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PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20

Modeling Fish Growth in LowDissolved OxygenRachael Miller NeilanPublished online: 13 Aug 2013.

To cite this article: Rachael Miller Neilan (2013) Modeling Fish Growth in LowDissolved Oxygen, PRIMUS: Problems, Resources, and Issues in MathematicsUndergraduate Studies, 23:8, 748-758, DOI: 10.1080/10511970.2013.815677

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PRIMUS, 23(8): 748–758, 2013Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2013.815677

Modeling Fish Growth in Low Dissolved Oxygen

Rachael Miller Neilan

Abstract: This article describes a computational project designed for undergradu-ate students as an introduction to mathematical modeling. Students use an ordinarydifferential equation to describe fish weight and assume the instantaneous growthrate depends on the concentration of dissolved oxygen. Published laboratory exper-iments suggest that continuous exposure to low concentrations of dissolved oxygen(DO < 6.0 mg/l) causes a reduction in the average growth rate of summer flounder,compared with the conditions growth rates in normoxia (DO ≥ 6.0 mg/l). Studentsuse data from these laboratory experiments to estimate parameters in the fish growthmodel. Subsequently, students answer biological questions using their calibrated model.Analyses and simulations may be carried out in Matlab, R, or Excel. Supplementalmaterials are available for this article. Go to the publisher’s online edition of PRIMUSto view the supplemental file.

Keywords: Mathematical modeling, computation, biology, fish.

1. INTRODUCTION

The goal of this computational project is to expose students to basic conceptsin mathematical modeling. These concepts include constructing a mathemat-ical model based on physical assumptions, collecting data, estimating modelparameters using data, and answering important real-life questions using a cal-ibrated mathematical model. This multi-disciplinary project is appropriate forundergraduate students in a calculus or mathematical modeling course as wellas students beginning to learn about ordinary differential equations.

The project asks students to use mathematics to explore a problem posedby biologists. Specifically, students are asked to:

Address correspondence to Rachael Miller Neilan, Department of Mathematicsand Computer Science, Duquense University, Pittsburgh, PA 15282, USA. E-mail:neilan@mathcs.duq.edu

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● use an ordinary differential equation to model fish growth;● read a published article describing laboratory experiments for measuring fish

growth during exposure to low concentrations of dissolved oxygen;● use computational software and data from experiments to estimate model

parameters;● answer important questions about dissolved oxygen levels and fish growth

using mathematics;● present methods and results professionally.

Section 2 of this article outlines the project as it would be presented to thestudents and it also includes brief solutions for instructors. The remainder of theof the article is guidance for instructors. To complete this project, students musthave access to computational software (such as Matlab, Excel, or R). Solutionsprovided in Section 2 were prepared in Matlab. A Matlab .m-file with detailedsolution techniques is accessible online as a supplemental document.

2. PROJECT

2.1. Background: Fish growth and dissolved oxygen

In this project, you will parameterize and simulate a mathematical model forfish growth in the presence of an environmental stressor. In particular, you willinvestigate how exposure to low concentrations of dissolved oxygen affects theinstantaneous growth rate of summer flounder. Dissolved oxygen is the oxy-gen dissolved in the water and it is necessary for aquatic life. Most fish prefernormoxia, which is defined here as dissolved oxygen greater than or equal to6 mg/l. Lower levels of dissolved oxygen may be stressful to fish and con-tribute to sublethal effects such as reduced growth or decreased reproductivepotential [1, 3–5]. Extremely low concentrations of dissolved oxygen is knownas hypoxia and is typically defined as a concentration of dissolved oxygen lessthan 2 mg/l. Large areas of hypoxia in coastal waters are detrimental to fishsurvival and can cause massive fish kills [2].

A frequent assumption made by biologists and modelers is that the weightof a fish changes at a rate proportional to the fish’s current weight. Let M(t) bethe weight (in grams) of a fish at time t (days). From this assumption about fishgrowth, we can write the differential equation

M′(t) = GM(t), M(0) = M0, (1)

where G is the instantaneous growth rate and M0 is the weight of the fish attime t = 0. If the growth rate G is constant, then the solution to this differentialequation is

M(t) = M0eGt. (2)

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Biologists would like to understand and quantify the effect of lowconcentrations of dissolved oxygen on the parameter G. Below, you are givendata from published laboratory experiments on summer flounder exposed tovarious levels of constant dissolved oxygen. Based on this data set, you willconstruct and parameterize a mathematical expression relating G in (2) tothe dissolved oxygen concentration. Subsequently, you will use the cali-brated model to answer important biological questions about the risks of lowconcentrations of dissolved oxygen to summer flounder.

2.2. Laboratory experiments

You should access and read the article “Ecophysiological responses of juvenilesummer and winter flounder to hypoxia: experimental and modeling analy-ses of effects on estuarine nursery quality” by Stierhoff et al. [4]. The articledescribes several laboratory experiments performed to examine the effect ofconstant, low concentrations of dissolved oxygen on summer and winter floun-der growth. In one experiment, 40 summer flounder were equally distributed infour tanks and assigned to one of four constant dissolved oxygen treatments:7.0, 5.0, 3.5, or 2.0 mg/l. During the experiment, the temperature of each tankwas held at 30◦C. Each fish was weighed at the beginning of the experimentand weighed again after seven days. Table 1 displays the data collected dur-ing the experiment. It is displayed here with permission from the authors. Allweights are in grams.

Table 1. This table displays the weight (grams) of 10 summer flounder in each of fourdissolved oxygen treatments (7.0, 5.0, 3.5, and 2.0 mg/l). Fish were weighed at thebeginning and on day 7 of the experiment. Data is from published lab experiments[4] and reproduced here with permission by authors

7.0 mg/l 5.0 mg/l 3.5 mg/l 2.0 mg/l

Initial Day 7 Initial Day 7 Initial Day 7 Initial Day 7

8.55 13.82 6.53 7.47 5.19 5.47 5.76 5.954.66 7.44 6.88 9.96 8.08 9.28 8.40 8.526.84 10.33 7.53 9.74 5.18 6.47 8.49 8.659.53 15.47 10.22 13.15 8.90 10.69 6.63 6.62

10.24 14.94 7.75 10.92 4.45 5.47 5.90 6.2410.31 13.86 5.49 7.04 8.02 9.01 8.34 8.83

6.08 8.97 4.76 6.26 4.44 5.09 2.73 2.816.67 10.10 8.70 12.62 3.72 4.66 5.03 5.076.30 7.86 5.50 7.21 8.22 9.21 6.20 6.094.72 6.14 4.90 7.30 5.63 6.36 4.94 died

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Modeling Fish Growth 751

2.3. Mathematical model

Part A: Data analysis

1. Using the data in Table 1, compute the instantaneous growth rate for eachfish in the four dissolved oxygen treatments.Solution: If each column in Table 1 is entered as a vector, then students caneasily calculate G for each fish in a particular treatment using the equation

G = ln(M(7)) − ln(M(0))

7,

where M(0) and M(7) are vectors representing the weight of the fish at theinitialization of the experiment and on day 7, respectively.

2. Compute the mean and standard deviation of the growth rates in eachtreatment.Solution: This can be done in Matlab using the commands mean() andstd(). In the 7.0, 5.0, 3.5, and 2.0 mg/l treatments, the mean growthrates are 0.0544, 0.0417, 0.0217, and 0.0031, respectively, and the standarddeviations are 0.0132, 0.0113, 0.0080, and 0.0036, respectively.

3. Plot the growth rate of each fish as a function of dissolved oxygen.Specifically, plot the growth rates on an x-y scatterplot with “Dissolvedoxygen (mg/l)” on the x-axis and “Growth rate (1/day)” on the y-axis.Solution: Students can plot growth rates as points using the plot()command. See Figure 1.

4. In one sentence, elaborate a hypothesis about the effect of lowconcentrations of dissolved oxygen on summer flounder growth from thedata.Solution: Students will observe that as dissolved oxygen decreases the meanand standard deviation of the growth rates also decrease.

Part B: Parameter estimation

1. Assume the summer flounder’s growth rate is unaffected during exposureto dissolved oxygen concentrations greater than or equal to 6.0 mg/l andis negatively affected during exposure to dissolved oxygen concentrationsof less than 6.0 mg/l. In your mathematical model, relate the growth rate(G) to the dissolved oxygen concentration (DO) using a piece-wise linearfunction f . Assume G = f (DO) where f is

f (DO) ={

c DO ≥ 6.0a × DO + b DO < 6.0

, (3)

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0 1 2 3 4 5 6 7 8 9−0.03

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Summer flounder growth rates vs. dissolved oxygen

Figure 1. Points on this plot represent the instantaneous growth rates of the summerflounder in each dissolved oxygen treatment (2.0, 3.5, 5.0, 7.0 mg/l). We assume thesummer flounder’s growth rate is unaffected during exposure to concentrations of dis-solved oxygen greater than or equal to 6.0 mg/l and is negatively affected duringexposure to concentrations of dissolved oxygen less than 6.0 mg/l. A piece-wise definedline-of-best-fit is plotted (color figure available online).

and a, b, and c are constants. Let c equal the mean growth rate in the7.0 mg/l treatment. Use the growth rates from the 5.0, 3.5, and 2.0 mg/ltreatments to determine the values of a and b in (3). Also, describe themeaning of the parameter b in terms of fish growth and dissolved oxygen.Solution: Students can use the polyfit() command to determine values ofa = 0.0129 and b = −0.0229. The parameter b is the instantaneous growthrate when dissolved oxygen is depleted (DO = 0 mg/l). Because b is nega-tive, the model suggests that summer flounder will decrease in weight whendissolved concentration of oxygen is close to zero.

2. On your scatter plot from Part A, plot the piece-wise defined function G =f (DO).Solution: See Figure 1.

Part C: Answering biological questions

1. Predicting the effect of exposures not tested in the laboratory: Suppose asummer flounder with initial weight 5.0 g is exposed to constant dissolved

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Modeling Fish Growth 753

oxygen 4.1 mg/l for 10 days. According to your growth model, what is theexpected weight the summer flounder after 10 days of constant exposure?Solution: Using G = f (DO) from (3) with DO = 4.1, students will find G =0.0299. Therefore, M(10) = 5.0e(10·0.0299) = 6.742. The summer flounderwill weigh approximately 6.7 g.

2. Risk Assessment: The Environmental Protection Agency (EPA) uses math-ematical models (similar to yours) to suggest dissolved oxygen standardsin coastal waters to protect aquatic life from acute and chronic effects oflow concentrations of dissolved oxygen [6]. Suppose you work for the EPAand are asked to recommend a lower bound on dissolved oxygen levels sothat constant exposure to low concentrations of dissolved oxygen causesno more than a 25% reduction in expected summer flounder weight gainover a 24-h period. According to your growth model, what would yourecommend?Solution: Let M0 denote the initial weight of the fish. Let M1(t) denotethe weight of the fish in normoxia (DO ≥ 6.0 mg/l). Let M2(t) denote theweight of the fish in low concentrations of dissolved oxygen (DO < 6.0mg/l). We must ensure that

0.75 ≤ M1(1) − M0

M2(1) − M0. (4)

Substitute M1(1) = M0ec and M2(1) = M0e(a·DO+b) with values of a, b, andc from Part B into (4). Students will solve for DO to find DO ≥ 4.96.Therefore, according to the model, dissolved oxygen should be no lessthan 5.0 mg/l in order to ensure no more than a 25% reduction in summerflounder weight gain over a 24-h period.

3. Non-constant exposure: Most laboratory experiments investigating theeffects of low concentrations of dissolved oxygen on fish expose fish toconstant concentrations of dissolved oxygen for a specified period of time.However, in nature, fish are likely to experience non-constant concentra-tions of dissolved oxygen. This is because dissolved oxygen fluctuatesthroughout the day as temperature changes. Also, fish move in and out ofareas with low concentrations of dissolved oxygen as they search for food.Therefore, it is more realistic to consider a fish growth model that accountsfor exposure concentrations that vary in time.

Suppose a summer flounder with initial weight 5.0 g is exposed to dis-solved oxygen concentration of 7.0 mg/l for 22-h per day. During the other2 h, the fish swims into an area with dissolved oxygen concentration of1.5 mg/l to search for food. How can you extend your growth model topredict the weight of a summer flounder during non-constant exposures?Provide details of any biological assumptions that you make in extendingyour model. What would your extended model suggest the weight of the

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summer flounder to be after 10 days of this particular non-constant expo-sure? (Hint: One idea is to let G to be a piece-wise constant defined by thehour of the day. )Solution: Students are encouraged to explore different ideas for solu-tion and choose one that they can support biologically. As the hintsuggests, one idea is to solve model equations on an hourly time-stepto accommodate the hourly changes in dissolved oxygen. Let DO ={DO1, DO2, DO3, . . . , DO240} be a sequence of values representing thedissolved oxygen concentration each hour for 10 days. Let M(0) = 5.0.At t = 1 h, M(1) = M(0)eG/24 where G = f (DO1).At t = 2 h, M(2) = M(1)eG/24 where G = f (DO2).At t = 3 h, M(3) = M(2)eG/24 where G = f (DO3).

...

At t = 240 h, M(240) = M(239)eG/24 where G = f (DO240).Students may write a “for” loop to find the value of M(240). For the valuesof a, b, and c given above, we find M(240) = 8.21 g. Figure 2(a) displaysthe daily non-constant dissolved oxygen profile and Figure 2(b) displaysthe expected weight of a summer flounder after 10 days of this particularexposure scenario. In constructing this solution, it is assumed that the sum-mer flounder’s growth rate is affected instantly by its environment. In otherwords, there is no lag in changes to the growth rate when the summer floun-der moves in and out of areas with low concentrations of dissolved oxygen.

2.4. Laboratory report

You should submit a typed lab report that includes the following three sections.

1. Summary of Part A: Display the scatterplot with the growth rates.Summarize the growth rates in each treatment with the mean and standarddeviation. State your hypothesis about the effect of low concentrations ofdissolved oxygen on summer flounder growth.

2. Summary of Part B: Describe the methods used to determine values of a,b, c in the formula G = f (DO). Describe the meaning of the parameter bin terms of fish growth and dissolved oxygen concentration. Display thescatterplot with growth rates and the function G = f (DO).

3. Summary of Part C: Present clear solutions to each of the three questions.State any biological assumptions that were made in constructing solutions.Also, comment in one to two sentences on the use of mathematical modelsto answer questions arising in other disciplines.

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0 5 10 15 200

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0 48 96 144 192 2404.5

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Figure 2. (a) The daily dissolved oxygen profile for the non-constant exposure scenariofrom Part C, Problem 3 and (b) the expected weight of a summer flounder (M0 = 5.0 g)after 10 days (240 h) of the non-constant exposure.

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3. IMPLEMENTATION

Implementation of this project requires an in-class tutorial on the computa-tional software, a brief discussion of the laboratory experiments and projectobjectives, and time for students to complete the project outside of class. Beforepresenting the details of the project, I have the students meet in a computerlaboratory for one class period for instruction on the computational software.I typically require students to use Matlab, but if it is not freely available tothe students at the University, then both R and Excel are excellent alternatives.A brief, yet sufficient tutorial in Matlab can be done in 50 min. The tutorialshould show students how to create and execute .m-files, enter data as arrays,apply functions to arrays, perform a linear regression analysis, and plot dataand lines. I provide students with a handout with the commands learned dur-ing the tutorial for future use, and I provide information in the handout aboutadditional commands such as if-then statements and for-loops. Students areasked to access (via the University’s electronic database) and read the articledescribing the laboratory experiments [4]. Subsequently, the students and I dis-cuss the experiments in class and I provide an overview of the project. Projectinstructions are posted online for the students to follow step-by-step. Studentsare given approximately 2–3 weeks to complete the project and submit a typedlaboratory report. Although discussion among students is encouraged, I expectall students to submit a copy of their own work.

My assessment of the laboratory reports is based on presentation, cor-rect use of software for calculations and plotting, and appropriate justificationof solutions. In particular, there is no unique solution to Part C, Problem 3.I give full credit to students who present a clear and reasonable solution withappropriate biological justification.

My experience in implementing multi-disciplinary projects has been pri-marily in the context of a Mathematics for Life Sciences course. This courseprovides students with basic statistics, calculus, and mathematical modeling.In such a course, this project is ideally introduced after students learn basicstatistics, including linear regression, and at the beginning of their study of thederivative.

4. CONCLUDING REMARKS

The goal of this project is to introduce students to basic concepts in mathe-matical modeling via a real-life example from the biological sciences. Studentsuse data from a published laboratory experiment to formulate and calibrate amathematical model for fish growth. Students gain experience with computa-tional software (such as Matlab, Excel, or R) and answer important biologicalquestions using mathematics. In addition, students create a professional lab-oratory report describing their methods and results. The biological example

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outlined in this project is easily understood by students in all disciplines andthe concepts outlined here apply to all types of mathematical modeling, e.g.,modeling in physics, finance, engineering, etc. In general, feedback from mystudents indicates that projects (like the one presented here) combining math-ematics, computation, and real-life applications are interesting and effectivetools for motivating students to study mathematics for the purpose of solvingproblems in other disciplines.

ACKNOWLEDGMENTS

I am grateful to Kevin Stierhott, Timothy Targett, and Kerrilynn Miller for theuse of their data.

REFERENCES

1. Landry, C. A., S. L. Steele, S. Manning, and A.O. Cheek. 2007. Long ternhypoxia suppresses reproductive capacity in the estuarine fish, Fundulusgrandis. Comparative Biochemistry and Physiology, Part A. 148: 317–323.

2. Ecological Society of America. Hypoxia. http://water.epa.gov/type/watersheds/named/msbasin/upload/hypoxia.pdf. Accessed 18 July 2013.

3. McNatt, R. A., and J. A. Rice. 2004. Hypoxia-induced growth rate reductionin two juvenile estuary-dependent fishes. Journal of Experimental MarineBiology and Ecology. 311: 147–156.

4. Stierhoff, K. L., T. E., Targett, and K. Miller. 2006. Ecophysiologicalresponses of juvenile summer and winter flounder to hypoxia: experimen-tal and modeling analyses of effects on estuarine nursery quality. MarineEcology Progress Series. 325: 255–266.

5. Thomas, P., M. S. Rahman, J. A. Kummer, and S. Lawson. 2006.Reproductive endocrine dysfunction in Atlantic croaker exposed to hypoxia.Marine Environmental Research: 62: 249–252.

6. U.S. Environmental Protection Agency. (2000) Ambient Aquatic Life WaterQuality Criteria for Dissolved Oxygen (Saltwater): Cape Cod to CapeHatteras. EPA-822-R-00-012. Washington, DC: Environmental ProtectionAgency.

BIOGRAPHICAL SKETCH

Dr. Rachael Miller Neilan received her Ph.D. in 2009 from The University ofTennessee, Knoxville. In graduate school, she studied optimal control theoryand mathematical biology. After graduation, Rachael worked as postdoctoral

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researcher in the Department of Oceanography at Louisiana State Universitywhere she developed models for individual and population-wide effects ofhypoxia on fish and shrimp. These models inspired the project presented inthis article. Currently, Rachael is an Assistant Professor of Mathematics atDuquesne University. She resides in Pittsburgh, PA with her husband (also amathematician) and their son (a mathematician-in-training).

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