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Dancing Under the Moonlight

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004356/00001

Material Information

Title: Dancing Under the Moonlight A Mathematecal Modeling Approach to Foraging Octracod
Physical Description: Book
Language: English
Creator: Correa, John
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2011
Publication Date: 2011

Subjects

Subjects / Keywords: Ostracod
Modeling: Foraging
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Ostracods are a widespread crustacean that in certain species produce bioluminescence and have a relationship with cycles of the moon. By setting traps to sample the foraging population, and finding the rate of foraging, I investigate the relationship between the lunar cycle and ostracod foraging activity, using experimental data collected at Cayos Cochinos, Honduras from July 15 to July 28 in 2010. I observed that there is a relatively strong correlation between the lunar cycle and ostracod foraging activity. A mathematical model is developed to describe this relationship further. With the model, I am able to describe how the lunar cycle can play a role in turning non-foraging populations to foraging populations, and vice-versa.
Statement of Responsibility: by John Correa
Thesis: Thesis (B.A.) -- New College of Florida, 2011
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Yildirim,Necmettin; Gilchrist, Sandra

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2011 C82
System ID: NCFE004356:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004356/00001

Material Information

Title: Dancing Under the Moonlight A Mathematecal Modeling Approach to Foraging Octracod
Physical Description: Book
Language: English
Creator: Correa, John
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2011
Publication Date: 2011

Subjects

Subjects / Keywords: Ostracod
Modeling: Foraging
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Ostracods are a widespread crustacean that in certain species produce bioluminescence and have a relationship with cycles of the moon. By setting traps to sample the foraging population, and finding the rate of foraging, I investigate the relationship between the lunar cycle and ostracod foraging activity, using experimental data collected at Cayos Cochinos, Honduras from July 15 to July 28 in 2010. I observed that there is a relatively strong correlation between the lunar cycle and ostracod foraging activity. A mathematical model is developed to describe this relationship further. With the model, I am able to describe how the lunar cycle can play a role in turning non-foraging populations to foraging populations, and vice-versa.
Statement of Responsibility: by John Correa
Thesis: Thesis (B.A.) -- New College of Florida, 2011
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Yildirim,Necmettin; Gilchrist, Sandra

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2011 C82
System ID: NCFE004356:00001


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DANCING UNDER THE MO ONLIGHT: A MATHEMATICAL MODEL ING APPROACH TO FORA GING OSTRACO D POPULATIONS By John Correa A Thesis Submitted to the Division of Natural Sciences New College of Florida In partial fulfillment of the requirements for the degree Bachelor of Arts Under the sponsorship of Dr. Necmettin Yildirim and Dr. Sandra G ilchrist Sarasota, Florida April 26, 2011

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ii Acknowledgements Thank you to my Mom and Dad for bringing me into this world. Thank you to Dale for being my big sister. Thank you to Dr. Yildirim for helping me throughout the entire process a nd showing me what I could draw from the hard work I had done. Thank you to Dr. Gilchrist for inviting me on the trip to Honduras in the first place, after my other plans for summer programs fell through, and for believing in me and setting me on the right path. Thank you to Dr. McDonald for helping me to stay on track at New College and to realize when I was running into a wall. Thank you to Dr. Hart for being incredibly reasonable with my work, and discussing any topic. Thank you to the New College Travel and Grant Foundation for supplying the money necessary for my research. Thank you to Johnathan Statz and Alexander Salisbury for going through the thesis process with me, and helping me with many formatting issues. I would also like to thank Direcci—n Gen eral de Pesca y Acuicultura (DIGEPESCA) for letting me access the park to perform my research, and to the people of Honduras for preserving this natural resource. For anyone I left out, I sincerely apologize, this has been a long and complex process that t ook more effort than just myself could give. John Correa

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iii Table of Contents Acknowledgements ________________________________ ______________________ ii Table of Contents ________________________________ _______________________ iii List of Tables and Figures ________________________________ _________________ v Abstract ________________________________ ______________________________ vii Preface ________________________________ ________________________________ 1 1. Introduction ________________________________ __________________________ 3 1.1 What is an Ostracod ________________________________ ______________________ 3 1.2 What does the Ostracod hunt, and w hat hunts the Ostracod? ____________________ 6 1.3 Relationship with the Moon ________________________________ ________________ 7 1.4 Who studies Ostracods? ________________________________ ___________________ 9 1.5 Mating Interactions ________________________________ _____________________ 10 2. Methods ________________________________ ____________________________ 12 2.1 Field Methods ________________________________ __________________________ 12 2.1.1 Attraction ................................ ................................ ................................ ................................ .......... 14 2.1.2 Isolating the Ostracods ................................ ................................ ................................ .................... 15 2.1.3 Retrieving the Ostracods ................................ ................................ ................................ ................. 16 2.1.4 Counting Samples and Release ................................ ................................ ................................ ....... 17 2.2 Mathematical Modeling and Simulation ________________________________ ____ 19 2.2.1 Mathematical Modeling ................................ ................................ ................................ ................... 19 2.2.2 Development of a Mathematical Model ................................ ................................ ......................... 21 2.2.3 Nondimensionalisation of Model ................................ ................................ ................................ .... 23 2.2.4 Steady State Analysis of the Model ................................ ................................ ................................ 24 2.2.5 Calculation of Illumination and Percent Darkness ................................ ................................ ........ 26 3. Resu lts ________________________________ _____________________________ 31 3.1 Field Data Analysis ________________________________ ______________________ 31 3.1.1 Observed Changes in Ostracods ................................ ................................ ................................ ...... 34 3.2 Simulation Results ________________________________ ______________________ 39

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iv 3.3 Modeling Results ________________________________ ________________________ 46 4. Discussion ________________________________ __________________________ 52 4.1 Field Data Relevance and Importance ________________________________ ______ 52 4.2 Field Methods ________________________________ __________________________ 53 4.3 Modeling Data Relevance and Importance ________________________________ __ 53 4.4 Future Testing ________________________________ __________________________ 55 Reference ________________________________ _____________________________ 57 Appendix ________________________________ _____________________________ 59 Appendix A: Field data collected at Cayos Cochinos, Honduras between July 15, 2010 and July 28, 2010 ________________________________ _______________________________ 59 Appendix B: Matlab code that estimates the parameters in the Hill function for lunar data __ 64

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v List of Tables and Figures Figure 1.1: ................................ ................................ ................................ ................................ ................... 3 Figure 1.2: ................................ ................................ ................................ ................................ ................... 5 Figure 1.3: ................................ ................................ ................................ ................................ ................... 9 Figure 2.1: ................................ ................................ ................................ ................................ ................. 13 Figure 2.2: ................................ ................................ ................................ ................................ ................. 14 Figure 2.3: ................................ ................................ ................................ ................................ ................. 15 Figure 2.4: ................................ ................................ ................................ ................................ ................. 16 Figure 2.5: ................................ ................................ ................................ ................................ ................ 17 Figure 2.6: ................................ ................................ ................................ ................................ ................. 18 Figure 2.7: ................................ ................................ ................................ ................................ ................. 22 Figure 2.8 : ................................ ................................ ................................ ................................ ................. 28 Figure 2.9: ................................ ................................ ................................ ................................ ................. 29 Figure 3.1: ................................ ................................ ................................ ................................ ................. 32 Figure 3.2: ................................ ................................ ................................ ................................ ................. 33 Figure 3.3: ................................ ................................ ................................ ................................ ................. 34 Figure 3.4: ................................ ................................ ................................ ................................ ................. 35 Figure 3.5: ................................ ................................ ................................ ................................ ................. 36 Figure 3.6: ................................ ................................ ................................ ................................ ................. 37 Figure 3.7: ................................ ................................ ................................ ................................ ................. 38 Figure 3.8: ................................ ................................ ................................ ................................ ................. 39 Figure 3.9 : ................................ ................................ ................................ ................................ ................. 40 Table 3.1: ................................ ................................ ................................ ................................ .................. 41 Table 3.2: ................................ ................................ ................................ ................................ .................. 42 Figure 3.10: ................................ ................................ ................................ ................................ ............... 43 Figure 3. 11: ................................ ................................ ................................ ................................ ............... 45 Figure 3.12 : ................................ ................................ ................................ ................................ ............... 46

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vi Figure 3.13: ................................ ................................ ................................ ................................ ............... 47 Figure 3.14: ................................ ................................ ................................ ................................ ............... 49 Figure 3.15: ................................ ................................ ................................ ................................ ............... 50

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vii Dancing Under the Moonlight: A Mathematical Modeling Approach To Foraging Ostracod Populations John Correa New College of Florida, 2011 ABSTRACT Ostracods are a widespread crustacean that in certain species produce bioluminescence and have a relationship with cycles of the mo on. By setting traps to sample the foraging population, and finding the rate of foraging, I investigate the relationship between the lunar cycle and ostracod foraging activity, using experimental data collected at Cayos Cochinos, Honduras from July 15 to J uly 28 in 2010. I observed that there is a relatively strong correlation between the lunar cycle and ostracod foraging activity. A mathematical model is developed to describe this relationship further. With the model, I am able to describe how the lunar cy cle can play a role in turning non foraging populations to foraging populations, and vice versa. Dr. Necmettin Yildirim and Dr. Sandra Gilchrist Division of Natural Sciences

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1 Preface Work began on this thesis a little less than 2 years ago when I went on a trip with Dr. Gilchrist and other students to Cayos Cochinos in Honduras. While I was there helping others on projects concerning coral, I would stay up at night and watch the beautiful light displays done by a small marine crustacean. By the end of the three weeks there I was collecting them in jars and observing their movements. I would soon find out that what I had been so interested in were ostracods, also called seed shrimp, creatures that are almost ubiquitous across the globe. However, these li ght shows that had inspired me to try and videotape them (failing numerous time) were fairly unique to the Caribbean. Other ostracod species emit bioluminescence for protection, but the ostracods of Cayos Cochinos (and the Caribbean) use these lights to si gnal each other for mating. Being the curious sort, I immediately found a paper [15], and was inspired. In this paper Dr. Morin modeled the mating interactions that another species was performing. This complemented my interest in mathematics as well, beca use the paper eventually came up with methods to predict how the ostracods signaled one another. Unfortunately I could not acquire the equipment that was used in Dr. Morin's paper, and I had little inclination to try and directly copy the methods tha t he had performed. Instead, I researched Vargula hilgendorfii and how they are collected. Being a far more popular organism to study, there was more published work concerning shrimp wrangling of V. hilgendorfii than of my specific ostracod. From this I ca me up with a method of collecting the ostracods of Cayos, and releasing them back into the water after I had a value for how many had been caught. This is where my thesis then begins, after I have collected my samples for 14 days, spanning the waxing and w aning of the moon around

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2 the full moon during approximately the middle of the process. This thesis then centers on using this data to create a model of the activity of these bioluminescent ostracods in relation to the light provided by the moon. My initial hypothesis is that the moon provides the primary light source for the environment at the times I was testing, and therefore the ostracods, which rely on light for survival and mating, will try to get to the traps only when the light from the moon is at a minimum.

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3 1. Introduction 1.1 What is an Ostracod Ostracods are marine crustaceans that have a hard shell, eight limbs, and two eyes. The total distribution of ostracods is quite large, as they can live in almost any body of water and some areas on land that temporarily become wet. Figure 1.1: Photo depicting the general size and appearance of an ostracod. This was taken from Gerrish [8], and depicts Photeros annecohenae The compound eye, a feature of some ostracods, likely developed separ ately from other arthropods. Ostracods have been studied for the development of their bioluminescent courtship behaviors. A section of the ostracods important to this paper is

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4 the group, the myocopids. There are over 6,000 species of myocopids around the w orld, all in different marine benthic zones. Within the Myodocopida is the family Cypridinidae. This is a diverse group in size, ranging from .8 mm to 3 cm and habitat ranging from more than 5000 meters to the shallow coral reefs. Although these groups are quite large, there are few comprehensive studies of ostracods in the laboratory [8]. The species discussed here is related to a heavily studied species, Photeros annecohenae [3] This ostracod, like the ones caught in Honduras, feeds on dead and decaying fish in the meiobenthos. This is a common feature of ostracods, but the main difference between P. annecohenae and other ostracods is something that our species does have, a mating system using biolumescence. Species in the Photeros genus all use biolumine scence; the ones in the Caribbean are known to use it for mating. In other genus', bioluminescence is used as strictly a defense system. It can scare away a predator or signal an alarm to other ostracods nearby to get away [8]. There are several genera that have luminescent ostracods, including Cypridina Vargula and the relatively new Photeros [3], that contains P. annecohenae Photeros is the genus that contains the ostracod that were collected in this experiment, and they are widespread in the Carib bean. They are benthic, meaning they stay at the bottom and often they use the bottom to hide and to feed on decaying organic matter. Cypridina on the other hand is more planktonic than Photeros and is far more affected by tides and currents as a result. Some species in Vargula only use bioluminescense for defense. Although all produce bioluminescence via luciferin and luciferase, Photeros appears alone in this paper [13].

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5 The following life cycle information on P. annecohenae was taken from Gerrish [ 8]. There are six instars observed in our well documented ostracod, P. annecohenae Females are larger on average. In cases where the size of the ostracod and other features do not obviously distinguish sex there are a number of other methods of identifica tion. After the fifth instar the males copulation limb becomes more apparent, and before this identification needs to be done by comparing eye, keel, and valve sizes of the ostracods. Figure 1.2: Stages of development in ostracods. Adult forms and juveni le forms have very different characteristics, and here it displays the array of sizes that are encountered [8]. The development time of eggs and embryos in ostracod females was 26 1.6 days with the time between mating and the eggs beginning develop ment in the brood chamber of the female was 7.84 1.06 days. The time to release from the brood chamber was about 18.4 0.52 days. At first, the eggs do not appear to change, and in most ostracods

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6 the fertilization does not occur until the eggs actually move into the marsupium(brood chamber, development chamber). The eggs are released in the brood chamber and after two to three days the embryos appear cloudy. By the ninth day in the chamber a clear difference between one side of the embryo and the other m ay be observed. By days 14 or 15 eye spots begin to form. In another two days the spots darken and will be formed into eyes by days 17 or 18. It is at this time that the light organ also forms in the embryo. At day 19 the embryos are essentially first inst ar ostracods waiting to be released. The average number of offspring in the brood of P. annecohenae is 12.9 0.43. Interestingly, it is possible for the females to produce multiple broods of offspring without re fertilization, suggesting that the sperm is s tored. This suggests parthenogenesis, but further genetic studies would be needed to confirm or deny that parthenogenesis is used. This new brood may be released 15 to 17 days after the first brood has been released. In development of the juvenile instars, it took 80 100 days for the ostracods to develop and each instar lasted an average of 18.9 .77 days. In the culture in lab, the adults survived up to 188 days. In comparison to other ostracods, this model species has a long life span (about a year) and de velops slowly (about 3 months). It also produces multiple broods of few offspring, and does not have a larval stage 1.2 What does the Ostracod hunt, and what hunts the Ostracod? The ostracods that are being discussed in this paper scavenge. Ostracod s as a whole however have a variety of different methods that they use for collecting and digesting food. For benthic ostracods though, the majority will feast on dead and rotting organic matter, usually from fish.

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7 An interesting story can be found i n a paper by Felder [7], which notes that luminescent ostracods were found near a rotting fish carcass, and this site was also visited by a ghost crab. The ghost crab ingested enough ostracods to become luminescent, thus causing interest by the field biolo gists who investigated and wrote the paper. In a way, this shows a typical scenario, where the ostracods scavenge and in the process happen to be consumed and predated upon. Ostracods are well liked prey of a variety of organisms, but especially fis h [1]. Ostracods avoid this predation through a variety of means. One way some ostracods avoid predation is with bioluminescence, as in the ones studied here, but there are others that survive being digested [1], and many others with a variety of defenses. The ostracods studied here may use their bioluminescence to ward off predators, but they may also hide in the substrate and benthic areas [8]. 1.3 Relationship with the Moon For adult ostracods there is a dark threshold where less than a third of the moon must be illuminated for foraging and reproductive behavior to commence [9]. This can also be reached 2 3 minutes before nautical twilight when there is no illumination from the moon. Juvenile of these ostracods were also nocturnally active, but did n ot show any response to lunar light. In addition, water velocity was important for the activity of the ostracods. Darkness can be considered a resource in a similar way that time, seasons, and location are viewed. This is not that strange, as darkness is a lready viewed as important in systems of other organisms, such as plants, which will not develop seeds and reproduce properly unless the correct amount of light and dark is occurring. In marine systems coral, invertebrates, and fish all can have a link to the cycle of the moon. Fish

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8 and invertebrates will use the cover of darkness to rise up in the water column for feeding. However, bioluminescent ostracods are particularly interesting because they need darkness for mating and communication. This puts them in a fascinating relationship with the lunar cycle. Other organisms share this threshold that can be crossed where they may become active, and still others are designed to mate in relation to the location of the moon, based around the safety of the darknes s. One could also view situations where the mating that can occur only at this specific hour will cause a large number of females to wait to copulate, thus causing large fluctuations in the number of active animals, as seen in Figure 1.1. All of these can occur in the natural setting, and produce similar results. In this study of the ostracod P. annecohenae there were several similarities to the conditions that were met in this study. First and foremost is that the moon was not measured at the time for luna r intensity. Instead, a model for the lunar intensity was used. This simply produces a curve that can be referenced and set so that it displays how much illumination should be present in the environment at the time[9].

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9 Figure 1.3: Role of moonlight in ost racod foraging [9]. 1.4 Who studies Ostracods? Ostracods are of interest to paleontologists, because their shells fossilize well, and can be observed clearly for many years [5]. This means that there are an extensive amount of studied ostracods tha t are extinct. Some paleological studies use ostracods to make predictions on the environment at the time, and thus are essential for some paleontology work[12]. Current studies of living ostracod populations often look at ostracod populations role in a larger ecological system. One way would be to look at all organisms that inhabit its environment, by sampling the sediment and collecting any organisms. This is done in

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10 some studies such as Coull [4] where many organisms were sampled over 11 years to measure the environmental health of the ecosystem being observed. Other times ostracods are studied as part of a unique environment, such as Timms [18], where ostracods could survive in a lake that often dried up, and was only filled with water during cer tain seasons. But ostracods are not only part of an ecological system, they can also tell us much about the system itself. Ruiz [16] talks of the ostracod as a possible sentinel organism, letting researchers know the health of an ecological system. Du e to the wide range of ostracods, Ruiz looked at as diverse locations as New Zealand, Japan, and parts of Western Europe. As mentioned, ostracods are part of the greater predator and prey system that exists, and so it is not hard to see why less ostracods in an environment would be a bad sign. The factors Ruiz discusses that can give a clue to the problems that might exist, concern age profiles of ostracods, population density, and other factors that would be measurable using the techniques in this paper. Finally, due to the unique nature of the bioluminescence that ostracods use, there has been extensive study into the associated chemical pathway. This can be noted in Nakajima [13], where luciferin and luciferase, the two chemicals that combine to cre ate the light, is laid out in chemical formula. 1.5 Mating Interactions Mating displays of ostracods in the Caribbean are very complex. They are actually more related to firefly mating displays than with mating displays of other marine animals. The s ignals are sent by shots of luciferin and luciferase that mix and react to produce light. The different species of ostracod can have different times between signals and

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11 different paths that are taken when the signal is produced. Males will compete with one another, attempting to steal mates from others, and the overall system of flashes can be modeled and can be simulated. These interactions are studied by Morin [15] and are very difficult to photograph in the wild. Due to the difficulty in observation a means of viewing and measuring the ostracod and their activity had to be devised. The end result was a simple and repeatable experimental method, and the development of a model from this data.

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12 2. Methods 2.1 Field Methods Initially the col lection of the Ostracods and measurement of the results was developed from reading websites of biologists who had been in the field. This started with the work of Oakley[14], which inspired the methods that were used, and lead to reading the work of Gerris h[9], which set out some field methodology to help design the methods used here. The development of the traps and trapping system was based on a number of steps. I. Attracting the ostracod II. Isolating only the ostracods and not other scavenging organism s III. Retrieving the ostracods IV. Counting and releasing the samples During data collection, the traps were dropped into the water for 15 minute periods at the start of every hour, and then analyzed over the next 45 minutes before being set for another round of sampling. This was performed at Cayos Cochinos (Figure 2.1). The samples taken with the trap tied to the dock, placing it on the ocean floor upright, but without slack in the line.

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13 Figure 2.1: A map of the area where the field data were collecte d. The dock and sampling area was at Plantation Beach Resort. Photo retrieved from http://www.moon.com/maps?filter0=75362. Figure 2.2 displays the dock under which samples were taken. Samples were taken at the third post from the end of the dock. Th e water environment under the dock had several fish surrounding it during the day, and had a set of rock at the end of the dock that were visited by octopi and rays on occasion. There were a number of bivalves here, as well as barnacles on the dock posts t hemselves. A small reef was located approximately 50 feet to the south, with a larger reef offshore.

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14 Figure 2.2: Dock beneath which samples were taken. This photo was taken and provided by New College student Julie Krzykwa. 2.1.1 Attraction Ostrac ods will scavenge off of dead and decaying matter in the benthic zone. In some works, such as Oakley [14], the bait used was liver, as it has a large number of attractive compounds. Liver was not available at the time of the experiment, so steak was allowe d to rot over the course of the day and then subsequently used at night (Figure 2.3). This worked as a suitable substitute, and the amount of ostracods caught seemed to mirror the results seen from other studies. Sampling began at 7 pm, and lasted 15 minut es, occurring at the start of every hour. The meat worked, being productive over the course of the night it was used.

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15 Figure 2.3: Trap with bait is depicted. Trap is 22.0 cm tall, 17.0 cm diameter at the opening, and holds and approximately 460 mL of wat er at a time. 2.1.2 Isolating the Ostracods The main concern with catching the ostracods was ensuring that only the ostracods could get at the bait, and that a sufficient number of ostracods could get into the jar for an accurate measure of the acti ve foraging that was occurring. The solution is surprisingly simple though due to Oakley [14]. He used a small opening, and he notes that others used a perforated jar lid to let the ostracods pass but nothing else. In this experiment netting was used that was approximately 5mm wide at the openings, and this allowed the passage of all sizes of ostracods (Figure 2.4). Even with this, a number of other creatures were caught in the traps, such as isopods and the occasional gastropod.

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16 Figure 2.4: Covered trap is depicted. Note the size of the mesh is approximately 5mm which is slightly larger than the average mature ostracod's size. 2.1.3 Retrieving the Ostracods The method used of retrieving the trap was a simple one. The trap was simply a jar with a n et over the opening, and a rope was tied around the top, as in Figure 2.5. The jar was lowered so that it rest on the ocean bottom, and was tied to a pole on the dock. Even on days with heavy current, the jar did not move far from the initial drop point. W hen the time period began, the jar was quickly sent down, and when the time was up the jar was retrieved as quickly as possible so as to catch the ostracods in the trap. Occasionally the jar would fall off the rope, and for this reason several jars were ke pt as emergency back ups. The loss of data points is noted, and there was only one night that was cut short from the loss of a trap.

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17 Figure 2.5: This is how the trap was tied and then lowered into the water, vertically. 2.1.4 Counting Samples and Releas e Initially the samples were going to be pipetted out of the jar and then expelled through a fine mesh netting. However, after trying this technique in the field, it was found to be too time consuming for the 45 minutes between trap retrieval and tra p placement. Instead, the wide mesh net used to let the ostracods in was removed, and then the bait removed as well. This left the ostracods and other creatures in the trap with water. Then a fine mesh net was wrapped over the top of the jar, as seen in Fi gure 2.6, and the water was poured out through the net (Figure 2.7).

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18 Figure 2.6: Fine mesh net, wrapped over the trap after meat removal and net replacement. This technique gave very good results, as the ostracods became not only very easy to move and handle, but also were easy to count and measure. First, the other creatures in the trap were noted and removed (typically isopods) and then the number of ostracods counted. Finally, the ostracods were moved into a tube for volumetric measurement. After de termining the volume, the ostracods were released into the water by dipping the tube into the water by the dock. This resulted in a very fast, yet accurate measure of the amount and ratio of older to younger ostracods in a sample.

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19 2.2 Mathematical Modeling and Simulation 2.2.1 Mathematical Modeling The samples collected at Cayos Cochinos have three main pieces of information, first the time that the sample was taken, the amount of ostracods collected, and measured the volume of the ostracods in the tra p. Additionally, there is the location of the moon, which can be retrieved from the lunar charts. By combining data, several facts can be gleaned. For example, there is, from the research presented, the idea that the ostracods have extremely different fora ging methods if they are adult or juvenile. The total number of the ostracods does not differentiate though. Therefore the total amount collected is equal to an unknown amount of adults and an amount of juveniles. This can be expressed as T ot a l P opul a t i on = A dul t P opul a t i on + J uve ni l e P opul a t i on ( 1 ) where adult population is the number of adults and juvenile population the number of juveniles. The volume of the ostracods is also made up of these va lues. We have the total volume, which is made up of the volumes of the adults and juveniles. If we can estimate an average volume for the ostracods then we can make an estimate of the amount of adults and juveniles that are in the sample. While this predic tion style is not performed in other papers, it can be used with the data collected here, but to what accuracy may be unknown. It is highly dependent on information of the population itself, information that is not absolutely known for the population studi ed. Therefore, estimation of the age distribution using volume will be not be used in development of the model. Looking at the rates (of ostracods entering the trap per hour) that were collected we can see that not every night has a similar curve, and some nights are more greatly influenced by environmental factors (Figure 3.2). These environmental factors were

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20 noted in the lab notebook at the time of sampling, and include various factors such as rain, thunder, cloud cover, and trap malfunctions. Rain and thunder may have an effect on the ostracods foraging activity, but they definitely prevented sampling on a few occasions. Cloud cover was noted and may influence the sampling as well. Spaces in the samples are noted where the trap malfunctioned (such a s falling off the line, meat being ineffective, etc.). Considering the many variables that are at play in this situation, a complete model of the ostracod's interactions may be out of reach. However, from considering the data presented, some important clue s can be noted for the creation of the model that we will be suitable for making some predictions. This model then will focus on predicting the ostracod population from the major factors that affected changes in the population in studies, and will attempt to ignore temporary environmental changes. Data analysis and model simulations were performed in Matlab[10] First the data collected from the site were transcribed into Matlab and the days and hours set so that they start from zero. One of the key fe atures was that at 6:00 everyday the ostracod count was zero, but at 7:00 the amount of ostracods would become detectable as they would start approaching traps. The data collected can also be viewed in different ways. For example, and this is perhaps the most important, after the traps were retrieved the amount of ostracods in the trap that were caught was recorded. This is the amount of foraging ostracods that would enter the trap over the course of fifteen minutes. Therefore, this is actually the fo raging rate if it is divided by the amount of time taken, and therefore can be thought of in a more model applicable process.

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21 In addition the data can be viewed over the course of several days, giving a model of the total reaction of the ostracod popu lation to the changes in moonlight. This was first done with regression analysis by Gerrish [9]. Among the variables that likely caused the changes in the ostracod interactions, darkness was the one that seemed to relate the most to the changes in foraging ostracods. This was done by first calculating % Darkness, which in the paper is different than the % Darkness used here. Calculations of percent darkness are shown in the subsequent section. 2.2.2 Development of a Mathematical Model If we look at th e environment that we are attempting to model we can analyze the system in terms of the change in the foraging rate. This rate of change in foraging population is equal to the difference between the rate of gain in foraging population and the rate of loss in foraging population [6], as illustrated below in Eq. ( 2 ) r a t e of c ha nge i n f or a gi ng # $ % & = r a t e of ga i n i n f or a gi ng # $ % & r a t e of l os s i n f or a gi ng # $ % & ( 2 ) If we consider this as a picture, depicting the non foraging population as NF and the foraging population as F, we have a picture such as NF ~ + f ( t ) ! # ! F ~ ( 3 ) Where is the rate constant for the return of the ostracods from the foraging population ( F ~ ), is the rate constant at which the ostracods go from non foraging population( NF ~ ) to foraging population. And f(t) is the darkness function which will

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22 increase the active foraging population of ostracods, by influencing the rate constant In terms of the total system we can look at it as Figure 2.7: This is a graphical representation of the model that is used. Darkness is the main effect on the exchange between the non foraging and foraging populations. The model here will focus on these effects, within closed system. We assu me the system is based on the movements of these two main populations that change between each other. Figure 2.7 represents a simplified environmental system that encloses two subpopulations of ostracods, and excludes affects that change the size of the po pulation, such as predation. The dotted line in Figure 2.7 represents a closed system, in that there are no more populations coming in and out of the system (this excludes predation from this system). Therefore the total population N T ot > 0 is the sum of the non foraging population and the foraging population, as in Eq. ( 4 ) NF ~ + F ~ = N T ot ( 4 ) Under these assumptions we can write a system of two differential equations, as

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23 d F ~ dt = ( + f ( t ) ) NF ~ # F ~ d NF ~ dt = ( + f ( t ) ) NF ~ + # F ~ ( 5 ) where the values of f(t), NF ~ and F ~ remain the same as in Eq. ( 3 ) 2.2.3 Nond imensionalisation of Model We can then divide both sides of Eq. (4) by N T ot to get NF ~ N T ot + F ~ N T ot = 1 F + NF = 1 ( 6 ) Where F = F ~ N T ot a nd NF = NF ~ N T ot ( 7 ) This states that for any set population of NF and F, we can sum them both equal to 1. We can then convert this so that it takes place in a system where the foraging and non foraging populations are set in relation to the total population. We can then multiply by N T ot to get d F ~ dt = dF dt N T ot d NF ~ dt = dNF dt N T ot ( 8 )

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24 However, we know the values of d F ~ dt and d NF ~ dt and therefore can re write this as dF dt N T ot = ( + f ( t ) ) NF ~ # F ~ dNF dt N T ot = ( + f ( t ) ) NF ~ + # F ~ ( 9 ) Dividing both equations in Eq. ( 9 ) by N T ot we get dF dt = ( + f ( t ) ) NF ~ N T ot # F ~ N T ot dNF dt = ( + f ( t ) ) NF ~ N T ot + # F N T ot ~ ( 10 ) which results in, through Eq. ( 7 ) dF dt = ( + f ( t ) ) NF # F dNF dt = ( + f ( t ) ) NF + # F ( 11 ) where NF and F are defined as in Eq. (7). 2.2.4 Steady State Analysis of the Model At a steady state the rates for the foraging population and non foraging populations will not change over time. To investigate the relationship between ostracods and the moon, we assumed a small fraction of the total population was foraging when darkness was at a minimum, namely f(t)=0. Let F and NF represent the steady states of the foraging and non foraging populations respectively. Therefore at this steady state we have dF dt = dNF dt = 0 that gives us

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25 0 = ( + f ( t ) ) NF # F 0 = ( + f ( t ) ) NF + # F ( 12 ) As we can see there is only and f(t). The function f(t) will be equal to zero wh en darkness was at a minimum which gives us 0 = NF # F 0 = NF + # F ( 13 ) We also have NF + F = 1 ( 14 ) So we can set NF = 1 F ( 15 ) and get ( 1 F ) + # F = 0 + ( + # ) F = 0 ( 16 ) and therefore by solving Eq. ( 16 ) we get the steady state F = + NF = + ( 17 ) We can now set and so that they control the populations at the steady state, when f(t)=0. If is set to be much smaller than then the steady state will have a small fraction of the population foraging. If is set much larger than then the steady state will have the foraging population close to the value of the total population.

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26 2.2.5 Calculation of Illumination and Percent Darkness The ostracod population should respo nd to environmental changes in the moon, specifically in darkness [9]. While the exact formula used for darkness is never specified in this paper, it is important to create one of our own and to then analyze it appropriately. While this is performed in the results section, the formation of the darkness can be explained and clarified at this juncture. For ostracods the amount of darkness seems to increase the ability to forage and to effectively hunt. In order to investigate this effectively the amount of darkness needs to be able to be quantified. Darkness is a function of time that we assume changes between zero and one hundred. Zero represents the situation where the moon is at maximum illumination, and one hundred represents a lack of lunar light. To create our darkness function, information is taken from meteorological sources. This gives us the lunar intensities, as well as the amount of time that the moon is up. We can use this lunar intensities to say what value of illumination the moon wi ll reach when it rises[11] From these values we can see that the moon is up for a certain period of time, and then it sets and is not visible (or visa versa). In the first case, where the moon is risen as the night starts, the equation that will be us ed is L ( t ) = V m Km n Km n + t n ( 18 ) We use a Hill function at this juncture, as it is a function that will allow us to manipulate the curve so as to refl ect the changes observed in the environment. This is not

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27 limited to information collected, but also information that is expected to occur, such as lunar intensity, something that would be difficult to measure in the field. L(t) represents the intensit y of the lunar light. Vm, Km, and n are adjustable positive parameters, selected from the observed and collected data. Vm is the maximum illumination provided by the meteorological resources, n is a positive integer, t is time, and Km is a rate that determ ines the point along the timeline that the curve changes. We can see that at time zero the value of L will be equal to Vm. As this Hill function increases with time t we can see that the function will reach a point where the moon sets and attempting to mat ch it to the associated moon values will require changing the Km value so that it lines the point where the function changes with the change in the data. Furthermore, the value of n will also be changed in order to align the rate of decrease in the lunar d ata.

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28 Figure 2.8 : This is a graphical representation of the decreasing Hill function given in Eq. ( 18 ) This decreasing Hill function was used to describe days in which t he moon was risen before nightfall and set during the course of the night. This Hill function was then set so that it changed between total light and total darkness over the course of only an hour, as in the environment. This will work for the first few days, until the full moon. Even at this point the function can still be used, but the time that the moon sets is beyond the range of the night time. However, there are a few days where the moon rises during the night. This means that the function will need to be changed so that the function rises with the night time. This change is made so that the function appears as so: L ( t ) = V m t n Km n + t n ( 19 )

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29 It should be noted that at time zero the function will be at zero as well. When the time goes to infinity the function will maximize to the value of Vm. Thus, the Km value again determines the point at which the function will change and can be recorded as a w ay of stating the hour the moon rises. Figure 2.9: A graphical representation of the Eq. ( 19 ) which is the ascending Hill function. This function will be used during nig hts where the moon rises over the course of the night. This function was then matched to the meteorological data found and made to simulate the changes that occurred in the environment. Finally the Hill functions can be integrated, as I L = L ( t ) dt 0 t m a x ( 20 )

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30 and will then supply the value IL, which is the value of illumination for that night. t m a x represents the maximum time, w hich would be the time of the sunrise. This then provides a good way of looking at the amount of light that is cast into the system on any one night. Then the percentage darkness can be calculated as % D a r kne s s = I L m a x I L I L m a x x 100 ( 21 ) where I L m a x represents the maximum illumination in the environment. With this calculated percent darkness we can then examine the same st udies that Gerrish [9] performed and apply them to our own data.

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31 3. Results 3.1 Field Data Analysis There were 14 days of data, with collections of volume and number of ostracods. The moon was also noted and its rise and fall marked on the data. First, there are several dates that must be noted for odd values, depending on environmental factors such as weather, and for problems with the trap collection. On the 19 th of July, day 5 the meat appeared to not be working as well as previously. New me at was taken and supplied to the trap but that night may have been off track because of this. The next day, the 6 th day, had a trap fall off the line at 8:00, eliminating this hour of testing. In addition, on the 23 rd the trap fell off the line at 7:00, an d so the knot that ties the trap onto the line was changed, and these issues decreased. The moon data, as displayed in Figure 3.1 denote the intensity of the moon on the y axis, and the x axis is the hour. Figure 3.2 shows the data on foraging rate collected, and shows the rate in ostracods per hour on the y axis, and hour sampled on the x axis.

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32 Figure 3.1: The graph that is on the left hand column at the top represents day 1, and the graph to its immediate right, in the right hand column, represen ts day 2. Day 3 is in the subsequent row in the left hand column, and proceeds in numbering. It can be noted that days 9, 10, and 11 are all close or are full moons. Hour is the hour from 6:00 PM, and lunar light level is the lunar intensity.

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33 Figure 3. 2: This figure displays the changes in average foraging rate over time. This plot shows the rate at which ostracods entered the trap at any specific time(Ostracods/hour). Beginning at the top in the left hand column in the first row is day 1, and to its ri ght, in the right hand column, is day 2. In the subsequent row below there is in the left hand column, day 3, and this continues in sequence.

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34 These data results are very similar to the data collected by Gerrish [9]. There are definite spikes in the volume when the moon has set. This suggests that the fully developed population of ostracods will only forage at these times. However, the number, which is far more dependent on the number of juvenile ostracods that are foraging, is much more likely to no t change with changes in the moon. 3.1.1 Observed Changes in Ostracods In this section, photographs are analyzed so that the physical characteristics of the observed species are noted. A field microscope was used to take the following photos. Some of these were done through a microscope vial that kept the ostracods free swimming, and others were taken with samples that were removed and photographed on the bench top. Physical features of the ostracods are described. Figure 3.3: This figure was the fi rst photo taken of the ostracods while on the island, at seven times magnification. This was taken with a microscope provided by Dr. Gilchrist; carapace and pigment spots can be clearly discerned. Range of size also becomes apparent quickly.

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35 The ost racods observed at Cayos Cochinos were quick moving, small, and hard to isolate. Getting a still picture of a free moving ostracod thus became a difficult yet important endeavor for this project. In Figure 3.3, the first photo taken is presented. Figure 3.4: This figure shows how to identify males and females, roughly, from a photograph, at seven times magnification. The best way is to note the copulatory limb, which is present on males, but not on females. In addition, females are typically larger than males. There is a startling lack of video of ostracods, especially considering their elegance in the water and their associated "light show". The reason for this becomes clearer as it is seen in the field that the light emitted by the ostracods is p erceptible to the human eye but is at a level that is very difficult for a camera to photograph. In Figure 3.4 we see different sexes of ostracod. The males have a copulatory limb that is underneath the carapace, making identification easier.

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36 Figur e 3.5: This figure was taken at seven times magnification, and gives an excellent view of the shape of the carapace of the ostracod. The eye pigment spots are very clear, and internal structure can be barely observed. The keel at the back stands out as wel l. Observing the ostracods in the viewing tube was difficult. An alternative would to take the samples from the water and examine in a dry environment. This type of viewing can be observed in Figure 3.5. Thankfully, because of the hard shell of the ostracod, the samples retain their shape well out of water, and the characteristics of the Photeros family can be seen. Notably, the shape of the shell and notch appear to agree with literature on shape, and the presence of bioluminescence in the samp le is a strong indicator that the samples were from this small family.

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37 Figure 3.6: Taken at twenty seven times magnification, this figure displays a close up view of the pigment spot. The location of the actual eye would be expected to the slight right of this spot. Also the detail of the carapace can be observed, showing the calcified structures that make ostracods such useful morsels in predators diets. Figure 3.6 is the next to be examined, and it was taken at the highest magnification possible This makes it difficult to observe the carapace shape, but gives an excellent view of other features. Of note here is the brilliance of and clarity of the eye pigment spots. From a distance it appears that the ostracods themselves have small eyes. These are actually the pigment spot, that are used in sight, but is not the eye itself [2]. To get a good look at the complete eye of the creatures, it would be necessary to dissect the samples.

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38 Figure 3.7: This figure shows a free swimming ostracod, at seven times magnification. This was done to ensure that the figures presented here did not show a changed or altered view of the organism. Although light, the limbs used in propelling the ostracod through the water can be seen here, as it searches for food, she lter, and mates. To return briefly to the water viewed ostracods, Figure 3.7 shows a singular free swimming ostracod. The free swimming nature of ostracods makes this species difficult to capture on camera, but also allows for their foraging. If we were dealing with a planktonic form rather than a benthic form of the ostracod, it would not be as easy to isolate as these free swimming benthic forms that are attracted to and ingest organic matter around them.

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39 Figure 3.8: This figure shows one of t he best detailed on the set, focusing mainly on the pigment spots which are so easy to identify. This allows for easier identification and analysis of the carapace. This was taken at seven times magnification. The incisor is also visible, and is easy to id entify in ostracods, and represents the potion of the mouthparts visible at this magnification. Finally, in Figure 3.8, there is an excellent view of the shape of the carapace. The notch on the end is a sign that it is in the Cypridinidae family and the genus Photeros [2]. 3.2 Simulation Results By taking the data collected, we may estimate parameters of the associated Hill functions. The graphs in Figure 3.9 depicts the Hill functions fitted to the illumination data using Matlab 's built in funct ion lsqcurvefit

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40 Figure 3.9 : Hill function fitted to known lunar data to simulate the changes in the environmental light that the sampled Ostracods experienced. This uses Eq. ( 19 ) on days 1 11 and Eq. ( 20 ) on days 12 14 to generate approximate graphs of the changes in the lunar intensity in relation to the hour of the day. The values in Tab le 3.1 and 3.2 are used for matching the environmental data obtained from observations in the field.

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41 In Table 3.1 and 3.2, the estimated parameter values are given by minimizing the sum of the difference squares between the data and the fitted Hill funct ions. Table 3.1 has the parameters that resulted for the decreasing Hill function for the first 11 days. Table 3.2 has the same parameters fit for the last three days. The Matlab code we used for calculating these estimations is provided in Appendix B. Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Vm 22 33 45 56 67 77 85 N 8 20 20 20 20 20 20 Km 2 3.50 4.50 4.50 5.50 6.50 7.50 IL 45.10 115.96 203.29 253.03 370.00 502.50 639.09 Day 8 Day 9 Day 10 Day 11 Vm 92 96 100 100 N 20 20 20 20 Km 8.50 9.50 10 0 100 IL 783.38 912.87 1000 800 Table 3.1: Estimated values of the parameters parameters in the Hill function given by Eq. ( 19 ) using the code given in Apendix B.

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42 Table 3.2: Estimated values of the parameters parameters in the Hill function given by Eq. ( 20 ) using the code given in Apendix B. Day 12 Day 13 Day 14 Vm 99 98 95 N 20 8 12 Km 1.50 1.50 2.50 IL 642.88 241.17 234.77 We calculated in Eq. ( 21 ) the illumination of the moonlight at night, and attempted to cl assify it in terms of the percent darkness that appeared using Eq. ( 21 ) Thus the percent darkness can then be calculated and compared to the foraging ostracod population. For calculating percentage darkness we need to find the illumination using Eq. ( 20 ) then we can calculate the percent darkness by following Eq. ( 21 ) Using the estimated parameters in Tables 3.1 and 3.2 we can find the illumination value for each day. To do this, we must take the integral of the illumination function over the observed t ime interval, Matlab 's quad function, which numerically evaluate a given integral using the adaptive Simpson quadrature, was used to calculate such integrals

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43 Figure 3.10: Determining the percent darkness and average foraging rate allows for the corre lation to be calculated, as in Gerrish [9], the correlation is seen to be fairly strong. The percent darkness was first calculated for each night, and the corresponding percent darkness values were set so that they could be compared to the values for the a verage foraging rate. To investigate if there is any correlation between darkness and foraging rate, we averaged the foraging rate collected over a particular night and plotted it with the percent darkness calculated for that particular night. Resu lts are shown in Figure 3.10, and suggests that there is a relatively strong correlation between the percent darkness and the averaged foraging rate with correlation coefficient R=0.6432. This shows an increase in the average foraging rate from 60 ostraco ds per hour to 180 ostracods per hour, which is about a three fold increase, from a lunar intensity of zero to one hundred percent.

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44 We also calculated the average foraging rate in an alternative way, relating it more closely with the percent darkness If we take the values over the course of the month, we can view any particular point describing the rate of ostracods entering the trap as being connected to a distinct value for the amount of darkness at that point. This can be separated by looking at t he time of the night that the sample was collected. If we take every rate point that has the same amount of darkness and average them together, we thus have a point that is the average foraging rate for when the darkness is at a particular level. The resul ts are shown in Figure 3.11, (Again, there is a relatively strong correlation(R=0.7848)), an increase ove the method outlines above. This shows an increase in the average foraging rate from 60 ostracods per hour to 240 ostracods per hour, which is about a four fold increase, from a lunar intensity of zero to one hundred percent. This is different than in Figure 3.10, as it describes the average for each night in relation to the percent darkness for that night in and of itself. Figure 3.11 shows the relation ship over the sample of the entire month. In this way, the value at 100 is the average of every point that was collected at 100 percent darkness, which occurred every night. Therefore, this represents a plot that averages the rates in relation to the darkn ess, not when they were sampled.

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45 Figure 3.11: This figure describes the average foraging rate at any percent darkness. This allows for a more condensed picture, one that hopefully more accurately depicts the changes seen. This was done by taking each v alue at a particular percentage of darkness and averaging them together. Thus, the value at one hundred percent darkness has the most samples. While this removes possibilities of more complex relationships(rise and fall after certain light is hit) it is be tter at determining the direct correlation of the average foraging rate and the percent darkness. Figure 3.12 shows the average foraging rate for each day that was sampled. The error bars show the standard deviation in the samples that were collecte d. This lets us look at the rate averages for every day in comparison to the darkness profile (green curve). The error bars here also show the variance that occurs in the foraging rate data. There are significant variations in the data except day 5. This d ay is set because of rain and trap malfunctions, so it had its data repeated and thus has a very small error bar shown, but actually reflects a smaller sample size.

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46 Figure 3.12: This figure shows the average taken for each day and shows that it can correlate closely to the changes expected and seen in the lunar cycle. The green line depicts the percent light observed on each day. The blue bars represent the average foraging rate for any particular day, and the red error bars r epresent the error in the average. The error then is fairly large, however, it still dictates the overall curve seen each lunar cycle. Day 5 should be noted, as it was set to a consistent value for calculation purposes. 3.3 Modeling Results This sec tion contains the results of the modeling that was performed using the model developed in Section 2. First the model that was developed was tested in a number of ways to see that the expected qualitative results in the theory were reproducible.

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47 Figure 3 .13: This figure shows how the darkness switches the foraging and non foraging populations(and vice versa) in the model described by Eq. ( 11 ) We investigated if dar kness switches the non foraging population to foraging population. Figure 3.13 was created by solving Eq. ( 11 ) numerically when the darkness function f ( t ) = 0 We first fixed a nd so that the steady state of the foraging population F is significantly smaller than the steady state of the non foraging population NF We used these values as an initial starting point, and solved our differential equation model numerically. The steady state appears as the finely dotted horizontal lines in the figure, defined by Eq. ( 17 ) As expected in the figure, since we started our simulation from steady state, the foraging population F and the non foraging population NF are not changing over time. In this simulation we mi micked the darkness by increasing the foraging rate constant The solid lines represent the results of this

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48 simulation, and are the foraging and non foraging populations. We started our simulations from the same initial foraging population of the steady state, but instead set higher at t = 0 (as marked by Darkness ON), till the halfway point in time(as marked by Darkness OFF), and then dropped back to th e value it had at the steady state. The foraging population increases quickly and reach a higher steady state. When the value of was returned to it's original steady state value, this caused a decrease in the foraging population that was not as rapid as the increase. This return to the steady state value of associated to the population that is independent of darkness changes, reaches the same value in the foraging population as at the steady state. Th e steady state may be reached in the environment where the darkness is constant and low, so the population of ostracods then do not come out to forage, such as during the full moon. These changes in are not realistic for the envi ronment however, as will not increase from one value strictly to another, instead it will gradually change, as in the Hill functions. This shows the basic reactions that the ostracod population is expected to have with changes i n light.

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49 Figure 3.14: This graph shows the effects of darkness (by increasing parameter ) on the foraging population over time in the model described by Eq. ( 11 ) wa s multiplied by 1 for the dotted red line at the steady state, then increased by 1, 1.5, 2, and finally 4 folds. Figure 3.10 and 3.11 show in the environment a three to four fold increase in the average foraging rate for changes from zero lunar int ensity to 100 percent lunar intensity. In Figure 3.14, we mimicked such increases in the foraging rate by increasing the parameter values up to four fold of its value at steady state. As the population is in a percentile, it is of note that for larger values the foraging population will increase more quickly, and will settle to higher steady states over time. In this Figure 3.14, the dotted horizontal line represents the value at the steady state, or

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50 Figure 3.15: This plot shows the inter relation between the foraging population in response to darkness produced by solving the model equation given by Eq. ( 11 ) numerically. The darkness profile is set in the corner as an increasing Hill function, described by Eq. ( 19 ) and then a decreasing Hill function, desc ribed by Eq. ( 20 ) As seen in this plot darkness is strongly correlated with the foraging population of ostracods. The quick increase in foraging population as a res ponse to increase in the values which mimics darkness in Figures 3.13 and 3.14 are probably not realistic. In Figures 3.13 and 3.14 we increased to represent the changes in darkness, but in Figure 3.15 we ut ilized the Hill functions as in Eq. ( 19 ) and Eq. ( 20 ) in order to provide a more realist ic change in the lunar intensity when f ( t ) 0 We defined f ( t ) as

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51 f ( t ) = at n b + t n i f t t ab b + t n i f t > t # $ $ % $ $ where a b and n are adjustable positive parameters. We adjusted the values of a b and n in order to create the function that rose evenly to a plateau and then decreased the lunar intensity evenly back to the origin. This was done to create a realistic profile for the changes in lunar intensity over time, and can be seen in the inserted graph in Figure 3.15. The ver tical dotted line in both graphs represents the point t* where the function shifts from an increasing Hill function to a decreasing Hill function. We started our simulation at the steady state when f(t)=0 and ran it with the lunar function over a time period. The result is shown in Figure 3.15. As seen the foraging population follows the lunar intensity over time. As in Figures 3.14 and 3.13 the increase is much more sharp than the decrease. This is due to the value of being increased with the value of f ( t ) during the increasing phase of the function, but in the decreasing phase the model is reliant on as the only draw away from the foraging population. Therefore, the increase follows very quickly, but the decreasing population leaves at a much slower rate. This increase and decrease occur similarly in the environment, as it changes slowly over the course of an hour.

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52 4. Discussion 4.1 Field Data Relevance and Importance The data collected suggest a relatively strong correlation between moonlight and the foraging of ostracods. Gerrish [9] calculated this correlation as R 2 = 0.509, while with our data we calculated R 2 =. 6432. Gerrish [9] also noted that the effect of the moon increases as summer approaches, thus it is somewhat expected that the relationship between the moon and ostracod foraging would increase, so the data in this thesis follows even mild assertions made by Gerrish [9]. While there were only a few days that had issues with inconsistent data in this thesis, these were either removed, or if they were necessary they were simulated, such as the data for day 5, see Figure 3.2. In this case the average tha t was expected was set and then the subsequent hour data was set at this average When sampling for day 5, the trap's bait was unreliable that day as it had lost its effectiveness. This can be noted in the average of the data, as the error bars come out as zero. The simulated data were not given great significance in this study, but it is a piece that was included as it helped in modeling and simulation. The other data that received edits have issues with rain and weather that not only made it impossible for the data to be reliable but impossible for the trap to be set without losing the trap itself. Perhaps the most important part of the data collection was that it occurred. There is not an abundance of data on ostracod foraging, and without the field data the conclusions drawn in this study could likely not have been made. The issues are with the data all being collected around the same time, during the summer, but reasonable estimates can

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53 be made for how the data will change with different time period s. In addition, environmental conditions were ideal, allowing for sampling to occur often. There is a conceivable situation where the environment could have made this thesis impossible. 4.2 Field Methods The methods that were used in this project wer e sufficient to detect the relationship between the moon and ostracod. This is clearly shown in Figure 3.10, as we have a strong enough correlation to investigate the effect of the lunar cycle. While this thesis may not have used the advanced sampling meth ods as used in Gerrish [9], the fact that results were similar suggests that this may be a method that requires less work than others, but gets as much information. 4.3 Modeling Data Relevance and Importance The intent of this thesis was to investiga te the relationship between ostracods and the lunar cycle. This led to the analysis of our data that shows that there is a relatively strong correlation, with the main intention of investigating this relationship through a qualitative model. The conclusion from the information stated is that the complex relationship viewed in the field can be described using a system of two differential equations. The model can show the qualitative relationship between ostracods and the moon, by showing the overall rel ationship that the ostracods have with lunar light. This includes the foraging population having a low foraging rate when the moon is full, but having an abrupt rise in the foraging rate when the moon sets. Included in this is that the ostracod population s foraging rate rises quickly but drops not as fast. Considering the data collected, there are few sampled situations that fall outside this activity. The foraging rate

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54 of course varies during the full moon, and does not stay completely constant as in the model, in that real life scenarios are not as smooth as the model presents. These are minor issues though in the understanding of the model itself. When the moon reaches its maximum, it sets the ostracods to a steady state, and the only change will then co me from the moon setting. While there are obviously changes in the wild that affect foraging other than the moon, it is still observed that the largest change in the rate does occur as the moonlight changes. The study of foraging is not a simple proposal [ 17], which can become considerably more complicated as more and more variables are considered. For example, in Vanlangevelde [19], herbivores are examined to analyze their foraging, and in some situations forage dependent on the quality of the grass itself something that may be ignored in the field initially. Therefore, many factors must be considered. There are many changes in the environment itself that influence nocturnal organisms. This is dependent on the moon, but in this situation describes a very close relationship at night to environmental factors. Much of the ostracod's existence is dependent on hiding during the bright points of the lunar cycle and foraging and mating during the times of darkness or less light. Therefore, this system is inc redibly sensitive to environmental factors that affect it (something that needs to be enhanced in future models), allowing us to consider that it may be applied to other environmental situations with nightly changes that affect foraging. For an example, th e situation where there is excessive moonlight, or man made light, is likely to change the situation for an ostracod so that it is under a high level of predation. While the model suggests that ostracods will stay away from these light sources, experimenta l data was important in determining if this is realistic behavior.

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55 Finally, the model brings up a question of investigating other lunar effects. While there are other studies in this area, the interactions vary greatly between species, suggesting that the night time environment changes significantly for several animals. As Ruiz[16] points out, certain measures of ostracods can tell us about the health of the environmental system the ostracod is a part of. This night time ecology could be studied effici ently by such methods described here, as well as looking at pollution and what changes it makes to the ostracods behavior. 4.4 Future Testing The first future problem that could be investigated from this study is the effect that changes in ostracod po pulation have on the activity of other species. While this depends on knowing exactly what species are affected by the ostracod foraging, this can be narrowed down to a reasonable number of observable organisms. These include crustaceans and small fish, an d although these would be significantly harder to capture they would be observed much more easily. A method of observing the active predators would have to be conceived, but there are already a few apparent options. The most obvious choice would be to obse rve the predators in the environment. From here, the building and use of a method to observe the predators from above is important, especially being able to see what these predators consume. When this information has been collected, a more accurate model w ould result from being able to predict the fish's foraging on the ostracods. The next question that comes up is what the influence of extreme environmental conditions could have on the ostracod population. While this can be perhaps performed in lab wi th the highest level of study, these extreme conditions also exist in the wild. As an

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56 example, there are sources of man made light that exist near the beach that likely force the ostracods into hiding, and traps could be placed to see how this level of man made lighting disturbs the foraging activity of the ostracods. By observing and noting how these changes in light affect ostracod foraging, information could be gained on how light affects other benthic crustaceans, and how to model other creatures that r ely on a strict relationship with the lunar cycle.

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57 Reference [1] Aarnio K. Mattila J. Predation by juvenile Platichthys flesus (L.) on shelled prey species in a bare sand and a drift algae habitat. Hydrobiologia 2000;440(1):347 355. [2] Cohen A.C., K ornicker L.S. Catalog of the Rutidermatidae (Crustacea: Ostracoda ) Deep Sea Research Part B. Oceanographic Literature Review 1987;34(9):781. [3] Cohen A.C. Morin J.G. Two New Bioluminescent Ostracode Genera, Enewton And Photeros (Myodocopida: Cypridini dae), with Three New Species from Jamaica. Journal of Crustacean Biology 30(1):1 55. 2010 [4] Coull B. Long term variability of estuarine meiobenthos: an 11 year study. Marine Ecology Progress Series 1985;24:205 218. [5] Curry B. Delorme D. Ostracode b ased reconstruction from 23 300 to about 20 250 cal yr BP of climate and paleohydrology of a groundwater fed pond near St Louis ,. Journal of Paleolimnology 2003;(1992):2003 2003. [6] Edelstein Keshet, Leah. Mathematical Models in Biology Philade lphia: Society for Industrial and Applied Mathematics, 2005. [7] Felder D.L. A Report of the Ostracode Vargula harveyi in the Southern Bahamas and Its Implication in Luminescence of a Ghost Crab, Ocypode quadrata (Fabricius, 1787). Crustaceana 1982;42(2): 222 224. [8] Gerrish G. Life cycle of a bioluminescent marine ostracode, Vargula annecohenae (Myodocopida: Cypridinidae). Journal of Crustacean Biology 2008;28(4):669 674. [9] Gerrish G. Morin J.G. Rivers T.J. Patrawala Z. Darkness as an ecologic al resource: the role of light in partitioning the nocturnal niche. Oecologia 2009;160(3):525 36. [10] MATLAB The Language Of Technical Computing: www.mathworks.com [11] Moonrise, Moonset and Moonphase for Honduras Tegucigalpa July 2010."Timeandda te.com. Web. 01 Apr. 2011. . [12] Mourguiart P. Holocene palaeohydrology of Lake Titicaca estimated from an ostracod based transfer function. Palaeogeograp hy, Palaeoclimatology, Palaeoecology 1998;143(1 3):51 72.

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58 [13] Nakajima Y. Kobayashi K. Yamagishi K. Enomoto T. Ohmiya Y. cDNA cloning and characterization of a secreted luciferase from the luminous Japanese ostracod, Cypridina noctiluca Bioscien ce, biotechnology, and biochemistry 2004;68(3):565 70. [14] Oakley, Todd. "Ostra blog 7 Trapping Ostracods." Evolutionary Novelties. Web. 01 Apr. 2011. . [15] Rivers T.J., Morin J.G. Complex sexual courtship displays by luminescent male marine ostracods. The Journal of experimental biology 2008;211(Pt 14):2252 62. [16] Ruiz, F. Abad M. Bodergat A.M. Carbonel P. Rodr’guez L‡zaro J. and Yasuhara M. Marine and brackish water ostracods as sentinels of anthropogenic impacts E arth Science Reviews v. 72 no. 1/2 (September 2005) p. 89 111 [17] Stillman R. Goss Custard J.D. Caldow R.W.G. Modelling Interference from Basic Foragin g Behaviour. The Journal of Animal Ecology 1997;66(5):692. [18] Timms B.V. A study of Lake Wyara, an episodically filled saline lake in southwest Queensland, Australia. International Journal of Salt Lake Research 1998;158(2):227 132. [19] Vanlangeveld e F. Drescher M. Heitkonig I. Prins H. Instantaneous intake rate of herbivores as function of forage quality and mass: Effects on facilitative and competitive interactions. Ecological Modelling 2008;213(3 4):273 284.

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59 Appendix Appendix A: Field d ata collected at Cayos Cochinos, Honduras between July 15, 2010 and July 28, 2010 Day 1 Date: 7 15 2010 Hour Volume (ml) Number 18:00 0 0 19:30 0.25 60 21:00 0.1 35 22:00 0.03 20 Day 2 Date: 7 16 2010 Hour Volume (ml) Number 18:00 0 0 19:00 0.07 27 20:00 0.12 102 21:00 0.08 32 22:00 0.1 37 23:00 0.05 35 Day 3 Date: 7 17 2010 Hour Volume (ml) Number 19:00 0.03 61 20:00 0.01 12 21:00 0.04 56 22:00 0.03 32 23:00 0.07 52 0:00 0.04 27

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60 Day 4 Date: 7 18 2010 Hour Volume (ml) Number 19:00 0.02 34 20:00 0.03 37 21:00 0.02 22 22:00 0.02 29 23:00 0.01 11 0:00 0.05 30 1:00 0.01 4 Day 5 Date: 7 19 2010 Hour Volume (ml) Number 19:00 0.05 40 20:00 0.01 1 21:00 0.01 4 22:00 0.01 2 23:00 0.01 1 0:00 0.01 3 1:00 0.01 4 2:00 0.01 3 Day 6 Date:7 20 2010 Hour Volume (ml) Number 19:00 0.05 65 20:00 21:00 0.01 11 22:00 0.01 12 23:00 0.02 28 0:00 0.07 48 1:00 0.06 42 2:00 0.12 120 3:00 0.05 40

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61 Day 7 Date:7 21 2010 Hour Volume (ml) Number 19:00 0.05 40 20:00 0.01 38 21:00 0.01 10 22:00 0.01 20 23:00 0.01 15 0:00 0.01 21 1:00 0.01 22 2:00 0.02 35 3:00 0.03 10 Day 8 Date:7 22 2010 Hour Volume (ml) Number 19:00 0.03 20 20:00 0.01 2 21:00 0.02 26 22:00 0.02 22 23:00 0.01 17 0:00 1:00 0.05 59 2:00 0.08 92 3:00 4:00 0.02 30 Day 9 Date:7 23 2010 Hour Volume (ml) Number 19:00 20:00 0.02 17 21:00 0.01 13 22:00 0.01 7 23:00 0.01 4 0:00 0.05 62 1:00 0.01 12 2:00 0.02 16 3:00 0.01 12 4:00 0.08 26 5:00 0 0

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62 Day 10 Date:7 24 2010 Hour Volume (ml) Number 19:00 0.01 4 20:00 0 0 21:00 0.01 3 22:00 0 0 23:00 0.01 5 0:00 0 1 1:00 0.01 10 2:00 0.01 5 3:00 0.01 5 4:00 0.02 13 Day 11 Date:7 25 2010 Hour Volume (ml) Number 19:00 0.02 21 20:00 0.01 10 21:00 0.01 12 22:00 0.02 18 23:00 0.01 5 0:00 0.01 7 1:00 0 1 2:00 0.01 5 Day 12 Date:7 26 2010 Hour Volume (ml) Number 19:00 0.25 191 20:00 0.02 27 21:00 0.03 30 22:00 0.06 44 23:00 0.03 32 0:00 0.02 33 1:00 0.01 2 2:00 0.01 2 Day 13 D ate:7 27 2010 Hour Volume (ml) Number 19:00 0.2 75 20:00 0.02 15 21:00 0.01 12 22:00 0.01 10

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63 Day 14 Date:7 28 2010 Hour Volume (ml) Number 19:00 0.01 10 20:00 0.03 41 21:00 0.02 25 22:00 0.01 10 23:00 0.01 12

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64 Appendix B: Matlab code th at estimates the parameters in the Hill function for lunar data %% Matlab code that uses lsqcurvefit to estimate %% the parameters in the hill function using lunar data. function ParameterEsitimation clc clf clear all %%Input: Lunar intensity and time and the funtion to fit %Outputs: Estimated parameters for the function to match the data %% ydata is lunat intensity data %% xdata is time ydata = 22*[1 1 1 1 0 0]; xdata = [0 1 2 3 4 5]; %% This line plots the data plot (xdata, ydata) pause %% Initial estimate for the parameters Xo = [10 6]; %%Lower and upper bound for the parameters LB = [0 0]; UB = [10 20]; %% This line calls Matlab's built in function lsqcurvefit [X,Resnorm]=lsqcurvefit(fcn, Xo, xdata, ydata, LB,UB,[],ydata) end %% Function that feeds into lsqcurvefit function f = fcn(Xo,xdata,ydata) Km = Xo(1); n = Xo(2);

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65 %% Function that we are fitting data to f = 22*Km.^n./(Km.^n+xdata.^n); plot(xdata,ydata, 'g' ,xdata,f, 'bo' ) pause(.1) end

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