Prey

(Pe) (P

Th) (Th =P

Th/Pe)

1

2

3

Mean



Place your mean values in the chart on the front board or overhead and then copy other studentvalues to fill in your own chart of the complete Type I response:

CLASS DATA TYPE IPrey Total Prey Consumed Handling Time/Prey

Density (N) (Pe) (Th =P

Th/Pe)

4

10

30

90

173

256

When all of the data has been reported, graph the class data in Fig. 3.

Question: Can you think of a predator/prey situation in which a Type I type of functional responsewould be expected?

–7–

11

To simulate this, we will clump the prey. When the predator encounters a disc in a clump, he or she knows to go back to the clump. Before starting this bout, all members of the team (except the predator who is blindfolded) should clump the discs into groups. The bigger the total number of discs, the bigger each clump should be.

Use the experimental protocol for Type II foraging and complete the following table.

(Note: Total Handling Time ( Th) is the cumulative time recorded by person b.) �

Place your mean values into the shared data sheet. Be sure to enter the correct value and the correct prey density. For “Type_FxnlResponse,” enter “3”.

On your own, use R to make a graph of the Type III functional response. �

Estimating a′ Our problem is that we want to estimate the value of a′ for the Type II functional response. The equation is:


Your Group Data Type IIIPrey Eaten Total Handling Time Handling Time/Prey

Trial (Pe) (P

Th) (Th =P

Th/Pe)

1

2

3

Mean




CLASS DATA TYPE IIIPrey Total Prey Consumed Handling Time/Prey

Density (N) (Pe) (Th/Pe)

4

10

30

90

173

256

When all of the data have been reported, graph the functional response in Fig. 4.

Wait for the everyone to complete the graphing at which time the instructor will help you with thenext problem: What is the attack rate for the disc eaters?

Estimating a0

Our problem is that we want to estimate the value of a

0 for the Type II functional response. Theequation is:

Pe =a

0NTtot

1 + a

0ThN

–9–

12

We know that this equation is not a straight line, because the equation is not in the form of a straight line:

y = mx + b �

The maximum predation rate is 1/Th and is the maximum value that Pe can have. Attack rate, a′ in the Type II equation, determines how steeply the curve rises with increasing prey density (N).

Our experiments have given us a set of values for Pe for each value of N, but not a′. We want to estimate a′, taking into account the fact that it depends on all of the prey densities we experimented with and the fact that there is statistical variation in our estimates at each particular prey density. This is why we asked you to repeat the disc experiment three times.

The General Idea: Using Straight Lines

Many statistical tests are useful for linear data, but are much more complicated when data are nonlinear, or curved. One possible idea to solve this problem is to try and convert the curvilinear Type II equation as written above into a straight-line equation. Then, we might hope, we could graph our data and draw a straight line through the data points. Once we have a straight line, we will have estimates for the slope and the intercept. Maybe (we hope), one of these two quantities will tell us what a′ is.

How do we know a scheme like this might work? Well, you might not know it, but this idea has worked in the past for quantitative biologists on different problems (e.g., Beer-Lambert Law, Newton’s Cooling Law, photosynthesis equation). So, it might work again. This is how science often happens: get some experience, then apply what worked before to the new problem. If it works, great; if not, well ... it’s back to the drawing board and take another approach.



Trial (Pe) (P

Th) (Th =P

Th/Pe)

1

2

3

Mean






4

10

30

90

173

256



Estimating a0



Pe =a

0NTtot

1 + a

0ThN

–9–

13

But the main point is: Have courage! Try it and see if it works!

The Messy Details

We want to re-arrange the Type II equation to be a straight line.

We notice that the numerator looks kind of like mx, where x = N and (a′Ttot) = m. But the denominator causes problems with that. So, let’s simplify the denominator by taking the inverse of both sides.

This last equation has alot of terms in it, but it has the form

So,

�Let’s now re-write this one more time to see if we have a straight line:

Compare this with a straight line: �

y = mx + b �



Trial (Pe) (P

Th) (Th =P

Th/Pe)

1

2

3

Mean






4

10

30

90

173

256



Estimating a0



Pe =a

0NTtot

1 + a

0ThN

–9–


How do we know a scheme like this might work? Well, you might not know it, but this ideahas worked in the past for quantitative biologists on di↵erent problems (e.g., Beer-Lambert Law,Newton’s Cooling Law, photosynthesis equation). So, it might work again. This is how scienceoften happens: get some experience, then apply what worked before to the new problem. If itworks, great; if not, well ... it’s back to the drawing board and take another approach.


The Messy Details

We want to re-arrange the Type II equation to be a straight line. We notice that the numeratorlooks kind of like mx, where x = N and (a0

Ttot = m. But the denominator causes problems withthat. So, let’s simplify the denominator by taking the inverse of both sides.

1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot

Let’s now re-write this one more time to see if we have a straight line:

1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)

Compare this with a straight line:

y = mx + b

We have a straight line when we let

x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–

14


, ,and ,whereSo, if we plot the class data as 1/Pe on the y-axis against 1/N on the x-axis, we should have data points that appear to fall along a straight line.

We will use R to calculate new columns in our dataframe. These new columns will be N-1 and Pe

-1, which represent our linear estimates for x and y respectively.

Then, we will use R to plot the data.

Finally, we will conduct linear regressions to get estimates of the slope (m) and intercept (b) for our data.

In the space below, record the estimates for the slope and intercept, then figure out how to calculate a’ and Th (per prey) from those estimates.




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–

y =




The Messy Details



1Pe

=1 + a

0ThN

a

0TtotN


y =A + B

C

=A

C

+B

C

So,

1Pe

=1

a

0TtotN

+Th

Ttot


1Pe

=1

a

0Ttot

1N

+Th

Ttot(5)


y = mx + b


x =1N

m =1

a

0Ttot

and

b =Th

Ttot

–11–

15

Functional Response Problems Using R

1. Use the first sample data found in the tab labeled SD1 to estimate the following. Assume that Ttot = 1day. Include the units on your estimates. �

a. The type of functional response b. The attack rate (a’) c. The handling time (Th) per prey

HINTS/STEPS

• Download the data from the shared file and save it on your computer in a place you can locate. For our purposes, I recommend downloading just the SD1 sheet and save it as a .csv file.

• Import the data into Rcmdr. If you have saved your file as a .csv, you can go to Data>Import data>from text file. When the dialogue box comes up, you could name your file SD1, and be sure to click on the button for Commas under the Field Separator Heading. Then click OK.

• To make a plot with the raw data, you can go to Graphs>Scatterplot. Select N as your independent variable and Pe as your dependent variable, and choose OK.

• To perform a test to determine whether your graph has significant linear and nonlinear trends, you can do a Polynomial regression

o Go to Statistics>Fit models>Generalized Linear Model o Double click on Pe to select it for your Response (or

Dependent) variable. It should show up in the box before the ~.

o Highlight the N, then click on the raw polynomial button. This creates the right side of your equation, with both N and N2 as predictors.

o Click OK to run the model and look to see if the estimates for poly(N)1 and poly(N) 2 are significant to determine if there is a significant linear and nonlinear trend in the data.

• To transform, or linearize, your data following the method to calculate the attack rate and the handling time, compute new

16

variables that are the inverse of Pe and N. o Go to Data>Manage variables in active dataset>Compute

new variables o Enter a new variable Name. For example, I chose InvN.

Its best to avoid spaces and symbols in your names. o In the Expression to compute box, enter 1/N. Click OK. o Repeat to create the inverse of the Prey eaten variable.

• Plot your transformed data to see if the relationship looks more linear. Can you estimate the slope and intercept by looking?

• Now you are ready to do a linear regression with your transformed data to get estimates for the slope and intercept.

o Go to Statistics>Fit model>Linear Regression. o Choose your Response (DV/y) and your Predictor (IV/x).

! Hint, look back at the equations if you aren’t sure what to choose.

o Choose OK. o The estimate of the intercept and slope are given in the

output table. Check to see if they are reasonable given the plot you made previously.

• Use the estimates of the slope and intercept to calculate a’ and Th.

2. Using the coefficient (estimates) you calculated in SD1 for Th, what is maximum capture rate when prey are extremely dense (per day)? Look at the original data to see whether this is a reasonable estimate.

HINT: Maximum Capture Rate=1/Th

3. Using the data below, estimate a and Th by transforming the data and plotting. From this plot, use your best guess to estimate the two constants. � You are responsible for entering the data into R yourself.


Holling Disc Equation Problems

Do the following. For full credit, show all of your intermediate steps and algebra, where required.

DUE: Your lab instructor will inform you when this is due.

1. Using the data below, estimate a

0 and Th by transforming the data and plotting. From thisplot, use your best guess to estimate the two constants.

N 2 4 8 16 32Pe 2 3.5 7 7.5 8

2. Transform the following equation into a straight line. Show your algebra steps and identifythe slope and the intercept. x is the independent variable and y is the dependent variable, qand r are constants.

y =qx

3q + rx

3. You perform a foraging experiment like the Holling disc equation in class on an organismyou’ve never studied before. You present the predator with prey densities that increase byunits of 5 (i.e., 5, 10, 15, 20, etc.) At each experiment, you observe that the predator eatsexactly half of the prey available.

(a) What is the equation for the functional response for this organism? [Hint: it is not TypeII.]

(b) What is the attack rate?

4. You are the head manager of the Wasatch National Park where cougar prey on deer. Yourjob is to manage the deer herd so that the cougars will maintain constant numbers. Supposethat for the cougar population to stay constant, each cougar must eat on average 3.75 deerper month. Using the values below, how many deer are needed in the Park? Units are basedon one month.

a

0 = 0.005 Th = 0.067 (about 2 days)

A calculator answer is not su�cient. Use the equations developed in class and use algebra tosolve for the answer. Show all your algebraic steps. [The solved problems distributed by theinstructor will be helpful.]

–14–

17

4. You perform a foraging experiment like the Holling disc equation in class on an organism you’ve never studied before. You present the predator with prey densities that increase by units of 5 (i.e., 5, 10, 15, 20, etc.) At each experiment, you observe that the predator eats exactly half of the prey available. �(a) What is the equation for the functional response for this organism? [Hint: it is not Type II.] �(b) What is the attack rate? �

5. Think about how you might create an equation (or more than one) that would describe how much plant tissue an herbivore would consume. What might that equation look like?

Hint: Plant Tissue Consumed is your response (y); what are your predictors?

6. In your own words, how would you describe what it means for data to be statistically significant? How does that relate to a p-value?

Documents

Holling Disc Lab - St. Olaf Pages...Holling “disc equation.” It was given this moniker because Holling generated it by having a blindfolded assistant pick up (“prey upon”)