Functional response simulation with nuts and bolts

Overview

Ecologists use models to explain how things, ranging from organisms to ecosystems, change in response to various factors. Because the natural world is very complex, models always make simplifying assumptions in order to be useful (and tractable). However, even with these assumptions these models offer clear, testable hypotheses about relationships and mechanisms that drive change. These models may be conceptual or mathematical, and they sometime scare students at first glance. This lab uses predation to introduce basic growth models and then explores a key assumption of many predator-prey models using a lab exercise. You should leave this exercise understanding how mechanisms that explain how organisms interact are translated to a modelling framework.

Connections to Other Labs

This lab can be coupled with the Photosynthesis and Pigment labs to consider how bottom-up and top-down forces interact and impact communities.

Objectives

Students should be able to

Define predation and explain its impacts on prey and communities
Discuss how the Lotka-Volterra model modifies logistic growth equations to consider predator-prey interactions
Define basic functional responses (Type 1,2,3) and identify them from provided graphs or data

Introduction

Predator - Prey Interactions

Predation, or the consumption of one organism (a prey) by another (a predator), is a common species interaction that plays a central role in regulating prey populations and structuring ecological communities. Due to its importance, ecologists have developed multiple models to explain how predator and prey populations interact. The Lotka-Volterra model, for example, expands the commonly used exponential growth model to consider how prey and predator populations grow.

The basic exponential growth model considers how populations that reproduce continuously grow over time. If we consider a population of N individuals, then the change in population size at any time (N’ or dn/dt, coming from a differential equation) is equal to

where

r is the growth rate (a parameter, or constant that ecologists determine using data; we will underline parameters in this lesson)
N is the size of the population; this changes over time!

In this model, the parameter r and the initial size of a populatoin (N0) determines how fast a population grows.

Notice these simplifications this model uses. It does not account for any differences among individuals in a population. It also does not account for population regulation, or the fact that populations can’t grow forever, or Allee effects, when small populations tends to go extinct for a variety of reasons such as difficulty finding mates or behavioral syndrome breakdown. However, we can modify this equation to account for population regulation in multiple ways. Intraspecific, density-dependent regulation, which models how growing populations end up competing for limited resources, modifies the exponential equation to become

Where

K is the carrying capacity, or the maximum population size an area can support for a given species (another parameter)

Notice how this modification (which we call the logistic growth model) works: when the population size is small relative to its carrying capacity, N/K approaches 0. At this point, the (1 - N/K) term approaches 1, so we growth similar to that predicted by the exponential growth model. However, as the populatoin grow and N gets close to K, N/K approaches 1 and (1 - N/K) is close to 0. At this point, the instantaneous rate of change approaches 0, which means the populatoin stabilizes. A common misconception is that this implies the population goes extinct (since growth is equal to 0); make sure you understand why this isn’t the case.

The exponential growth model also can be modified to consider species interactions. If we consider N to be a prey species, we can model its growth as

Where

P is a predator population
α is the rate at which predator capture prey upon encountering them.

If P = 0, then we are left with prey growing exponentially. If we introduce predators to the system, prey population size increases following the exponential equation but decreases due to lethal encounter predators. The number of prey lost to predation is impacted by how “good” predators are at capturing prey (α ) and by how often prey and predators encounter each other. As either population size increases, we would expect more encounters!

We can connect the size of prey population to the predator population, that feeds off it. We model the predator growth as

First, note the growth of the predator population is directly related to prey capture (α* N * P). We add another parameter to consider how predators convert prey to new individuals (f). Since predators can’t grow forever, we add a death parameter (d), so that we see more predators dying as the population increases.

This set of coupled predator-prey equations allow ecologists to consider how these two populatoins impact each other. Analysis shows that populations following these equations show cycles of growth. When both populations are small, the prey population can grow (remember, just like exponential growth!). As prey populations increaes, the predator populatoin also begins to grow. Eventually, the predator population gets so large it begins to reduce the prey population, which eventually leads to both populations declining (and then we start over!). Notice that these coupled equations mean that that the dynamics of the prey and predator population resemble each other but the predator population tends to “lag” behind the prey population.

In addition to not considering differences among organisms such as size or stage, which might impact predator-prey interactions, this simple model makes a number of additional assumptions. For example, iIt doesn’t consider that predators may eat other things (these populations are tightly coupled!) or that prey may use energy to avoid predation. It also assumes that individual predators will consume more prey as they become available. If the total number of prey lost to predation is

, then the per-capita (individual) rate of prey consumed by a predator is

where

Pc is the number of prey consumed per individual predator

This relationship between prey density and predator consumption is known as a functional response, and this specific linear relationship is a Type 1 response. However, this relationship may not be realistic for several reasons. For example, imagine are really hungry and love to eat burritors (vegetarian-style if that’s your preference). If you get one burrito, maybe you could eat one. If I give you five, maybe you could eat 5. If I gave you 1000, could you eat a 1000? Probably not, and for several reasons.

First, you (and predators) may eventually get full! Given the fact acquiring prey takes energy, predators may cease feeding once they are sated. Second, eating isn’t an instantaneous act. You need to get the burrito, unwrap it, and consume it . Even if you had a room full of burritos and never got full, there is some limit where you can’t eat more burritos per day due to the inherent handling time.

Predators operate in a similar way: they have to find and handle their prey. So, we can modify our per-capitat predation model to account for these specifics by changing it to:

Where

Ts is how much time a predator spends searching for prey

To follow our burrito example, even if we assume our predator is only consuming prey, the search time (Ts) is limited by how much of the total time (T) is spent handling prey (or unwrapping burritos). We can find this by multiplying a handling time, Th, or the amount of time it takes to unwrap a given burrito, by the number of burritos that were consumed). We can thus say

If we solve the equation for Pc , we get

To match the Type I assumptions that a predator is always eating, we can also let total time (T) equal 1 and thus remove it from the equation. We call this a Type II functional response. This relationship exists for predators that actively search for and handle prey. Examples may include a fox hunting rodents (that it must dig out of holes) or an otter finding and opening shellfish. This means that as a prey density increases, predator consumption eventually levels off! Likewise, a Type 1 response may be more realistic for filter feeders that spend very little time actively searching or handling prey.

Today for lab we’ll focus on creating a Type II functional response. However, you should be aware another classic option exists. Both Type 1 and Type II responses assume predators are randomly searching for a single prey. However, predators may instead eat multiple prey items. If these prey items live in different habitats, they may tend to overconsume one prey when its common (since its clumped together and more likely to be found in rapid bursts). Similarly, predators may key in to certain prey traits (using a search image) during searches and thus be more likely to find common prey. Both of these modifications lead to a Type III response. We won’t derive this, but its shape should be understood. When compared to a Type I curve, rare (low density) prey are less likely to be consumed, while high density prey are more likely. Along with the fact predators eventually spend all their time handling prey (and thus reach a maximum consumption), this response curve is s-shaped and resembles a logistic growth curve you may have seen in class.

Functional Responses

Methods

Explore the equation

To better understand the equations, download or copy the following spreadsheet. Using the Compare Responses tab, manipulate the parameters values (cells highlighted in yellow) to determine how they change Type I and Type II functional responses.

Functional Response

In this sheet, the parameters are highlighted in yellow. Changing these values will update the expected number of prey consumed (highlighted in green). In this sheet total time T can be considered the proportion of time a predator spends searching (f we assume a predator spends all their time searching, we can set T to 1 (default when you download the sheet)). Th is thus the proportion of their time the predator spends handling a single prey. If Th is small, the two responses look very similar! As Th gets larger, the Type II curve plateaus off more quickly as the predator starts to spend most (or all) of their time handling prey and thus reaches a maximum consumption. Make sure you understand these relationships! For example, what does changing Th do? What does it mean?

Simulations

Now you will simulate predation to parameterize a Type II functional response curve.

Materials

Device to measure and mark search area
Nuts with bolts fully threaded to top
- DO NOT USE LAG/WOOD SCREWS OR OTHER SCREWS WITH SHARP ENDS!

Procedure

In this lab we will work in groups of at least 2 students to simulate a predator searching for prey at various densities. One student in each group will search for prey that have been dispersed across a set area at a given density. If weather or indoor space allows, the simulation can be carried out in a local green space (check website for lawn openings in Madison Square Park) or indoor area (hallway or classroom with chairs moved out of the way). In these settings the predator will “search” for prey (bolts with nuts attached) by shuffling through the area until they step on a prey item. Predators should look forward for safety but not use their eyes to search for prey on the ground. Once they encounter a prey, they can pick it up and consume it by removing the nut from the bolt WITHOUT SPINNING IT FREELY. In this way the bolt represents the prey and removing the nut represents the time required to handle the prey (e.g., remove inedible portions). At this point the predator can hand the prey item to their group members. The other group members' jobs are:

Watch for the safety of the predator
Record how many prey items the predator captures
Immediately replace the “captured” prey with another “uneaten” prey randomly dispersed in the search area. This ensures prey density is remaining constant throughout the simulations.
Fully rethread the nut for future use as “prey”

If working in a large area all predators can search in a shared space; predators should move slowly and be careful. If large areas are not available, the exercise can be modified with the prey items spread across a flat surface (e.g., table-top); in this scenario the predator searches the area by moving one finger pointing straight down around the surface until they encounter prey.

The predators should search for prey at various densities for specified time periods. For example, prey could be dispersed at ratios of 3, 6, 9, 12, and 15 prey per predator, and each simulation could last 3 minutes.

Detailed Procedure

Form groups and designate roles including a “predator” and support roles (“handler”, "data recorder", "prey return specialist") in order to
- Watch for the safety of the predator
- Record how many prey items the predator captures
- Immediately replace the “captured” prey with another “uneaten” prey randomly dispersed in the search area. This ensures prey density is remaining constant throughout the simulations.
- Fully rethread the nut for future use as “prey”
To gather information on handling time, have the predator remove the nut from the several bolts for practice (without spinning the nut around the bolt). After some practice, record how many prey the predator can “handle” in one minute. Divide one minute by the number of prey handled (record up to a “quarter” handled prey by considering how unthreaded the last nut was) to determine the average handling time per prey. Record this in hundredths of a minute (keeping units the same throughout is essential for interpretable results).
Follow class/instructor’s directions to run simulations. For each simulation, scatter prey (bolts with nuts fully threaded to to the top) around the search area. Have the predator “search” the area for the same amount of time for each simulation; for rest of exercise we assume total search time is 3 minutes. Make sure you replace prey that are captured by your predator. For each simulation, record the prey density and total number of prey captured.

Using data from the first round of the simulation only, calculate the predator’s attack rate. TIP: Read through these instructions, then use the spreadsheet to help with calculations.
- Multiple the pre-determined handling time (found previously) by the number of prey consumed for a given simulation to determine a total handling time.
- Since 3 minutes were spent in each simulation, the search time is equal to 3 minutes minus the total handling time.
  - This is why recording handling time in minutes is so important! If you use seconds instead you will find answers that don't make sense!
- By rearranging our starting equation

we can derive

Note N above is equal to the prey density per predator in our lowest simulation (3 is recommended above). For our recommended levels, this becomes

For help with this calculation, check out the Chart Actual Data tab in the spreadsheet linked belows.

Functional Response

You can update the main parameters on the sheet (highlighted in yellow) including
- Time spent in each simulation round
- Prey density at lowest simulation
- Prey captured in lowest density simulation
- # of prey "eaten" in one minute trial
After you update these fields, the sheet will calculate your handling time and attack rate and use your collected data to plot the number of prey captured at each prey density (if desired, class averages may also be used for graphs).
- If your Type I graph isn't linear something is wrong! Make sure "Treat labels as text" is NOT checked in the Customize graph

On the same graph, the sheet will plot the expected number of prey captured based on your parameters for a Type I and Type II functional response. Assume 0 prey would be caught if prey density was 0.

Questions for review

Which response type (I or II) best matches your actual data? Why?
If two nuts were placed on each bolt, what would this simulate? How would it impact your graph?

Tips

Adding multiple data series to a Google Drive graph

To add multiple data series, let the first column be your desired independent variable (for x-axis). Dependent data series can be placed in adjacent columns. All columns can have identifying labels. Select all the columns you desire to graph, then insert a chart using the same procedure from the Summarizing Data Lab. The only modification required is that you may need to specify which series is the X-axis depending on your browser defaults.

Select all columns you desire graphed and then insert chart.

If needed, designate the appropriate series as the x-axis.

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