Functional responses with lionfish data

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 you 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 (data-based with Sheets) - student

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, which is the default value 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, change the parameters to determine

What does changing Th do? What does it mean?
What does changing alpha do? What does it mean?
For this graph, also consider why we can not consider type 3 curves.

Functional responses in lionfish: an experiment

We will now explore functional responses by considering the impact of invasive lionfish. This will allow us to use real data from experiments to compare functional responses and also consider how experiments impact data interpretation.

Non-indigenous invasive species are considered one of the major threats to biodiversity. Generalist invasive predators have a direct effect on their ecological communities by their direct and indirect impacts on food webs (David et al. 2017), and these impacts are compounded by other threats to biodiversity (Doherty et al. 2015). Examples include Burmese pythons in the Florida Everglades, feral and domestic cats, the harlequin lady beetle (Harmonia axyridis) in North America, and lionfishes in the Western Atlantic Ocean (focus of this lab).

Lionfish photographed in Curacao in 2010. LASZLO ILYES from Cleveland, Ohio, USA, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons

Pterois volitans, or the red lionfish, is a coral reef fish from the Indian Ocean and western Pacific Ocean, and has rapidly established in the Gulf of Mexico, the Caribbean region, and in the Atlantic Ocean along the southeast US coast (Hare and Whitfield 2003). A popular tropical fish in the aquarium trade (though now banned in Florida), it is thought to have been intentionally or accidentally released off the coast of southeast Florida, rather than catching a ride in ballast water like other marine invasive species (Hare and Whitfield 2003).

A few characteristics of lionfish facilitate its rapid expansion in its new range. As a venomous species, lionfish have very few natural enemies in their new habitat. Lionfish eat mostly other fish, but also invertebrates such as crabs and shrimps. They are voracious in their appetites. A study conducted in the northern Gulf of Mexico documented at least 77 coral reef taxa from the contents of lionfish stomachs (Dahl and Patterson 2014). Altogether, the invasion of the western Atlantic by lionfish is a major threat to marine biodiversity.

Eradication of lionfish from the western Atlantic is likely impossible. A more practical goal is to try to reduce the negative impact of lionfish. Many management approaches have been and continue to be implemented, e.g., lionfish spearfishing tournaments encourage a concentrated effort at reducing the population size of lionfish (Ulman et al. 2022). We can apply our knowledge of the predator-prey interaction to help figure out the best approaches for reducing the negative impact of lionfish.

Dr. Monica McCard and her colleagues conducted an experiment to examine how lionfish responded to the density of prey (McCard et al. 2021). The researchers measured predation rates in trials with only one prey species at a time. They also conducted trials where two prey species were mixed together in different proportions. The prey species studied were brine shrimp (Artemia salina) (AS), dwarf white shrimp (Palaemonetes varians)(PV) and a marine gammarid (Gammarus oceanicus)(GO). They gathered 14 lionfish juveniles (pre-reproductive individuals) of similar size for their trials. All trials took place in 45 L aquarium tanks.

Read the journal article related to this project.
- McCard, Monica, Josie South, Ross N. Cuthbert, James W. E. Dickey, Nathan McCard, and Jaimie T. A. Dick. “Pushing the Switch: Functional Responses and Prey Switching by Invasive Lionfish May Mediate Their Ecological Impact.” Biological Invasions 23, no. 6 (June 1, 2021): 2019–32. https://doi.org/10.1007/s10530-021-02487-7.

For the single-species trials, the researchers set up a tank with one of 16 density levels: 2, 4, 6, 8, 12, 16, 20, 25, 30, 35, 40, 45, 50, 55, 60, and 70 prey individuals per tank. For each of these single-species trials, the scientists released one lionfish into the tank to hunt and capture prey for three hours, then counted the number of remaining prey individuals to calculate the number of prey eaten during the trial. Each density level was repeated seven times, with different lionfish each time.

From the single-species trials, choose any low density (e.g., 2, 4, 6, 8) and high density (e.g., 50, 55, 60, 70) level, and one of the prey species. Draw a diagram (i.e., a cartoon or schematic) of the aquarium tank with prey and lionfish to show how a low-density and a high-density trial was set up for that species. Be sure to differentiate among the lionfish and prey species. You can use symbols and/or colors (or draw little shrimp/gammarids and fish).

For the mixed-species trials, Dr. McCard and her colleagues chose to focus on P. varians (PV) and G. oceanicus (GO). In these trials, the researchers set up nine different prey ratios: 45 PV:5 GO, 40 PV:10 GO, 35 PV:15 GO, 30 PV:20 GO, 25 PV:25 GO, 20 PV:30 GO, 15 PV:35 GO, 10 PV:40 GO, and 5 PV:45 GO. Like the single-species trials, each ratio was repeated seven times, with different lionfish each time. For each of these trials, one lionfish was released into the tank to hunt and capture prey for one hour. During the one-hour trial, the researchers tallied which prey species were eaten, and added back prey individuals to maintain the prey ratios throughout the trial.

From the mixed-species trials, choose one prey ratio, and draw a diagram to show how a trial was set up. Be sure to differentiate among the lionfish and the two prey species. You can use symbols and/or colors (or draw little shrimp/gammarids and fish).
Fill out the table below to identify the response and predictor variables and other useful information from the experiment. The response variable, also called the dependent variable, of an experiment is what was measured. The predictor variable, also called the independent variable, is what we think causes the response variable to change in value. There can be more than one response and predictor variable in an experiment. Other information from the experiment may need to be included in graphs to help viewers understand differences among trials or species. A few rows are filled out as an example.

Table to complete

Can you identify any graphs from the McCard article that show data in ways similar to what you expected?
What functional responses do you expect to see for the single-species and the mixed-species trials? Why? Justify your hypotheses!

To see what the data actually show, we'll graph the data. The data from the experiment is provided below. Note there are 3 tabs.

lionfish_consumption_sheets version

First, let's focus on the the data from the single prey trials. The single prey tab contains the number of supplied prey in a given experiment and the number of consumed prey for each prey species. Note there are multiple trials for each prey density for each prey species (in other words, row 2 has data from 3 experiments: one where 2 A. salinas were supplied, one where 2 G. oceanicus were supplied, and one where 2 P. varians were supplied).

For the single prey trials, we will create graphs to compare expected outcomes assuming type 1 and type 2 functional responses and compare them to observed data. These graphs will build on (and use some of the same formulas) your earlier work exploring the equation.

One prey species (A. salinas) is completed for you as an example. Note the values in yellow came from the McCard et al. article (look at table 1). The total time was set to 1 (as noted above), so Th is thus the proportion of their time the predator spends handling a single prey. Alpha was estimated using our data.

To compare the outcomes, plot the number of prey actually consumed against the number supplied. This is what we actually observed! Then add in our estimates assuming Type 1 and Type 2 functional responses (for help on adding multiple series to a graph, see Data Summaries in Google Sheets .

Note the formulas for the type 1 and type 2 functions contain a minimum function (different than what we saw when exploring the equation). Why? Try removing these (you can duplicate the sheet, remove it from the first appearance in row 2, and drag down...).
Which functional response best matches the data?
Repeat this exercise for other 2 prey species (make graphs for each). You can duplicate the tab (click the small downward pointing triangle next to the tab name and select Duplicate), but you'll need to update the data and parameters (yellow cells).

Next considet the data from the mixed-prey trials. Note the data is provided in 2 tabs. The first, multiple prey trials-data, shows the raw data. This includes the prey ratios, the number of prey eaten for each species, and calculated outcomes including the proportion of prey supplied in a diet and proportion of prey consumed. Note a given experiment is represented on 2 rows (rows 2 and 65 give data on the different species for a single experiment under this format).

The second, multiple prey trials-data updated, simply reformats the data to show a focal prey percentage in the diet- this will allow both prey species to be graphed on a single graph, and the empty cells are ok (they account for the matched experiments).

For this graph, you should plot proportion of focal prey supplied in diet on the x- axis. You can also add it to the y-axis.

What does this relationship look like? What does it represent?

You can also plot (on the same graph) te proportion of PV (P. varians) and the Proportion of GV (G. oceanicus) eaten.

Which functional response best matches the data? What does it mean?

Today we learned about how the types of functional responses are characteristic of different kinds of predators (e.g., Type I and filter feeders). In a few sentences, summarize why different types of functional responses could arise for the same predator in different habitats.
Dr. McCard and her research team published a paper on their lionfish experiment. In this paper, they claim that the functional response of lionfish in real communities of multiple prey species has relevance for biodiversity: lionfish would have a smaller negative impact on the populations of prey species that are rare. Do the findings we examined in this activity today support their claim? Why or why not?
This experiment on lionfish studied functional responses in a relatively simple system of one lionfish individual plus one or two prey species. Real coral reef communities that have been invaded by lionfish have much higher densities of this invasive species. How might the shape of lionfish functional responses differ in coral reef communities with few lionfish compared to many? Explain your reasoning.

Links to related papers:

McCard, Monica, Josie South, Ross N. Cuthbert, James W. E. Dickey, Nathan McCard, and Jaimie T. A. Dick. “Pushing the Switch: Functional Responses and Prey Switching by Invasive Lionfish May Mediate Their Ecological Impact.” Biological Invasions 23, no. 6 (June 1, 2021): 2019–32. https://doi.org/10.1007/s10530-021-02487-7.