Stochastic Modeling of Orb-Web Capture Mechanics Supports The

STOCHASTIC MODELING OF ORB-WEB CAPTURE MECHANICS SUPPORTS

THE IMPORTANCE OF RARE LARGE PREY FOR SPIDER FORAGING SUCCESS

AND SUGGESTS HOW WEBS SAMPLE AVAILABLE BIOMASS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Samuel C. Evans

December, 2013

STOCHASTIC MODELING OF ORB-WEB CAPTURE MECHANICS SUPPORTS

THE IMPORTANCE OF RARE LARGE PREY FOR SPIDER FORAGING SUCCESS

AND SUGGESTS HOW WEBS SAMPLE AVAILABLE BIOMASS

Samuel C. Evans

Thesis

Approved: Accepted:

______Advisor Department Chair Dr. Todd A. Blackledge Dr. Monte E. Turner

______Committee Member Dean of the College Dr. Randall J. Mitchell Dr. Chand Midha

______Committee Member Dean of the Graduate School Dr. Steven C. Weeks Dr. George R. Newkome

______Date

ABSTRACT

Strong selective pressures can be exerted by events that occur extremely rarely and unpredictably during an organism's lifetime. The importance of such rare events may elude detection if the fitness consequences are not immediately observable, such as in the form of missed foraging opportunities. For orb-weaving spiders, fitness may depend almost exclusively on securing one or a few large, rarely-encountered, difficult-to-capture prey. Here, we present a stochastic individual-based model simulating foraging, growth, and survival of various-sized spiders in environments varying in distribution of biomass among prey sizes. We use this model to assess the degree to which foraging success is determined by the outcome of a small subset of foraging opportunities, and ascertain the architectural and biomechanical properties most crucial to deciding the outcomes of these rare events. Although our deterministic model suggests spiders should, on average, gain the most biomass from small prey sizes, spiders in stochastic simulations grew the most by capturing a single large and difficult-to-capture prey comprising the majority of their diets. The mechanics involved in stopping and retaining flying prey were more important in determining foraging success compared to those involved in encountering and contacting prey. Spiders lost the raw majority of biomass they encountered by failing to stop prey. However, prey retention exhibited the highest rate of biomass loss—spiders lost over 90% of successfully stopped biomass by failing to retain prey, but failed to stop only 40-80% of prey biomass their webs successfully contacted. Our results support the rare large prey hypothesis of Venner and Casas (2005), and reinforce the hypothesis that iii orb webs are pervasively selected for their potential to arrest large amounts of energy.

However, certain factors such as prey availability and the biomechanics of prey retention in webs warrant further investigation, as these may be crucial to the plausibility of alternative foraging strategies.

DEDICATION

To the scores of people who, as I struggled to rein in the thoughts culminating in this thesis, supplied the scaffolding around which my character was built in ways I never expected.

ACKNOWLEDGEMENTS

Steve Weeks and Randy Mitchell provided insight and support as SCE's thesis committee members. Michael Barton, Jeremy Prokop, Zhong-Hui Duan, and Charles van

Tilberg contributed computational resources. Mohammad Marhabaie, Bor-Kai Hsiung,

Gaurav Amarpuri, Thomas Beatman, Shagun Sharma, and Andrew Shall contributed helpful comments on the manuscript. Rafael Maia provided insightful discussions of the model and particularly valuable support with R syntax.

TABLE OF CONTENTS Page

LIST OF FIGURES ------viii

CHAPTER

I. INTRODUCTION ------1

II. MATERIALS AND METHODS ------9

The model ------9

Prey availability ------9

The spiders ------13

Prey capture process ------14

Prey-web contact ------15

Prey stopping ------16

Prey retention ------18

Biomass assimilation ------19

Metabolism and survival ------19

Deterministic model ------21

Simulation and analyses ------22

III. RESULTS ------24

IV. DISCUSSION ------33

REFERENCES ------39

APPENDIX ------44 vii

LIST OF FIGURES Figure Page

1 Diagram of a typical orb web ...... 5

2 Biomass and abundance distributions of prey simulated in the model ...... 10

3 Flowcharts representing (a) a single spider in the model, and (b) the daily prey capture process for that spider ...... 12

4 Deterministic distributions of percent of total captured biomass acquired from each prey size ...... 25

5 Histograms of spider growth in the stochastic model ...... 26

6 Relationships between spider growth and single largest prey captured ...... 27

7 Scatterplots showing positive relationships between spider growth and percent of total captured biomass from the single largest prey captured ...... 28

8 Relationships between spider growth and the size difference between the largest and second-largest individual prey items captured...... 29

9 Fate of total biomass encountered by spiders ...... 30

10 Proportion of prey biomass entering each capture stage that was lost vs. sustained to the next stage ...... 31

11 Relationships between total biomass encountered and total biomass stopped by webs, for all surviving spiders ...... 32

viii

CHAPTER I

INTRODUCTION

Much of evolutionary ecology research seeks to describe the selective forces that influence how organisms acquire resources to maximize fitness (Stearns 1977; Houston et al. 1988). Such traits might be shaped primarily via events that occur extremely rarely and unpredictably during an organism's lifetime. Aberrant climatic events can drastically alter distributions of phenotypes within entire populations (Brown and Brown 2011).

Recent discoveries of novel genera subsisting on nutrient bonanzas from rarely- encountered deep-sea whale and wood falls indirectly suggests that such organisms can only reproduce on a detectable scale in such rare instances (Amon et al. 2013; Bienhold et al. 2013). Moreover, organisms possessing phenotypes that are well beyond adequate for surviving and foraging in “average” scenarios are of curious interest because the likely high energetic costs of such phenotypes would presumably render them inefficient and have negative relative fitness consequences, yet these persist in some lineages

(Gaines and Denny 1993; Foelix 2011). However, a rare event that determines individual foraging success may elude detection as a significant selective force because its consequences for the forager may not be immediately observable.

Spiders that build viscid orb webs (Araneae: Orbiculariae: Araneoidea) are particularly useful model organisms for testing hypotheses of natural selection shaping traits via rare, cryptic events. These spiders represent a diverse, worldwide-distributed

1 clade of several thousand species ranging from 1mg to over 1g in adult body size

(Blackledge et al. 2009). As sit-and-wait foragers, orb-weaving spiders provide a complete blueprint of their foraging strategy in the characteristics of their orb webs.

Moreover, these spiders rebuild their webs daily, affording the opportunity to alter foraging effort in response to exogenous factors (Blackledge et al. 2011). Therefore, more complete and accurate measurements can be made of these organisms' life history traits and their underlying mechanics. Indeed, much data exists regarding orb-weaver energetic investment in foraging (Peakall and Witt 1976; Sherman 1994), foraging potential (Venner and Casas 2005; Blackledge and Zevenbergen 2006; Cranford et al.

2012), and changes in these with respect to ontogeny and environment (Nakata and

Ushimaru 1999; Nakata 2008; Abrenica-Adamat et al. 2009; Sensenig et al. 2011).

Orb-weaving spiders are typically characterized as generalist predators whose fitness depends on capturing a broad range of prey, from numerous smaller to few larger individuals. However, recent evidence suggests orb-weaving spider fitness may depend primarily on securing one or a few large, rarely-encountered, difficult-to-capture prey— an event that may never occur in an individual spider’s lifetime. Venner & Casas (2005) used simulations based on prey capture data from Zygiella x-notata in the field to show that, while availability of small prey supplemented spider survival rate, only spiders that captured large prey were able to reproduce. In a meta-analysis across 31 orb-weaver species, Blackledge (2011) showed that prey greater than 2/3 the length of the spider accounted for 17% of the number of prey captured but 85% of the total estimated biomass assimilated by spiders. The impressive mechanical properties of spider webs might therefore be a product of strong natural selection for capturing rare, large prey.

Successful acquisition of large prey by spiders is very rarely observed

(Blackledge 2011). Distributions of insect prey in environments are generally highly skewed towards numerous small and few large individuals (Rogers et al. 1977; Schoener

1980; Chacon and Eberhard 1980; Finlay et al. 2006). In addition to being less abundant than smaller prey, larger prey are heavier and typically fly much faster (Lewis and Taylor

1967), and so have exponentially greater kinetic energy. As a result, larger prey are expected to be much more rarely encountered and exponentially more difficult to capture.

Therefore, it may seem counterintuitive to think that orb-weaving spiders depend disproportionately on capturing these rare, large, highly energetic prey (Venner and

Casas 2005; Blackledge 2011). While each individual large prey represents a much higher biomass payout than does a single smaller prey, the high abundance and low kinetic energy of small prey would seem to be an easier route to maximizing biomass intake. Therefore, proper evaluation of the rare large prey hypothesis entails understanding the mechanisms that might constrain adaptive foraging strategies to targeting specific prey types. This in turn requires understanding the trade-offs among various web properties such that no single web can maximize performance in capturing prey of all sizes (Blackledge 2012).

For an orb-weaving spider to forage successfully, its web must encounter, contact, stop, and retain airborne prey. Orb webs are generally planar elliptical structures composed of a frame and radii of tough, stiff dragline silk, and a spiral of thinner, more pliable capture silk coated in droplets of viscid aggregate glue (Figure 1). Apart from microhabitat conditions, the planar area of the web largely determines its prey encounter rate (Foelix 2011). The distance between consecutive capture spirals, termed “mesh

3 width”, influences the likelihood of a prey item contacting the web as opposed to flying between capture spirals untouched (Eberhard 1986). Upon contact with a prey, the radii of the web do nearly all of the work to dissipate its kinetic energy (Sensenig et al. 2010), while the capture threads deform to cradle the prey and allow the aggregate glue to adhere to and retain it (Swanson et al. 2007) long enough to be immobilized by the spider. Therefore, different-sized prey are liable to escape capture differently: smaller prey by flying through the web untouched; larger prey by breaking through the web or struggling free from its viscid capture spiral.

Because altering web properties to increase encounter and contact rates generally comes at the cost of poorer stopping and retention performance (Blackledge 2012), there is a general trade-off in the ability to capture numerous small versus few large prey.

During web construction, the spider completely exhausts its supply of capture silk

(flagelliform thread and aggregate glue) but not its supply of radial silk (Eberhard 1988).

In theory, it could build a larger web that increases encounter rate and maintains or even increases stopping potential, but at the cost of diminished contact and/or retention performance (Blackledge and Eliason 2007). To avoid reduced contact rates with smaller prey, the spider could spin thinner capture threads. However, the capture spiral must be strong enough to withstand loads imposed by wind without extensive entanglement, and must exceed the breaking energy of its glue droplets so as to avoid extensive damage

Figure 1: Diagram of a typical orb web, showing the frame, hub and radii composed of stiff dragline silk, and capture area composed of a spiral of pliable flagelliform silk coated in regularly-spaced droplets of viscid aggregate glue. The capture spiral spacing, termed “mesh width,” influences the likelihood of a prey item contacting the web as opposed to flying through untouched. The radii do nearly all of the work to dissipate prey kinetic energy, while the capture threads and glue act to retain it.

5 incurred by escaping insects (Liao et al. 2009; Agnarsson and Blackledge 2009). Even if contact rate can be maintained, the web would still perform more poorly in retaining larger prey, as the thinner capture threads would still be weaker and the aggregate glue spread over a larger area. Furthermore, including more radial threads to maintain stopping potential would require additional web-building time, during which the spider expends more energy, is exposed to predators, and cannot forage. The spider would then need to navigate a larger area to secure stopped prey, which on average increases the time to prey immobilization, thereby affording prey more time to escape (Nakata and

Zschokke 2010; Zschokke and Nakata 2010). In all, encounter rate would have to be high enough to offset all these costs, and the spider would be restricted to recouping these costs primarily via capturing small prey.

Given these trade-offs inherent in the functional morphology of orb webs, the seemingly counterintuitive foraging strategy of targeting rare large prey could be plausible for multiple reasons. First, a spider’s foraging environment might not contain enough biomass packaged in small prey for it to survive and reproduce, even if small prey are numerically dominant. Venner & Casas (2005) extrapolated from their field data on prey capture by Zygiella x-notata that the largest prey (>10mm in length) accounted for only 3% of the number of prey captured, but provided an average of 60 times more biomass to spiders than did all other captured prey. Therefore, in order for a spider to subsist on a foraging strategy targeting small prey, small prey would have to comprise the vast majority of biomass available in the environment. Second, a web capable of capturing a sufficient amount of biomass via small prey may not be energetically tractable or even physically possible to build, for the reasons outlined in the previous

6 paragraph. Even if small prey represent the majority of available biomass in the environment and are therefore more numerically abundant than larger prey by multiple orders of magnitude, maximizing biomass intake via capturing small prey may still require a large yet sieve-like web with very tight capture spiral spacing, such that nearly all small prey encountered are captured. Both scenarios are unlikely. The Energetic

Equivalence Rule suggests, albeit indirectly, that any given size class in an environment contains a relatively equal amount of biomass (White et al. 2007) but the degree to which local environments vary from this standard has long been debated (Marquet et al. 1995).

Furthermore, as the web becomes damaged and saturated with small prey over the course of a foraging bout (Shear 1986), subsequently-encountered prey will be lost such that webs are unlikely to sustain near-100% capture rates of encountered small prey. Finally, even if the integrity of a dense mesh width can be sustained, such a web would be more visible to flying prey, which may actively avoid the web (Craig 1986).

The study we present here aims to more extensively evaluate the importance of rare events for foraging success, via an individual-based model synthesizing ecological and biomechanical data to simulate spider foraging, growth, and survival. We simulate foraging by spiders of initial sizes representative of the range found in nature, and among environments varying in the distribution of available biomass along a gradient of small- to-large prey size classes. Our approach is novel in that it incorporates assumptions regarding the biomechanics that govern prey capture, and provides a framework for further, more detailed examination of the mechanisms determining foraging success,

7 which can be used to address a variety of questions beyond this study’s scope of evaluating the rare large prey hypothesis.

CHAPTER II

MATERIALS AND METHODS

The model

We developed a model simulating foraging by orb-weaving spiders to evaluate the rare large prey hypothesis, by examining the relationship between spider foraging success (defined as amount of mass gained) and the relative contributions of various- sized prey to the total amount of captured biomass. Each iteration consists of a spider that forages and rebuilds its web daily for up to 30 days, in an environment of constant available biomass density packaged among prey of different sizes (Figure 2). Foraging success is quantified as the change in the spider's mass over this duration. Figure 3 gives flowcharts outlining the process of the model.

Prey availability

We modeled three prey environments in which available biomass was packaged disproportionately in smaller prey (“small-biased”), evenly across prey sizes (“even”), or disproportionately in larger prey (“large-biased”) (Figure 2a). As we suggest in the introduction, distribution of available biomass among prey sizes might greatly influence the balance of trade-offs in web function, as different prey sizes are most likely to escape

Figure 2: Biomass and abundance distributions of prey simulated in the model. (a) shows the percent of total available biomass contained in each prey size class, which determines prey numerical density. (b) shows the numerical percent abundance of each prey size class, which serve as the probability distributions of prey encounter in a given event.

10 at different stages during the prey capture process. Prey found in orb webs range from tiny arthropods 1mm or less in length, to large insects and even small vertebrates

(Blackledge 2011; Brooks 2012). Therefore, prey in the model occupy size classes from

1 through 100 mm in length, in 1 mm increments, which all but encompasses the full size range of known insect species (Finlay et al. 2006). For convenience, we characterize prey

“size” using length because this is the data most commonly given in the ecological literature. We estimate the wet mass mp of prey, necessary to calculate kinetic energy when flying, via a length-dry mass regression of tropical and temperate insects (Schoener

1980),

(1)

in which lp is prey length, and wet mass is obtained by multiplying by 4, a conversion based on percent body water content measurements of 16 orders of temperate flying insects (Studier and Sevick 1992).

All three prey environments are identical in total biomass density B, but differ in how this biomass is packaged among prey of various sizes. The fractional number of prey in each size class is

(2)

where is the total available biomass contained in prey of length lp and is the wet mass of a single prey of that length (Figure 2a). From this, the proportional abundance of

each prey size in the environment, , is

(3) ∑

Equation (3) produces the probability distribution of encountering a given-sized prey

(Figure 2b).

Figure 3: Flowcharts (a) representing a single iteration (spider) in the model, and (b) the daily prey capture process for that spider. A spider builds a web, then encounters a certain number of prey, for each of which contact, stopping, and retention is decided sequentially. The spider’s mass changes based on capture success/failure and a metabolic cost. The spider rebuilds its web daily based on its current mass, and must remain at or above a minimum mass in order to survive. The iteration terminates when the spider starves to death or forages for 30 days.

The spiders

Because “large” and “small” are relative descriptors whose meanings depend on the size of the predator relative to its prey, we modeled three different size classes of spiders foraging in each prey environment. A spider begins at an initial mass of 50, 400, or 750 mg, chosen to reflect a representative range of adult spider sizes, which vary from less than 1 mg in the Mysmenidae to upwards of 1 g in the large Nephila spp. (Foelix

2011; Blackledge et al. 2011). Regressions from a dataset of 22 representative orb- weaver species reveal strong positive relationships between spider mass and silk and web characteristics (Sensenig et al. 2010), and are thus used to calculate these characteristics

2 in the model. Spider mass ms scales with web area (cm )

(4) silk volume (the volume of silk used in the web, mm3)

(5) thread thickness (thickness of silk threads, defined as thread cross-sectional area,

µm2)

(6) and silk toughness (the amount of energy a silk fiber can absorb before failure, µJ)

(7)

From these properties, the geometric properties of the web can be calculated. Capture thread length (the total length of the capture spiral, mm) is

, (8)

13 where it is assumed that the capture spiral comprises 50% of the silk in the web. Given the web area A and the capture thread length CTL, mesh width (the distance between consecutive capture spirals, mm) is

√ . (9) ( √ ) ( )

Here, the capture area is assumed to be 90% of the web’s total area, as the hub does not contain sticky capture threads and therefore does not directly facilitate prey capture

(Herberstein and Tso 2000).

Prey capture process

We model prey capture as a discrete four-step process. The spider encounters a number of prey based on the size of its web and the density of prey in the environment.

For each encountered prey item, web geometry and prey size determine whether the prey item contacts the web or escapes by flying through untouched. Then, the prey’s kinetic energy is compared to the web’s “stopping potential” (i.e. the estimated amount of kinetic energy it can dissipate before failing) to determine whether the prey item is stopped by the web or escapes by breaking through. Lastly, the prey item’s size and kinetic energy and the web’s mesh width determine whether the prey item is captured. For each prey encounter “event,” the spider gains or loses mass based on whether it captures the prey item, its ability to process a captured prey item, and its metabolic requirements.

The number of prey the spider encounters in a day, N, is

∑ (10)

the product of its web area and the numerical density of prey in the environment

(denominator of equation 2). While it is generally accepted that larger webs encounter more prey (Shear 1986; Foelix 2011), some studies suggest characteristics other than planar web area, such as ultraviolet reflectance and position relative to substrate, can influence prey encounter rates (Eberhard 2013; Peng et al. 2013). However, given that such effects have yet to be quantified consistently, prey encounter rate in the model is assumed to scale linearly with web area.

For each encounter, the size of the prey item is selected at random according to the size-abundance distribution of prey in the environment (Figure 2b). As per the definition by Chacon and Eberhard (1980), capture success requires the web to contact, stop, and retain the prey item. For each capture stage, the spider has a certain probability of success, which depends on relevant characteristics of web and prey mechanics.

Success/failure is determined sequentially for each capture stage by comparing the probability to a randomly-generated number between 0 and 1, to decide whether the spider acquires the prey (Figure 3b).

Prey-web contact

Prey that encounter the web can either contact the web or escape by flying through gaps in the capture spiral. Here, prey are assumed to be spheres with diameters equal to their "lengths", as relationships between length and shape vary widely among 15 taxa of flying prey (Sample et al. 1993; Eberhard 2013). Ortega-Jimenez and Dudley

(2013) show that the electrostatic charge difference between flying prey and webs draws the capture spiral toward incoming prey, which could facilitate contact with prey smaller than the web’s mesh width. However, how this affects the relationship between contact rate and ratio of prey size to mesh width has yet to be investigated empirically, and is likely negligible for all but the few smallest prey sizes. Therefore, contact probability

pcontact is

l p  p , (11) contact MW

where is prey diameter (“length”) and MW is the mesh width of the web.

Prey stopping

We estimated the probability of the web stopping each insect as the ratio of the total kinetic energy that could be dissipated by the web compared to the flight energy of the prey. We estimated the average speed of each insect from a size vs. flight speed regression by Lewis and Taylor (1967), so that prey kinetic energy was

( ) (12)

It is worth noting that the data of Lewis and Taylor (1967) reflect maximal flight speed, whereas prey may often be flying at a fraction of their maximum speed and therefore contact webs at a fraction of their maximum kinetic energy. We ignore this for simplicity, but it is nonetheless potentially important in making exceptionally large insects

“stoppable” by orders of magnitude, in rare instances when they are moving slowly through the air.

We then estimate the ability of a web to dissipate prey energy based on the material properties (volume and toughness) of the silk. High-speed video analysis of web-impacts and modeling both show that approximately only 1/4 of the total area web does work to dissipate this energy (Blackledge et al. 2011; Cranford et al. 2012). The stopping probability pstop is therefore

SV  ST pstop  . (13) 4KE p where silk volume SV and silk toughness ST give the energy dissipation potential of the entire web.

Temperature and air turbulence can affect insect flight (Dudley 2002), and it has been suggested that web position relative to obstacles may facilitate capture by slowing down large prey (Eberhard 2013). Therefore we maintain this ratio as a reliable proxy for stopping probability, rather than deciding all prey with kinetic energy greater than 1/4 the web’s stopping potential as escaping. An alternative estimate of stopping probability could involve calculating KEp using a randomly-determined fraction of maximum speed, though that would require further assumptions such as minimum flight speed, and would add complexity to the model ancillary to the goal of this study.

Prey retention

Stopped prey often escape from webs by detaching from the viscous glue droplets on capture threads (Rypstra 1982; Nentwig 1982; Blackledge and Zevenbergen 2006).

The mechanics of prey retention are still poorly understood, and likely consist of a complex interplay among prey morphology, spider behavior, and microclimate conditions

(e.g. dust, humidity) that affect the adhesive potential of the glue (Opell and Schwend

2009). That said, the web’s potential to stick to prey is determined both by glue adhesion per se and by energy absorption by extensible capture threads (Blackledge 2012). Mesh width varies proportionally to the number of capture threads attached to a prey item via glue droplets, and is likely very important to successful prey retention (Blackledge and

Eliason 2007). However, as discussed earlier in this study, it is not yet well understood how or even if a spider might be able to manipulate this geometrical feature to its advantage by spinning thinner capture threads to build a web with narrower mesh width.

Furthermore, while Sensenig et al. (2010) document a significant relationship between spider mass and adhesive potential per web area, spider mass explains only a small fraction of variation in stickiness per capture area (adjusted R2=0.189), and so is excluded

from our model as a predictor. Therefore, retention probability pretention is calculated simply by comparing the density of adhesive capture threads to the prey’s size and kinetic energy,

l p 10 pretention   1 , (14) MW KE p SAp where the first term gives the fractional number of threads contacting the prey, and the second term gives an approximation of the kinetic energy applied to the web by the 18 struggling prey. Here, SAp is the fraction of the prey’s surface area that is contacting the web, calculated as ¼ of the surface area of a sphere with diameter l.

Biomass assimilation

For every prey captured, the spider gains a portion of that prey item’s biomass according to an assimilation efficiency assumption. Venner and Casas (2005), in a controlled experiment, found that Zygiella x-notata orb-weavers convert an average of

78% of their prey’s dry biomass to their own body mass. Because our model uses prey wet biomass, assumed to be four times estimated prey dry biomass, we make the simplifying assumption that spiders assimilate 20% of prey wet biomass, which is roughly equal to one-fourth the rate found by Venner and Casas (2005) upon measuring prey dry biomass. Indeed, spiders obtain some mass from prey water content, though how this water is used and lost is not well understood and likely depends on a host of microclimate conditions.

Metabolism and survival

Because spiders cannot survive indefinitely without capturing prey, we modeled the ability for spiders to starve prior to the culmination of the simulation. The spider continually loses mass according to a metabolic cost, and must maintain a minimum mass

in order to avoid starvation. For every prey encounter event, the metabolic cost mmet , calculated as mass lost due to metabolism for that event, is

3 m 4 m  s , (15) met 30  N

where ms is the spider’s current mass and the denominator is the product of the duration of the model (number of days) and the number of prey to be encountered N during the current day. Metabolic cost is assumed to follow Kleiber's law, which posits that metabolic rate is a function of current body mass raised to ¾ (Kleiber 1947; Brown et al.

2004). Indeed, larger orb-weaving spiders have proportionally lower metabolic rates

(Greenstone & Bennett 1980; Shear 1986). However, no study to date has comparatively quantified the daily mass lost due to metabolic cost, including during web construction, across orb-weaving spiders (but see Peakall and Witt 1976). Therefore, after each event, the spider’s mass is updated as

ms  ms 0.2mp  mmet , (16) whereupon capturing a prey item, the spider gains mass equal to 20% of the prey item’s  wet mass. Otherwise, the spider simply loses mass equal to mmet.

In order to approach a realistic probability of starvation, spider starving threshold begins at 80% of the spider’s initial mass, and increases linearly to reflect a growth requirement such that, after 30 days, the spider must be of at least 120% its original mass to have survived. While no direct evidence suggests spiders must maintain an ever- increasing minimum mass in order to survive within an intermolt period, there is evidence to suggest they have accelerating dietary requirements throughout their ontogeny (Higgins and Goodnight 2010) and are limited by specific nutrients (Blamires

2010). However, the precise nutritional requirements of orb-weaving spiders are poorly

understood and likely vary among species. Therefore, we justify the use of this increasing

threshold in place of a precise understanding of nutritional requirements.

The day ends when the spider has encountered N prey (as per equation 10), and a

new day begins with the spider building a new web based on its current mass (equations

4-9), provided it has not starved. Some evidence suggests spiders may reduce web

investment following large gains in mass (Sherman 1994), while others suggest web size

closely tracks change in body mass (Blackledge and Zevenbergen 2007). Here, we invoke

the simplifying assumption that spiders build a web with characteristics determined by

their mass at the beginning of each day.

Deterministic model

A deterministic version of the model is required to evaluate the importance of

stochasticity, as capturing rare large prey in a given event is highly improbable.

Deterministically, the biomass obtained from each prey size is

D  0.2 p  p  p  p  m , (17) l p lp contact stop retain p

where D is the product of that prey’s mass, proportional abundance, contact, stopping, l p  and retention probabilities, and spider assimilation efficiency. From this, the expected

 biomass payout from each prey size is determined for 30 days of foraging. The

deterministic model excludes the assumption of starvation below a minimum mass

threshold.

Simulation and analysis

For each combination of initial spider size and prey environment in the stochastic model, 100 000 individuals were simulated, which was the minimum number of replicates necessary for all treatment combinations to produce an adequate number of surviving spiders (at least roughly 100) for analysis. For each spider, we recorded the following data: initial and final mass, number of days survived, and, for each prey size class, the number of prey encountered, contacted, stopped, and retained. These data allowed us to calculate the total amount of biomass captured as well as lost due to failure at each stage of the prey capture process.

Using spider growth (change in spider mass) as our measure of foraging success, we employed a variety of exploratory analyses to interpret our data. To evaluate the importance of rare large prey in the stochastic model, we examined the relationship between foraging success and i) size of the single largest prey item the spider captured, ii) percentage of total captured biomass consisting of that single largest prey, and iii) difference in size between the largest and second-largest individual prey captured. To understand how webs capture and lose available biomass, we examined the proportion of total encountered biomass captured versus lost at each capture stage. We also evaluated the proportion of biomass entering each capture stage (contact, stopping, retention) that was lost versus sustained to the subsequent stage. Finally, we used simple linear regressions to evaluate the relationship between amount of biomass encountered and amount of biomass stopped, to understand whether spider foraging success depended on encountering more biomass or simply succeeding in capturing enough of what was encountered. We used these methods to draw conclusions regarding the relative

22 importance of different prey sizes, and the limiting stages and therefore mechanics of successful prey capture.

All simulations and analyses were conducted in R versions 3.0 and 3.0.1, using the following open-source packages: compiler, multicore, plyr, reshape2, ggplot2, gridExtra, data.table, and sqldf.

CHAPTER III

RESULTS

The deterministic model suggests spiders derive the most biomass from small prey under all scenarios (Figure 4). However, according to the deterministic model, medium-sized spiders foraging in large-biased prey environments and large spiders in all environments do not consume enough biomass to survive. In all scenarios of the stochastic model, over 50% of spiders gained less mass than in the deterministic model, yet a small fraction of spiders were able to gain many times their initial body mass

(Figure 5). There was a negative relationship between initial spider mass and mass gained; small spiders tended to gain the most mass on average, and large spiders the least.

Similarly, spiders in small-biased prey environments gained the most mass, while those in large-biased environments gained the least. However, the frequencies with which spiders gained more than 400mg were remarkably consistent across spider size class.

Small spiders exhibited near-100% survivorship regardless of prey environment

(Figure 5). Prey environment more substantially influenced medium and large spider survival; medium spider survivorship was >90% in small-biased environments, 60% in even environments, but only roughly 10% in large-biased environments. Roughly one- third of large spiders survived in small-biased environments, with less than 5% survivorship in even environments and only 0.15% in large-biased environments.

Figure 4: Deterministic distributions of percent of total captured biomass acquired from each prey size. Biomass captured is calculated as the product of the mass, proportional abundance, and capture probability of that size class, and assimilation efficiency by the spider. Gradients are composed of individual lines representing spiders ranging in size from 1mg (blue) to 1000mg (red) in 1mg increments. Deterministically, spiders of all sizes capture the most biomass from smaller prey.

In all scenarios, spiders gaining the most mass did so by capturing a single large prey item—much larger than any other prey item they captured and accounting for the majority of their total captured biomass. Spider growth was positively correlated with the size of single largest captured prey item (Figure 6), percent of total captured biomass consisting of that single largest prey item (Figure 7), and difference in size between the largest and second-largest prey item they captured (Figure 8). Spiders in the large-biased 25 prey environment exhibited the greatest range of prey capture. Those gaining the most mass did so via a single prey accounting for upwards of 85% of their diet, while some of those starving captured very few prey (Figure 7). The few spiders that acquired more than

50% of their biomass from a single prey item and still failed to survive were large spiders in even and large-biased prey environments.

Figure 5: Histograms of spider growth, faceted by spider size class (rows) and prey distribution (columns). Dashed blue lines represent minimum growth required to survive the duration of the model; spiders to the left of these lines did not survive the full 30 days of simulation. Dashed orange lines represent deterministic growth, (cont’d)

26 which most spiders in stochastic simulations fell short of while a few greatly exceeded.

Insets show the tail end of distributions (x-axes match those of full distributions).

Figure 6: Relationships between spider growth and single largest prey captured, faceted by spider size class (rows) and prey distribution (columns). Boxplots representing each prey size are composed of medians (horizontal lines), lower and upper quartiles (ends of boxes), and 5 and 95 percent quantiles (vertical lines). Grey curves represent the amount of biomass the spider assimilated from its single largest prey. Note that most spiders’ growth depended disproportionately on the single largest prey they captured.

Figure 7: Scatterplots showing positive relationships between spider growth and percent of total captured biomass consisting of the single largest prey captured. Dashed blue lines represent minimum growth required to survive the duration of the model; spiders below these lines did not survive the full 30 days of simulation.

Figure 8: Relationships between spider growth and the size difference between the largest and second-largest individual prey items captured. Boxplots are composed of medians

(horizontal lines), lower and upper quartiles (ends of boxes), and 5 and 95 percent quantiles (vertical lines). Note that spiders foraged most successfully by capturing a single prey item much larger than any other they captured.

All spiders lost the vast majority of their biomass by failing to stop or retain prey.

While the majority of encountered biomass was lost by failing to stop prey (Figure 9), nearly all spiders failed to retain 90% or more of the biomass their webs succeeded in stopping, compared to a much more variable 40% to upwards of 80% of contacted biomass lost due to prey breaking through the web (Figure 10). Biomass lost by failing to

29 stop prey was positively correlated with amount of biomass packaged in large prey, such that spiders in large-biased environments failed to stop the most biomass and the highest proportion of successfully contacted biomass.

Finally, the relationship between amount of biomass encountered and amount of biomass successfully stopped was negligible in all scenarios (Figure 11). Biomass encountered accounted for less than 10% of the variation in biomass stopped.

Figure 9: Fate of total biomass encountered by spiders. Webs either fail to contact, fail to stop, fail to retain, or capture prey. Boxplots are composed of (cont’d)

30 medians (horizontal lines), lower and upper quartiles (ends of boxes), and 5 and 95 percent quantiles (vertical lines). Most biomass is lost when webs fail to stop flying prey, followed by failure to retain stopped prey.

Figure 10: Proportion of prey biomass entering each capture stage that was lost vs. sustained to the next stage. Proportions were calculated based on the amount of biomass entering each stage. Error bars represent middle 95% quantile ranges. While most spiders lost the raw majority of biomass encountered because their webs failed to stop prey, all spiders failed to retain at least 90% of the biomass their webs succeeded in stopping, compared to failing to stop only 40-80% of contacted prey.

Figure 11: Relationships between total biomass encountered and total biomass stopped by webs, for all surviving spiders. Blue lines are simple linear regressions, with associated

R2 values. Although all relationships are significant at p < 0.05, amount of biomass encountered explains very little variation in biomass stopped. This suggests that spiders foraging successfully did not necessarily do so because they encountered more biomass, but because they succeeded in highly improbable events.

CHAPTER IV

DISCUSSION

Spiders forage most successfully by capturing a single large, rare, difficult-to- capture prey item. Stochastic simulations show that spiders never far exceed the average biomass intake from smaller prey (i.e. the amount suggested by the deterministic model), but they do occasionally far exceed the average biomass intake from large prey sizes. The binary nature of capture success/failure present in the stochastic model, coupled with the high payout from large prey, make capturing large prey the only route to maximizing foraging success. Indeed, spiders capturing nearly all the small prey they encountered still did not forage as successfully if they failed to capture any large prey.

Though we observed some differences in foraging success among spider sizes and prey environments, spiders in all scenarios took the same route to maximizing foraging success. Smaller spiders tended to grow more on average than did larger spiders, due to the higher proportion of available prey that are “large” relative to small spiders, and because those prey impact webs with less kinetic energy relative to bigger spiders’ “large prey”. Similarly, average growth was highest in small-biased environments and lowest in large-biased environments for each spider size class. However, growth distribution among the most successful foragers was remarkably similar in all scenarios (Figure 5, insets), with the exception of small spiders in large-biased environments. Finally, the proportion of captured biomass obtained from large prey positively correlates with spider

33 size and the amount of biomass packaged in large prey (Figure 6). This consistent pattern suggests natural selection should pervasively favor both the architectural and material traits of orb webs that best allow spiders to capitalize on improbable prey capture events.

Moreover, selection to capture large prey may be strongest on larger spiders and spiders foraging in environments where large prey are biomass-dominant.

Inferring selective pressures on orb web design requires knowing how prey are encountered and lost at various stages of capture in addition to observing what is successfully captured (Eberhard 2013). The results of our model (simulating prey capture as a four-step process of encounter, contact, stopping, and retention) suggest that stopping and retaining prey limit foraging success much more than encountering or contacting prey. Nearly all spiders lost the majority of prey biomass because their webs failed to stop flying prey (Figure 9). Of the biomass that spiders succeeded in stopping, upwards of 90% was subsequently lost when webs failed to retain prey (Figure 10).

Whereas webs fail to stop only larger prey, prey of nearly all sizes can escape webs at the retention stage. Spiders foraging most successfully were able to stop and retain a large prey item that most other spiders in their size class encountered but failed to stop. The amount of biomass a spider encountered correlated only slightly with the amount of biomass stopped by the spider’s web (Figure 11), suggesting that “being lucky enough” to stop and retain at least a single large prey item was more important than encountering those large prey more frequently. Taken together, these results indicate that foraging success can be most improved through aspects of web architecture and silk performance that enhance stopping and retention of prey.

In our model, retention potential is simplified as a function of the number of sticky threads contacting a prey item (determined by prey size and mesh width of the orb) and the kinetic energy applied to the web by that struggling prey. Capture spiral spacing directly influences a web’s ability to adhere to prey (Blackledge and Zevenbergen 2006;

Blackledge and Eliason 2007). But in reality, microclimate conditions, web geometry, glue droplet spacing and molecular structure, prey surface morphology, and prey and spider behavior all interact to determine whether a web retains a prey item long enough for the spider to secure it (Opell and Schwend 2007; Sahni et al. 2010, 2011; Harmer et al. 2011). Most of these factors lack the quantitative understanding necessary to be included in a model such as ours. However, stickiness of capture spiral glue does scale with spider size (Sensenig et al. 2010, 2011), though we did not directly model glue stickiness. If we were to include size-specific glue performance, we might expect larger spiders to better retain large prey items—and therefore grow larger—than they did in our current model.

The relationship between a prey’s size and characteristics such as mass and kinetic energy has long been debated (Schoener 1980; Eberhard 2013). We assumed in our model that prey of a given size always presented the same mass to spiders, though

Eberhard (2013) suggested that mass of some large insects could be vastly overestimated.

For a given “size” (conceptually, length or surface area), lighter insects would be easier to capture, but would provide less biomass to the spider. Therefore, we would still expect spiders to forage most successfully by capturing the most massive individuals, with one possible exception being environments where long and slender insects are unusually biomass-dominant.

We also assumed prey to be flying at maximal speed, although prey in nature may often fly at speeds below their maximum potential (Dudley 2002; Eberhard 2013).

Therefore, large prey might be striking webs with less kinetic energy than simulated in our model. However, lowering the exponent of our length-mass and length-flight speed relationships would simply make all prey—including large, highly profitable individuals—uniformly easier to capture. Alternatively, if we modeled prey flight speed to vary stochastically, we might expect foraging success to be maximized via capturing large yet slow-flying prey. Such a result would still support the rare large prey hypothesis, but might change our interpretation of how selection is acting on web design and performance. The impressive overall strength of orb webs might be an adaptation to both the need to stop highly energetic prey (Sensenig et al. 2013), and a safety factor for resisting damage from mechanical stressors such as wind gusts and repeated impacts with prey and substrate. Indeed, the relative strengths of these two selective pressures might vary among species foraging in environments with different physical risks. Regardless, even with prey flying at lower speeds, our model would still suggest that spiders forage most successfully from capturing a single rare large prey; the most successful foragers would simply be capturing even larger prey.

Our simplifying assumption that metabolic rate scales with spider mass raised to the ¾-power falls well within the exponent range suggested by empirical studies, though there is considerable variation among spider species (Glazier 2009; Kawamoto et al.

2011). Our results suggest that mass lost to metabolism is relatively negligible for very successful foragers, but might be more important for spiders needing to survive bouts of poor foraging success. Larger spiders lose more mass than do small spiders over the same

36 period of time, and therefore die slightly sooner if they fail to capture prey. This result likely reflects added pressure on larger spiders to capture biomass more promptly to avoid starvation. Furthermore, data from Glazier (2009) suggest a positive relationship between adult spider size and ontogenetic change in mass-specific metabolic cost, such that the metabolic cost for individuals of larger spider species increases more dramatically during ontogeny. Though the data of Glazier (2009) include spiders outside of the orb-weaving clade, these results suggest that, in addition to biomass availability, metabolic constraints might contribute to an upper limit on orb-weaving spider body size.

Our findings support the rare large prey hypothesis proposed by Venner and

Casas (2005) and generalized across the araneoid orb-weavers by Blackledge (2011), but the phenomenon of natural selection shaping organisms to succeed in rare events exists outside of orb-weaving spiders. For many organisms, capitalizing on rare-but-valuable resources or surviving aberrant events is crucial for fitness, and these events may represent a critical selective force in the evolution of organismal performance. For instance, many plants grow to resist loads well in excess of typical day-to-day requirements, with the most dramatic disparities found in environments where rare but extreme loads, such as waves and high winds, can be experienced (Gaines and Denny

1993; Denny 1995; Vogel 2003). Likewise, many forest understory plants suspend germination and growth until they can take advantage of rare tree-fall events, upon which they rapidly grow to reproductive maturity (Attiwill 1994). On a broader scale, Venditti et al. (2010) suggest that the majority of speciation occurs in a stochastic, punctuated fashion due to rare events that culminate in reproductive isolation. In all, rare events might impose disproportionately strong selection on organisms because the outcomes of

37 such events vary more widely in their fitness consequences than does the cumulative outcome of many common events. Indeed, mechanical constraints may exclude some organisms from maximizing success rate in common events, thereby leaving success in rare events as the only route to maximizing resource acquisition and therefore fitness.

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APPENDIX

#################################################### ### READ THE FOLLOWING BEFORE EXECUTING THE CODE ### ####################################################

# written and executed in R versions 3.0 and 3.0.1 # updated 14 November 2013 # Author: Samuel C. Evans

### CODE USED IN GENERATING THE RAW DATA PRESENTED IN THIS THESIS # This script allows reproduction of the exact raw data presented in # this thesis. This script does not include the code used in # restructuring the raw data to allow for visualization of # the figures presented.

### ANNOTATIONS/COMMENTS # The code is annotated to allow it to be more easily read and # understood. Comments are preceded with a ‘#’. # Some commands are ‘commented out’ (kept from being executed by # adding a ‘#’ to the beginning of the line). These commands are not # necessary to be used in generating the data, but may be found useful # if one wishes to further explore the model.

####################################################

### I. LOAD THE FOLLOWING REQUIRED PACKAGES # Note: if a package is not installed, you will need to install it. # Simply execute `install.packages(“packageName”)` to do so. require(compiler) require(multicore) require(plyr) require(reshape2) require(data.table)

# Suggested packages for data visualization #require(sqldf) # enables selectively importing data using sql syntax #require(ggplot2) # plotting figures #require(gridExtra) # plotting multiple figures in one image file

### II. ESTABLISH PREY CONDITIONS and GLOBAL SPIDER PARAMETERS { # Sets total prey biomass available in the 44

# environment at any given moment. # This should be thought of as "biomass density" # since the amount of biomass a spider encounters # will be a function of this constant *and* the size of the # spider's web. totalPreyBiomass <- 5000 # mg

## Creates prey vectors and dataframes

PREY_L <- 1:100 # vector of prey size classes (mm)

# Creates a variable to correct all references # to index of prey size, # for use if the smallest prey size # is set to be greater than 1 mm. ifelse(min(PREY_L)==1, min.preymm <- 0, min.preymm <- min(PREY_L) - (min(PREY_L) - 1))

# Creates dataframe of the three prey size-abundance distributions PREY.PML <- data.frame( smallBiased <- rev(PREY_L)/sum(PREY_L) , even <- rep(1/length(PREY_L),length(PREY_L)) , largeBiased <- PREY_L/sum(PREY_L) ) PREY.PML[1] <- PREY.PML[1] - 0.5 * (PREY.PML[1] - PREY.PML[2]) PREY.PML[3] <- PREY.PML[3] - 0.5 * (PREY.PML[3] - PREY.PML[2])

# Creates dataframe of biomass distributions. # Each element is the amount of biomass (in mg) that is # packaged in a prey size class PREY.ML <- PREY.PML * totalPreyBiomass

## Creates prey physical parameters.

# mass (mg) PREY_M <- 0.022 * (PREY_L^2.4) * 4 # kinetic energy PREY_KE <- 0.5 * PREY_M * ((0.3196 * PREY_L + 0.0909)^2) # surface area contacting web (cm^2) PREY_SA <- pi * ((0.5 * PREY_L * 0.1)^2) # kinetic energy per surface area contacting web PREY_KEA <- PREY_KE / PREY_SA

# Creates the relative number of prey in # each size class in the environment. # This will be used to calculate the # percent abundance of each prey size class, # so that the probability of encountering # a certain-sized prey can be decided. 45

NUM_PREY <- colSums(apply(PREY.ML, 2, function(x) x/PREY_M))

# Creates the prey proportion abundance distributions. PREY.A <- mapply('/', PREY.ML/PREY_M, NUM_PREY)

# Creates the prey cumulative abundance distributions. PREY.CA <- apply(PREY.A, 2, cumsum)

# Biomass loss factors # (used in calculating metabolic rate # of form y = cM^e, according to Kleiber's Law) spiderBLC <- 1.0 # biomass loss constant (c) spiderBLE <- 0.75 # biomass loss exponent (e)

# Minimum proportion of spider's initial mass # required for survival starveThreshold <- 0.8

# Growth requirement factor (per event). # This determines the slope of the line # that represents the minimum mass the spider # must maintain to survive. spiderGR <- 0.4

# Assimilation Efficiency: proportion of a # captured prey item's biomass # that is converted to spider mass. spiderAE <- 0.2

# Sets the duration of the model. numDays <- 30

# Initializes switch to preschedule jobs across threads. # (used when many replicates are simulated at once) opts <- FALSE }

### III. DETERMINISTIC MODEL FUNCTION # (no random variation in prey capture) det <- function(env, size, web) {

initSpiderMass <- size # initial spider mass (mg) spiderMass <- initSpiderMass # current spider mass (mg)

# "Web density", which can be varied to test hypotheses concerning # how spider web size scales with spider mass. # In the results presented in this thesis, # web density is always 1. 46 webDensity <- web

# Minimum mass required for the spider to survive (which is # not needed here in the deterministic model, and so # is commented out). #starve <- starveThreshold * initSpiderMass

# initializes day and event counters event <- 0 #numEventsLife <- 0 day <- 1

# Sets the status of the spider (alive/starved) #status <- 1 # 1 = alive

# LOOP SIMULATING A DAY IN THE LIFE OF A SPIDER # while (day <= numDays) {

# spider mass at beginning of day (mg) spiderMassDay <- spiderMass

## WEB CHARACTERISTICS, calculated from # Sensenig et al. 2010 JEB data # Each day, the spider builds a new web # with characteristics calculated below, # according to its mass at the beginning of the day.

# Web Area (cm^2) webArea <- 10^(0.3135 * log(spiderMass, base=10) + 1.93) * webDensity

# Silk Volume (mm^3) silkVolume <- 10^(-3.02 + 0.805 * log(spiderMass, base=10))

# Thread Thickness (um^2) threadThickness <- 10^(-0.696 + 0.717 * log(spiderMass, base=10))

# Length of Capture Thread (mm) captureThreadLength <- 0.5 * (silkVolume / (threadThickness * 10^-6))

# Mesh Width (mm) meshWidth <- (10 * (0.9 * sqrt(webArea / pi))) / ((captureThreadLength / 10) / (pi * (0.9 * sqrt(webArea / pi)) - 1))

# Silk Toughness (uJ) silkToughness <- 10^(2 + (0.13 * log(spiderMass, base=10))) * 10^3 47

## PREY CAPTURE PROBABILITIES FOR EACH CAPTURE STAGE contact <- PREY_L / meshWidth contact[contact>1] <- 1 stopping <- (silkVolume * silkToughness) / (4.0 * PREY_KE) stopping[PREY_SA > webArea/4] <- ((silkVolume * silkToughness) / ((webArea / PREY_SA[PREY_SA > webArea/4]) * PREY_KE[PREY_SA > webArea/4])) stopping[stopping>1] <- 1 retention <- 10 * (1 / PREY_KEA) * PREY_L / meshWidth retention[retention>1] <- 1 ##

# Number of "events" (i.e. prey to be encountered) for the day numEventsDay <- round((webArea / 1000) * NUM_PREY[env])

# Calculates the biomass to be gained for the day biomass <- 0 for (i in 1:length(PREY_L)){ biomass <- (biomass + spiderAE * PREY.A[i,env] * (contact[i] * stopping[i] * retention[i]) * PREY_M[i] * numEventsDay) }

# LOOP: WHEN PREY MEETS WEB # for(i in 1:numEventsDay){

# Calculates metabolic cost for the event metabolicCost <- ((spiderBLC * spiderMass^spiderBLE) / (numDays * numEventsDay))

# Adjusts spider mass spiderMass <- (spiderMass + (biomass/numEventsDay) - metabolicCost)

# Adjusts the starving threshold for the end of the event. # (not needed here unless you wish to track survival in the # deterministic model) #starve <- starve + (spiderGR * ((initSpiderMass / numDays) # / numEventsDay))

}

# AT THE END OF THE DAY... day <- day + 1 48

} # ends when 30 days have passed (or spider starves #if starvation is applied)

# DATAFRAME OUTPUT # output <- c(env, initSpiderMass, spiderMass - initSpiderMass) output }

### IV. STOCHASTIC MODEL FUNCTION # This is the function simulating foraging and survival # of a single spider. The model is run by calling a function # that in turn calls this function repeatedly # for a specified number of spiders. StochasticModel <- cmpfun(function(env, initSpiderMass, webDensity) { ### PART 1: SETTING UP INITIAL CONDITIONS FOR A SINGLE SPIDER # Initializes variables used to track spider traits, and # creates matrix where final data will be stored.

# Initializes variable to track spider mass spiderMass <- initSpiderMass # initial spider mass (mg) minSpiderMass <- spiderMass # lowest mass reached during simulation maxSpiderMass <- spiderMass # highest mass reached

# Sets minimum mass required for the spider to continue surviving starve <- starveThreshold * initSpiderMass

# Initializes day and event counters event <- 0 numEventsLife <- 0 day <- 1

# Initializes data matrix (this will be the data # written to file for analysis). data <- matrix(0, ncol=17, nrow=length(PREY_L))

# Defines vectors of data matrix data[,1] <- env # prey environment # prey environment: # 1 = "small-biased" # 2 = "even" # 3 = "large-biased" data[,2] <- initSpiderMass # initial spider mass data[,8] <- PREY_L # prey size classes (lengths, in mm)

### PART 2: LOOP SIMULATING A DAY IN THE LIFE OF A SPIDER # This is the main component of the StochasticModel function, # where spider foraging is simulated. The loop iterates until 49

# the spider starves to death or forages for 30 days. while (day <= numDays && spiderMass >= starve) { # spider mass at beginning of the current day spiderMassDay <- spiderMass

# Web Area (cm^2) webArea <- 10^(0.3135 * log(spiderMass, base=10) + 1.93) * webDensity

# Silk Volume (mm^3) silkVolume <- 10^(-3.02 + 0.805 * log(spiderMass, base=10))

# Thread Thickness (um^2) threadThickness <- 10^(-0.696 + 0.717 * log(spiderMass, base=10))

# Length of Capture Thread (mm) captureThreadLength <- 0.5 * (silkVolume / (threadThickness * 10^-6))

# Mesh Width (mm) meshWidth <- (10 * (0.9 * sqrt(webArea / pi))) / ((captureThreadLength / 10) / (pi * (0.9 * sqrt(webArea / pi)) - 1))

# Silk Toughness (uJ) silkToughness <- 10^(2 + (0.13 * log(spiderMass, base=10))) * 10^3 ##

# Number of "events" (i.e. prey to be encountered) for the day numEventsDay <- round((webArea / 1000) * NUM_PREY[env]) numEventsLife <- numEventsLife + numEventsDay

## PREY CAPTURE PROBABILITIES FOR EACH CAPTURE STAGE contact <- PREY_L / meshWidth contact[contact>1] <- 1

stopping <- (silkVolume * silkToughness) / (4.0 * PREY_KE) stopping[PREY_SA > webArea/4] <- ((silkVolume * silkToughness) / ((webArea / PREY_SA[PREY_SA > webArea/4]) * PREY_KE[PREY_SA > webArea/4])) 50 stopping[stopping>1] <- 1 retention <- 10 * (1 / PREY_KEA) * PREY_L / meshWidth retention[retention>1] <- 1 ##

# Number of each prey size expected to be encountered # during the day. # (Remember that, because this is a stochastic model, # spiders can encounter more or fewer prey of a certain size # relative what would be probabilistically expected) data[,9] <- data[,9] + (PREY.A[,env] * numEventsDay)

## LOOP: PREY CAPTURE EVENT # This loop simulates each prey capture event. # i) The spider encounters a prey item; # ii) the prey item either escapes or contacts the web; # iii) if contacted, the prey item either escapes or # is stopped by web; # iv) if stopped, the prey item either escapes or # is retained by web, # v) if retained, the spider gets a meal # (20% of the prey's mass). # Spiders lose a little bit of mass every event, regardless of # prey capture success/failure. # The loop moves on to simulate the next event # when the prey either escapes or is captured. # The loop ends when all the prey for the day # have been encountered. for(i in 1:numEventsDay){

# Calculates metabolic cost for the event metabolicCost <- ((spiderBLC * spiderMass^spiderBLE) / (numDays * numEventsDay))

# Randomly selects prey item for the event, # based on the prey abundance distribution in the environment. rn <- runif(1, 0, 1) # random number between 0 and 1

# Selects the size (length) of the prey item. itemL <- PREY_L[which(PREY.CA[,env] >= rn)[1]] # sets the mass of this prey item. itemM <- PREY_M[itemL-min.preymm]

# Counts the number of prey of this size # that the spider's web has encountered data[itemL-min.preymm,10] <- data[itemL-min.preymm,10] + 1 # Updates the number of prey of this size *expected* # to be contacted data[itemL-min.preymm,11] <- (data[itemL-min.preymm,11] + 51

contact[itemL-min.preymm])

## Prey capture process (contact, stopping, retention).

# random number to be used in deciding contact rn <- runif(1, 0, 1) # if web contacts prey item if(contact[itemL-min.preymm] > rn){ # Updates the number of prey of this size that # the spider's web has successfully contacted. data[itemL-min.preymm,12] <- data[itemL-min.preymm,12] + 1 # Updates the number of prey of this size *expected* # to be stopped data[itemL-min.preymm,13] <- (data[itemL-min.preymm,13] + stopping[itemL-min.preymm])

rn <- runif(1, 0, 1) # if web stops prey item if(stopping[itemL-min.preymm] > rn){ # Updates the number of prey of this size that # the spider's web has successfully stopped. data[itemL-min.preymm,14] <- data[itemL-min.preymm,14] + 1 # Updates expected number retained data[itemL-min.preymm,15] <- (data[itemL-min.preymm,15] + retention[itemL-min.preymm])

rn <- runif(1, 0, 1) # if web retains prey item if(retention[itemL-min.preymm] > rn){

# Updates the number of prey of this size that # the spider's web has successfully captured data[itemL-min.preymm,16] <- (data[itemL-min.preymm,16] + 1) # Checks for largest prey captured data[,17] <- max(c(itemL, data[1,17])) # Updates spider mass spiderMass <- ((spiderMass - metabolicCost) + (itemM * spiderAE)) maxSpiderMass <- max(spiderMass, maxSpiderMass)

}else{ # if web does not retain prey

spiderMass <- (spiderMass - metabolicCost) minSpiderMass <- min(spiderMass, minSpiderMass) }

}else{ # if web does not stop prey

spiderMass <- (spiderMass - metabolicCost) 52

minSpiderMass <- min(spiderMass, minSpiderMass) }

}else{ # if web does not contact prey

spiderMass <- (spiderMass - metabolicCost) minSpiderMass <- min(spiderMass, minSpiderMass) } # EVENT ENDS

# Adjusts the starving threshold for the end of the event starve <- starve + (spiderGR * ((initSpiderMass / numDays) / numEventsDay))

} # DAY ENDS

day <- day + 1

} # ENDS WHEN SPIDER STARVES OR 30 DAYS HAVE PASSED

### PART 3: RECORD DATA data[,3] <- spiderMass # final mass data[,4] <- minSpiderMass data[,5] <- maxSpiderMass data[,6] <- day data[,7] <- round(max(0, (spiderMass - initSpiderMass)) / 10^(0.4373 * log(initSpiderMass, base=10) - 1.1747)) # number of eggs # num.eggs.dbc <- round(max(0, (spiderMassDBC - starve)) # / 10^(0.4373 # * log(initSpiderMass, base=10) - 1.1747))

data <- as.data.frame(data) data })

### V. FUNCTION TO ITERATE StochasticModel OVER A NUMBER OF REPLICATES ManySpiders <- cmpfun(function(nrep, env, size, web, batch) { set.seed(batch) mclapply(rep.int(size, nrep), function(x) StochasticModel(env, initSpiderMass=x, webDensity=web), mc.cores=numCores, mc.set.seed=TRUE, mc.preschedule=opts) })

### VI. SCRIPT TO EXECUTE MODEL

# Specifies the batch of data to be generated. # This number also serves as the seed for random number generation, # so if you want to run multiple batches, you must change this number # every time, or you will get identical results. # In this thesis, simulations were run in 10 batches (1 through 10) # of 90,000 spiders (10,000 spiders for each combination # of size class and prey environment). batch <- 1

# Specifies the number of replicates (i.e. spiders) to be simulated. # Each spider has 100 lines of data, so 10,000 spiders will give you # a one-million-row dataset, which is quite large. This is why # running multiple batches is advised, unless you are # running on a PC/server with plenty of RAM (at least 32 GB). nrep <- 10000

# Detects the number of cores (threads) available to use # (for parallel execution). # If you do not wish to use all the cores # on your computer, you must change # `multicore:::detectCores()` to the # number of cores you want to use. numCores <- multicore:::detectCores()

## Executes simulation of a single batch, # including each spider size class (‘size’) # and prey environment (‘env’). system.time({ for (size in c(50,400,750)) { for (env in 1:3) { # Executes simulation data <- as.data.frame(rbindlist( ManySpiders(nrep, env, size, web=1, batch))) gc() # cleans up memory

# Names data columns names(data) <- c("spider", "env", # prey environment "initial.mass", # initial spider mass (mg) "final.mass", # final spider mass (mg) "min.mass", # lowest mass reached by the spider (mg) "max.mass", # highest mass reached by the spider (mg) "days.survived", # number of days spider survived "num.eggs", # number of eggs spider produced (if survived) "prey.mm", # prey size (mm) "det.numE", # expected number of each prey size encountered "numE", # actual (stochastic) number encountered 54

"exp.numC", # of prey encountered, expected number contacted "numC", # actual number of each prey size contacted by web "exp.numS", # of prey contacted, number expected to be stopped "numS", # actual number of each prey size stopped by web "exp.numR", # of prey stopped, number expected to be retained "numR", # actual number of each prey size retained by web "largest.preymm") # size (mm) of largest prey captured

# Writes the data to a comma-separated text file. # MAKE SURE TO SET THE DIRECTORY TO WHERE YOU WANT # THE DATA TO BE SAVED. setwd("filepath") write.csv(data, file=paste(batch, env, size, nrep, "rawdata.csv", sep="-"), row.names=FALSE, fileEncoding="UTF-8")

# Removes data (recommended for memory management) rm(data) gc() } } })