Students often have difficulty understanding the underpinning mechanisms of natural selection because they lack the means to directly test hypotheses within the classroom. Computer simulations are ideal platforms to allow students to manipulate variables and observe evolutionary outcomes; however, many available models solve the scenario for the users without revealing the evolutionarily significant calculations. I developed a simplified bioenergetics model of a hammerhead shark for teaching natural selection that allows the users to manipulate variables and see the impacts of modeling while solving for the evolutionary consequences. Students generate variation within the population by controlling cephalofoil widths and swimming speeds of an individual, which affect its ability to detect and capture prey at the expense of energy lost as drag from swimming. The trade-off between energy gained from successful predation and energy lost from metabolic expenditures dictates rates of reproduction. By manipulating a subset of factors that influence differential reproductive success, students gain an improved understanding of natural selection.

## Introduction

*Vision and Change in Undergraduate Biology Education* (AAAS, 2011) identified *evolution* and *structure and function* as the two most important core concepts for biological literacy. Assessments, however, have frequently shown poor student proficiency in evolutionary topics (Alters & Nelson, 2002; Anderson et al., 2002). Furthermore, in traditional classroom settings, where students do not have the ability to test hypotheses experimentally, student comprehension and retention of evolution is low (Gardiner, 1998; Alters & Nelson, 2002; Johnson & Lark, 2018). Several authors have proposed that students move beyond the terms *know* and *understand* when assessing competency and advance to higher-level outcomes, including *analyze*, *compare*, *predict*, and *model* (Fu et al., 2009; AAAS, 2011; National Research Council, 2012). AAAS (2011) further identifies *ability to use quantitative reasoning* and *ability to use modeling and simulation* as two of the top three core competencies students must learn about scientific inquiry. Johnson and Lark (2018) call for selection experiments with digital organisms as guided investigations of evolutionary mechanisms.

Computer simulations allow students to simplify complex systems by manipulating variables, making comparisons and predictions, and testing hypotheses within the classroom. Several simulations for evolution are available for purchase or free download. While they may be effective in showcasing evolutionary concepts, and are often based on classic studies, they often lack transparency when calculating the evolutionary consequences or outcomes. For example, it is a common technique to let the user manipulate an organism's color or habitat, and then the computer solves which variants have higher fitness without revealing the assumptions of the model or the mathematical calculations that mimic life. This “black box” approach has the potential to minimize quantitative reasoning and prevent the student from using simulated data to predict, analyze, and test evolutionary concepts directly.

I developed a simplified model of hammerhead shark bioenergetics for demonstrating opposing evolutionary constraints and natural selection that is transparent in its assumptions, calculations, intermediate steps, and results. A fundamental assumption of fish bioenergetics models is that the calculations are performed on an individual because they give a more accurate response to environmental variables than population-level models (Kitchell et al., 1977); simulations can be repeated and results can be interpreted to represent the population. Here, the user is able to generate variation within the population by altering head widths (structure) and swimming speeds of an individual, which affect its ability to locate and capture prey (function). Although increasing head width and swimming speed improve prey detection, they cause an increase in drag that escalates the energy required to swim. Students are able to apply quantitative reasoning while making predictions about how changing these variables alter both prey detection and metabolic demands. As the students incrementally alter these variables, the model continuously updates and reveals the results, allowing the students to evaluate their predictions and investigate opposing constraints. Using the bioenergetics approach, the shark can reproduce only if there is remaining energy after the metabolic demands of producing a head and pushing it through water have been paid. These variations within the population result in differential reproductive success, which is the driving force in natural selection.

## The Hammerhead Shark

Hammerhead sharks are charismatic megafauna characterized by their laterally widened rostrum, called the cephalofoil. The cephalofoil is dense with ampullae of Lorenzini, which allow the shark to detect the electromagnetic fields of its prey. Mello (2009) hypothesized that the widened cephalofoil aids in prey detection by allowing the shark to scan a larger electromagnetic field. Additionally, the elongation of the rostrum provides wide eye spacing, producing acute binocular vision for visual detection of prey (McComb et al., 2009). Furthermore, the wide spacing of nostrils on the rostrum allows the shark to determine the direction of scent trails (Kajiura et al., 2005).

I chose to model the winghead (*Eusphyra blochii*) because it is the most basal of the hammerhead species (Lim et al., 2010) and has the highest rostrum-to-body ratio, ranging from 0.4 to 0.5. In contrast, that of the great hammerhead (*Sphyrna mokarran*) ranges from 0.20 to 0.25. Hammerhead evolution has incrementally reduced cephalofoil width in the more derived species (Lim et al., 2010). By investigating the upper limits of cephalofoil widths and associated costs, students will understand the opposing evolutionary constraints that give natural selection direction, and they can see why “bigger is not always better.” Wingheads eat a variety of fishes and cephalopods. Once sexual maturity is achieved at ~1.0 m in length, an average female winghead gives live birth to 6–25 pups/year.

## Suggested Teaching Narrative

Instructors are recommended to give the students background context before using the bioenergetics model. Particular attention should be paid to the equations that calculate the energy budget. I distribute a “pre-lab” exercise of these concepts and equations that students must complete before using the computer model. The following narrative is the basis of the one I assign and may serve as a starting point.

## Opposing Constraints

If the widened rostrum aids in prey detection, is a wider rostrum better? A wider rostrum should allow the shark to survey larger volumes of water per unit time than narrower rostrums. So, why is the rostrum-to-body ratio limited to 0.4–0.5 for wingheads?

Phenotypic performance is limited because every form is a balance of strengths and weaknesses. Water is 775 times denser than air; therefore, moving through it requires considerably more energy than moving through air. Since wingheads are constantly swimming in search of prey, the energy gained from digesting prey must exceed the energy spent pursuing and capturing the prey. Every object that moves through a fluid, such as air or water, is met with resistance, called drag. The drag equation for all objects is F_{D} = ½ρμ^{2}C_{D}A (Equation 1), where F_{D} is drag force measured in Newtons (N), ρ is the mass density of the fluid (kg/m^{3}), μ is the flow velocity (m/s), C_{D} is the drag coefficient, and A is the area of the object (m^{2}).

The combined surface areas of the cephalofoil and body directly contribute to the total drag the shark experiences while swimming. The widened rostrum protrudes laterally off the head of the shark, adding to the drag the shark encounters when swimming. Subsequently, a trade-off exists between gains from the rostrum during prey detection and the drag generated during prey pursuit. Therefore, there is a compromise of not being too small and limiting foraging success but not being too large and risking massive expenditures; this concept is known as *opposing constraints*.

## Bioenergetics Modeling

Bioenergetics is the study of energy movement and transformation through biological systems. This concept is modeled for fish in the equation C = R + W + G (Equation 2), where C is consumption, R is respiration, W is waste, and G is growth. A typical carnivorous fish, such as a winghead, has a baseline bioenergetics budget of 1 unit consumed = 0.44 units of respiration + 0.27 units of waste + 0.29 units of growth (Kitchell et al., 1977).

As soon as food enters the digestive tract, a flat percentage is lost as waste. Respiration sustains the organism, which includes basal metabolism, movement, prey capture, and the action of digesting the meal. Basal metabolism is fixed, based on shark size, but the others are variable, depending on swimming speed and number of prey consumed. After the costs of waste and respiration have been calculated, remaining energy can be invested in growth. Reproduction comes out of the growth budget because gonads have to develop and the developing young require energy. If the organism cannot balance metabolic demands, it cannot reproduce and may die.

The energy trade-off of energy gained from prey and metabolic expenses will determine the amount of energy available for reproduction. Sharks with higher drag have less energy left over for reproduction, so they will have lower differential reproduction. Sharks with decreased prey detection will also have less energy for reproduction. Natural selection will favor the shark body plans that maximize the energy gained to energy lost because, all other things being equal, those individuals within the population should have the greatest reproductive success.

## Using the Bioenergetics Model

### Distribution

The model (Figure 1) is a Microsoft Excel file that has been tested on several versions of Excel for both Windows and Macintosh. It is not compatible with mobile apps and other spreadsheet applications. It is provided “as is” and is free to download and distribute from https://goo.gl/jnPUJ9 (through Weebly.com). I will also share the model when contacted by email. The model has three spreadsheet tabs: Bioenergetics Model, Model Assumptions, and an optional Graph Builder. The first two tabs are locked to prevent user alterations to the equations and source code, but I will distribute an unlocked version upon direct request.

### Model Assumptions

While the model is based on values published in the literature, certain parameters have been simplified to aid student understanding of the core concepts (Figure 2). The adaptive peak the model solves for reflects the actual predominant phenotype and average birthrate in the natural population. Deviations from this represent phenotypes with lower fitness.

All variables have been scaled for a shark with a body length of 1.0 m. To minimize confounding variables and variation, the body and cephalofoil of the shark are assumed to be cylinders without fins and with diameters of 0.2 m when calculating surface area; however, the drag calculated from their movement reflects the actual shape of the shark. The shark body has the drag of a fusiform body with a drag coefficient of 0.04 and the cephalofoil has the drag of a wing with a drag coefficient of 0.09. Drag is calculated according to Equation 1. Since drag is often calculated as N/s, the model transforms the values into KJ to fit into the bioenergetics equation. Basal metabolic requirements of 65 KJ/day per kilogram of mass are extrapolated from Lowe (2002), Dowd (2006), and Bouyoucos et al. (2017). The basal metabolism of a 5.0 Kg body (sans head, 1 Kg/m) remains fixed, but as the cephalofoil widens at 1.0 Kg per every 0.20 m (scaled to one-fifth of the body mass), the shark's total basal metabolism will increase accordingly. Swimming speed, prey sampling area, and prey detection of wingheads are based on the work of Barousee (2009) and Mara (2012). For simplicity, the model assumes that every 100 m^{3} of water sampled yields one prey item, and the shark cannot eat more than two prey per day. Each prey item yields 650 KJ, and fractional prey cannot be consumed. The bioenergetics model of C = R + W + G (Equation 2) from Kitchell et al. (1977) has been modified to assume that respiration (R) is the sum of both basal metabolism and energy lost to drag. As either or both the cephalofoil size and swimming speed change, the model adjusts prey detection, prey consumption, basal metabolism, waste, body drag, cephalofoil drag, and growth. The shark can survive and reproduce only if the growth values are positive; the more positive the growth values are, the more pups it will birth. Each pup birthed requires 6.5 KJ, and fractional pups cannot be produced.

## Data Entry & Results

The user enters values for Rostrum:Body Ratio and Swimming Speed using clicker arrows in the orange box in the Bioenergetics Model tab found in columns A and B, rows 1–5. The smallest ratio allowed is 0.20 and the slowest swimming speed is 0.40 m/s; upper limits beyond reasonable biological measurements are possible, and students are encouraged to try them. After these values are input, all other values in the model are automatically populated using Equations 1 and 2.

As the user manipulates Rostrum:Body Ratio, values in the gray Rostrum Measurements boxes will change, including Rostrum Width, Rostrum Surface Area, Rostrum Mass, and Total Shark Mass (rows 9–11). Once Swimming Speed is altered, values in the gray Foraging Effects boxes will change, including Sampling Ability, Volume Sampled, Prey Encounters, and Consumption (rows 13–15). These foraging values are synergistic with rostrum size because changes in rostrum width will influence prey detection and consumption. As both Rostrum:Body Ratio and Swimming Speed are manipulated, values in the gray Bioenergetics Values change (rows 17–19). Specifically, Rostrum Drag Loss and Body Drag Loss adjust as differently sized cephalofoils swim at varying speeds, and Growth changes synergistically on the basis of energy gains from prey capture and losses from drag. Basal metabolism changes only according to total shark mass and is independent of swimming speed. Waste is a fixed rate based on total prey consumption and does not change with swimming speed or varying drag.

Final results are shown in the yellow Evolutionary Results boxes (columns C–E, rows 1–5). Students will learn how their manipulations of Rostrum:Body Ratio and Swimming Speed ultimately influence prey capture and total energy expenditures. If the shark runs an energy deficit, the shark will ultimately die. If the shark harvests sufficient energy, it will be possible to reproduce with differing total offspring. Drag calculations from Equation 1 are also listed so that students can compare the drag between the cephalofoil and body while assessing the cost and benefit of the sensory investment. Students may also use the optional Graph Builder tab to visualize changes in their variables (Figure 3). They can enter Rostrum:Body Ratio and Pups Birthed into columns A and B, respectively, and then plot other values of their choosing, including drag and other metabolic expenditures. The graph will automatically populate.

## Learning Outcomes

After using the winghead bioenergetics simulation, students should be able to use modeling to analyze, compare, and predict the following:

**Variation within populations.**By changing cephalofoil size (morphology) and swimming speed (behavior), individuals vary within the population. Each variant incurs different energy expenditures and gains.**Inheritance.**Sharks in this model have the potential to reproduce if energy gains exceed energy expenditures. The successful parental phenotype will be inherited by its offspring.**Natural selection.**Not all individuals survive, and those that do will have differential reproductive success reflective of the number of pups birthed. Traits are passed on in unequal rates, and this is a nonrandom process. Individuals that maximize energy gain to energy loss in their environment, in terms of prey detection and drag, will have the highest reproduction. The phenotype with the greatest reproductive success will be the most common in the population.

## Student Reasoning While Using Models

Computer simulations are often complex and need scaffolding to enhance student reasoning (Löhner et al., 2005; Sins et al., 2005). To guide student reasoning, Sins et al. (2005) outlined five steps in the process: *analyze*, *use inductive reasoning*, *quantify*, *explain*, and *evaluate*. The hammerhead model was intentionally made to be simplistic, in terms of only manipulating rostrum size and swimming speed, to reduce confounding variables and enhance student comprehension of the most essential concepts while exploring opposing evolutionary constraints. When using this shark model, instructors should guide students how to deconstruct the model into its individual variables of rostrum size, swimming speed, prey detection, energy gained, energy spent, and pups birthed. Then students should hypothesize how those variables interact and ultimately dictate energy budgets for growth and reproduction. As students manipulate the model variables, they will see quantitatively how each parameter changes. From those data, they should explain how phenotypic variation affects the opposing constraints of prey detection and drag, especially the upper and lower limits of each. Moreover, they should connect how all of the model variables are related and drive differential reproduction, which is central to natural selection. Finally, students should extrapolate the model results and apply them to concepts of natural selection, including comparing the model results to observed results in nature and applying them to other organisms and systems. Ultimately, these align with the AAAS (2011) core concepts of *evolution* and *structure and function* and the core competency of *ability to use quantitative reasoning*. Furthermore, the use of this model could be assessed as skills-based learning, meeting the core competency of *ability to use modeling and simulation*.

## Student Reception

This model has been used during five separate terms, to date, in a college course that was recently revised with funding by the National Science Foundation. Students in this class also used other evolutionary models. On two separate occasions, the overall course was independently assessed per National Science Foundation mandate. When compared to the commercial models, ~75% of the students ranked this shark model as of similar quality. Anonymous student comments about the shark model were overwhelmingly favorable and complimented the model's ability to reinforce concepts of natural selection.

## Alternative Models

While the model I produced is free to use and distribute, instructors may wish to build their own. To do so, at minimum a similar model must contain applications of Equation 1, Equation 2, and simulated prey detection and capture. Fish bioenergetics is an extensive field, and data are widely available. Jørgensen et al. (2016) provide a contextual overview.

This material is based on work supported by the National Science Foundation under grant no. NSF DUE-IUSE 1504662. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.