Predator-prey models with nuts and bolts

Overview

Ecologists use models to explain how systems, 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. Models offer clear, testable hypotheses about relationships and mechanisms that drive ecosystem change. Models may be conceptual/qualitative (e.g. food webs, energy flows) or mathematical/quantitative (e.g. population growth models), and they sometimes scare students at first glance. This lab uses predation to introduce a basic species interaction model, and then explores a key assumption of this model in lab exercise.  You should leave this exercise understanding how processes influencing species interactions are translated to a modeling 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

• Modify the exponential growth equations to illustrate predator-prey interactions

• Define basic functional responses (Type 1,2,3) and identify them from provided graphs or data

Introduction

The Lotka-Volterra predator-prey model

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 a given time t can be represented with the following differential equation:

where

•  r is the growth rate (a parameter, or constant that ecologists determine using data)

• N is the size of the population (note that this changes over time!)

If the results of this model are plotted on a graph of population size (N) over time (t), it creates a J-shaped curve:

Notice these simplifications this model uses.  It does not account for any differences among individuals in the population. It also does not account for population regulation (e.g. the fact that populations can’t grow forever) or Allee effects (when small populations tend to go extinct due to difficulty finding mates or behavioral changes).  

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 growth equation to become:

where

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

This is the logistic growth model. If the results of this model are plotted on a graph of population size (N) over time (t), it creates an S-shaped curve:

Notice how this modification works:

• When the population size is small (N is close to 0), the carrying capacity term approaches 1, and growth is similar to the exponential growth model

• When the population size is large (N is close to K), the carrying capacity term approaches 0, and the population stabilizes 

  • Note: 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 can also be modified to consider species interactions, such as consumption of prey by predators. If we consider N  to be the population size of a prey species, we can model its growth as:

Where 

• P is the size of a predator population

• a is the attack rate, or the rate at which predators successfully capture prey upon encountering them.

If P = 0, then we are left with prey (N) growing exponentially. If we introduce predators (P) 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 (a) and by how often prey and predators encounter each other. As either population size increases, we would expect more encounters!

We can also use the prey population size (N) to model the predator population (P) that feeds off it:

Note that:

• The growth rate of the predator is directly related to prey capture rate (aNP), as described in the last equation

• We add another parameter b to represent the conversion efficiency, which measures how many prey an individual predator needs to eat on average to produce one offspring

• Since the predator population can't grow forever, we add the parameter d, or the death rate of the predator; more predators die as the population size increases.

This set of coupled predator-prey equations allow ecologists to consider how these two populations impact each other simultaneously. Analysis shows that populations following these equations show cycles of growth. When both populations are small, the prey population can grow; as prey populations increase, the predator population 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 the cycle repeats.  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.

Functional responses

In addition to ignoring differences among individual organisms (such as body size or life stage) that might impact predator-prey interactions, this simple model makes a number of additional assumptions.  For example, it doesn’t consider that predators may eat other things, or that prey may engage in behaviors to avoid predation. It also assumes that individual predators will consume prey at a constant rate as more become available. If the total number of prey consumed by predators is aNP, then the per-capita (individual) rate of prey consumption 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, shown on the graph below.  

However, this relationship may not be realistic for several reasons. For example, imagine you are really hungry and love to eat burritors.  If you get one burrito, you could probably eat one. If I give you five, maybe you could eat five. If I gave you 1000, could you eat a 1000? Probably not, for several reasons!

First, you (like predators) will eventually get full!  Because acquiring prey takes energy, predators may cease feeding once they have enough.  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 above which you would be unable to eat more burritos in a day due to the inherent time it takes to process each one (what ecologists call "handling time").  

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

Where 

• T is the total proportion of its time a predator spends searching for and and consuming prey 

• Th is proportion of time a predator spends handling a single prey

If we match the Type I assumption that predators spend all of their time hunting and eating (T=1) and solve the equation for Pc, we get

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 burrowing rodents or an otter finding and opening shellfish. This means that as a prey density increases, predator consumption eventually levels off, as shown in the graph below.

We’ll focus on simulating Type I and Type II functional responses in the activities below. However, you should be aware another classic option exists. Both Type I and Type II responses assume predators are randomly searching for a single prey.  However, many predators may instead eat multiple species of prey. If these prey live in different habitats, predators may tend to over-consume 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 learned search image), and thus be more likely to seek out common prey. 

These modifications lead to a Type III functional response. We won’t derive an equation for 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. When combined with the fact predators eventually spend all their time handling prey (and thus reach a maximum consumption, as in the Type II response), an S-shaped is produced, resembling the logistic growth curve shown above.

In the lab exercises below, we will explore how these models work through activities that simulate predator-prey dynamics. First, we'll simulate the classic Lotka-Volterra model under the assumptions of a Type I functional response. Then, we'll simulate handling time to illustrate the Type II response.

Part 2: Compare Functional Responses

Before conducting our next set of simulations, use the second tab of the spreadsheet ("Part 2: Compare Responses") to develop a better understanding of the parameters a and Th (values 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 hunting and eating (If we assume a predator spends all their time on this, we can set T=1, the default when you download the sheet).  Th is thus the proportion of their time the predator spends handling a single prey item.  If Th is small, the Type I and Type II responses look very similar!  As Th gets larger, the Type II curve flattens out more quickly as the predator starts to spend most (or all) of its time handling prey, thus reaching a maximum consumption. 

Questions

1. Upload a copy of the graph where the difference between the Type I and Type II functional response is small. What parameter values did you use?

2. Upload a copy of the graph where the difference between the Type I and Type II functional response is large. What parameter values did you use?

3. What does changing a do? What does it mean?

4. What does changing Th do? what does it mean

Part 3: Handling Time

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

Materials

• Instruments to measure and mark search area (yard stick, flags, traffic cones, etc.)

• Bolts with nuts fully threaded to top (do not use screws with sharp ends!)

• A container to collect bolts

• A timer

Procedure

1. Form groups of 4-5 students and designate roles:

   • Prey Handlers (2 students) – disperse prey in the environemnt and rethread nuts to replenish prey supply

   • Data Recorder (1 student) – keep time and record how many prey the predator captures in each sampling round

   •  Predator (1 student) – capture, handle, and "consume" prey from the environment

2. Calculate Th. Have the predator remove the nut from several bolts (without free-spinning the nut around the bolt). After some practice, time the predator unscrewing a nut three times and record each attempt in the appropriate spreadsheet box (in seconds). The sheet will then calculate the average handling time Th, highlighted in blue.

3. Conduct the first simulation. The Prey Handlers should scatter the prey (bolts with nuts fully threaded to the top) around the search area. The Predator will then search the area for a 3 minute period. In this exercise, we will keep the prey density constant; as soon as a prey item is captured, the Prey Handlers should replace it with a new prey item (it can be in a new location). Record the total number of prey captured after 3 minutes.

4. Calculate a (read through these instructions, then use the spreadsheet to help with calculations).  Recall that the prey consumption rate Pc can be defined as:

Questions

1. Upload a copy of your completed graph. Which functional response (Type I or Type II) best matches your actual data. Why?

2. If two nuts were placed on each bolt, what would this simulate? How would it affect each of the following:

   • Handling time (Th)

   • The maximum number of prey captured per predator

   • The shape of the functional response curve

3. What aspects of real predator–prey interactions were not captured in this simulation? How could you modify the activity to include one of those aspects (e.g., prey behavior, spatial refuge, alternative prey, competing predators)?

4. What is one scenario where a model like this could be useful to accurately predict changes in predator and prey populations over time?