Promoting Quantitative Skills in Introductory Classes Using Optimal Foraging Theory and a Model-Assisted Activity.

INTRODUCTION

Mathematical models scientifically describe many natural processes. However, scientific theory is often taught to undergraduate students without presenting the mathematical basis, especially for lower-level classes. Adding quantitative lessons to introductory classes furthers student exposure to applied mathematics and helps students understand the interdisciplinary nature of science while simultaneously promoting critical thinking. Therefore, they are highly appropriate for undergraduate classes. Similarly, demonstrations with student engagement (1) and model-assisted activities (2, 3) enhance learning and can promote positive opinions of science (4).

Most courses that cover introductory ecology concepts address predatory-prey relationships but omit why certain organisms are included or excluded from a species’ diet. Optimal foraging theory can be used to describe this decision. Here, I present an active learning module designed to explain predator-prey dynamics, highlighting the underlying mathematical basis. This module contains a lecture (described) and an activity (Appendix 1). The foraging theory module was added to this lesson because foraging theory mathematics may appear daunting but are relatively straightforward and intuitive concepts. Thus, this is an ideal example to help break down math-adverse sentiments and build students’ skills and confidence. Additionally, it is a versatile model that can be used to understand behavioral ecology in a range of systems. Lastly, many students are familiar with food webs from high school biology, so the module provides an opportunity to go into more depth on a familiar topic.

Expose students to mathematical models

Build quantitative skills

Reinforce connections between biology and math

Reinforce key concepts around predatory-prey interactions

Address common misconceptions

Address concerns over the assumptions of the model

PROCEDURE

Lecture

The lesson begins with food webs and the concept of trophic transfer. Then we shift from the traditional approach by asking how an organism successfully hunts prey. Stephens and Krebs (5) provide a comprehensive review of the foundational foraging theory literature, including the quantitative models.

The foundations of foraging theory involve three required steps for successful hunting—encountering, attacking, and capturing a potential prey. I first describe this verbally and then introduce the equation (Fig. 1A). I emphasize that failure at any one stage guarantees overall failure and this is reflected in the mathematics (i.e., by entering values for the equation in Fig. 1A: 100%*100%*0%=0%). If a prey is well camouflaged so the predator never encounters it, then the probability of ingesting the prey must be 0. On the other hand, detecting the prey does not guarantee ingestion.

FIGURE 1

Equations and their descriptions. A) The three steps needed for successful hunting: encountering, attacking, and capturing a potential prey. During the presentation, I define and describe each term in an ecological and mathematical context. I emphasize that failure at one stage equates failure to ingest the prey, but success at one stage does not guarantee successful ingestion. B) Equation of the rate of energy gain of the predator for an ingested prey. I present the mathematical equation, term by term, and the associated definitions, and then summarize the equation. C) Comparison illustrating that the decision of whether or not to attack a new prey item is a function of the comparative energy gain between the new prey and the predator’s current diet. In the lecture, I emphasize that the decision to attack one prey item may differ between individuals or across time, depending on the current state of the organism. A prey that can yield moderate energy gains will be attacked when encountered if the predator has been consuming low energy items, but a moderate-energy prey item will not be attacked if the predator’s current diet primarily consists of high-energy-gain foods.

I then pose the question, assuming all of these three necessary steps are met and a prey item is ingested, what is the rate of energy gain? Here, I provide the mathematical model and explain each of the terms verbally alongside the discreet variables (Fig. 1B).

Tip 1: Ask students to predict the impact of increasing handling time on the rate of energy gain. Relate this to the mathematical equation (increasing the denominator).

Next, we explore factors that impact the decision of whether or not to attack a new prey item (Fig. 1C). The decision is a function of the amount of energy an organism is set to gain from including the new food source. If the new prey results in a net energy gain greater than that of the current diet, then the new prey item should be attacked. If the gain is less than that of the current diet, then the new prey should not be attacked.

Assumptions of the model include that the predator is omniscient: it knows the handling time and energy content of prey it has not yet handled or attacked and can compare this to the energetic gains of its current diet. These assumptions often cause students to question or dismiss the model. To reinforce the model, I have created an active learning activity which models the key concepts of foraging theory.

Activity

Each group of 4 receives:

90 M&Ms (or other small edible items such as oyster crackers) (“Fun packs” of M&Ms typically contain 17 M&Ms)

45 plastic eggs (or other sealable container)

3 plates

Handout with description of the activity (Appendix 1)

Provide six packs of M&Ms and make the groups set up each scenario (Fig. 2). Student “A” oversees the setup and timing and is the recorder. Student “B” receives 30 normal M&Ms. Student “C” receives 30 M&Ms, each contained inside a plastic egg, while student “D” receives 15 “free” M&Ms and 15 “contained” M&Ms. Students will attempt to consume as many of their own M&Ms as possible in 10 seconds. For each M&M they hunt, they will need to see, attack, and consume it, but they can only engage with one M&M at a time.

FIGURE 2

Activity set up. A) Set up for student B, with 30 M&Ms placed on a plate. B) Set up for student C, with 30 M&Ms each contained within a plastic egg. C) Set up for student D, with 15 M&Ms on the plate and 15 individually contained within plastic eggs.

Tip 2: This activity can be done in groups of three, but having the fourth group member accommodates any allergies (M&Ms are not considered peanut-safe) or preferences to not consume the food.

Tip 3: This activity can be a demonstration with three students. If conducting a demonstration, have students make predictions on the outcome to increase student gains (1).

Student B does not eat all of the M&Ms in the allotted time.

Student C eats the fewest M&Ms.

Student D eats more M&Ms than student C but less than Student B. Student D typically first consumes the “free” M&Ms and moves on to the prey source with the higher handling cost only after depleting the “easy” prey source.

Class-level discussion

After students have discussed the questions, have a larger course-level discussion on each question relating their findings back to optimal foraging theory. Some of the key points:

Even with the free M&Ms, generally student “B” does not consume all the M&Ms available. Handling time, while minimal, inhibits the ability to exploit all of the available resources. This addresses the common misconception that a predator can consume all of the available prey items. All components of hunting (searching, attacking, ingestion) have a time cost, and the organisms had to expend energy to maintain a basal metabolic rate during foraging.

Student “C,” with all M&Ms inside the eggs, ate the least, because the contained M&M prey exhibited high handling costs that increased the time spent capturing the prey.

Ask “D” students whether they consumed more of the free M&Ms or the contained ones? Which did they attack first? How did they decide which to attack? Did they calculate the net energy gain of both and then decide? If not, then how after encountering multiple prey items did they reach the decision on which to attack? Of course no one did the math, but each student made a decision, generally one that maximized their energetic gains. This helps address the concerns about the assumption of omniscient predators comparing the net gains of each prey item.

Tip 4: If any group had atypical results, then survey the larger class. Use this to emphasize the value of replication and larger sample sizes.

CONCLUSION

I created a lecture and simple group activity (or demo) that increases student exposure to mathematical modeling and ecological theory and can be added to any class discussing predator-prey dynamics. This lesson and activity can be added as a module for most introductory biology, ecology, or environmental science classes and can be used in small or very large classes with very inexpensive, easily obtained supplies.

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