Predation and community on a complex surface : toward a fractal ecology

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Predation and community on a complex surface : toward a fractal ecology
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Lowen, Robert Glen, 1962-
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Entomology and Nematology thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Entomology and Nematology -- UF   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 2001.
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Includes bibliographical references (leaves 321-347).
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Also available online.
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Printout.
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Vita.
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by Robert Glen Lowen.

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PREDATION AND COMMUNITY ON A COMPLEX SURFACE: TOWARD A FRACTAL ECOLOGY














By

ROBERT GLEN LOWEN












A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2001

























This dissertation is dedicated to the memory of Dr. William L. Peters














ACKNOWLEDGMENTS

I wish to gratefully acknowledge the help and leadership of my major professor and committee chair Dr. Dale H. Habeck. I also wish to thank Dr. Gary R. Buckingham who has been a co-chairman in fact if not in name. This work could not have been done without the financial and logistical support of these two. I wish to thank the other members of my committee, Dr. Frank Slansky Jr. for sticking with me throughout, and Dr. Benjamin M. Bolker and Dr. Charles E. Cicbra for filling in when I needed them. I also wish to thank Drs. Jon C. Allen, and C.S. "Buzz" Holling who encouraged me to think in nonlinear terms.

I extend special thanks to Dr. John R. Strayer and the late Dr. W.L. "Bill" Peters, without either of whom I would not have come to the University of Florida. I wish to thank Dr. Jerry F. Butler and all the people in his lab for providing the mosquito larvae, with special thanks to Diana Simon and Karen McKenzie. I wish to thank Dr. James Cuda and his assistant Judy Gillmore who provided laboratory space and logistic support. I also wish to thank the numerous lab assistants who helped in ways both small and large.

I wish to acknowledge the assistance of Myrna Litchfield and Debbie Hall who kept me straight when it came to University paperwork. Jane C. Medley aided the production of images used in the survey. Steve Lasley provided computer support at all levels. I thank the rest of the students and staff of the Department of Entomology and Nemnatology; the Division of Plant Industry; and the Center for Aquatic and Invasive Plants.


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I wish to thank my parents for moral and financial support and for always

encouraging me to achieve more. Above all else I wish to thank my wife Catherine for her love, support and boundless patience. Finally I wish to thank my stepchildren Erin and Bryan Brooks without whom this would have been a far less interesting ride.











































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TABLE OF CONTENTS
Rne

ACKN OW LED GEM EN TS ............................................................................................... iii

ABSTRACT ..................................................................................................................... viii

CHAPTERS

I INTRODUCTION ......................................................................................................... 1

2 REVIEW: COMPLEXITY AND COMMUNITY ........................................................ 6
Arena .............................................................................................................................. 7
Infinite Coastline ................................................................................................... 7
M athem atics of N oise ........................................................................................... 9
Euclidean W orld and its M onsters .......... *"**"*""*****"**"******'******"*'***'*"" ... ** ....... I I
Pow er Law s and a M ultitude of Dim ensions ...................................................... 16
M easure of the M onsters ..................................................................................... 18
M easure of N ature .............................................................................................. 21
Fractal Geom etry of N ature ................................................................................ 24
Anim al Com m unity ..................................................................................................... 26
Passive Physical Response .................................................................................. 26
Behavioral Response ........................................................................................... 28
Trophic/Energetic Response ............................................................................... 29
Com petition Response ........................................................................................ 32
Succession in Tim e or Space .............................................................................. 35
Sum m ary ...................................................................................................................... 38

3 PERCEPTION OF COMPLEX SURFACES .............................................................. 40
Rules for Box-Counting ............................................................................................... 40
Apparent D im ension ........................................................................................... 41
M inim um Size of Existence ................................................................................ 43
All Things are Relative ....................................................................................... 45
Rule of Averaging ............................................................................................... 49
Testing Real-W orld Validity ........................................................................................ 53
Introduction ......................................................................................................... 54
M aterials and M ethods ........................................................................................ 54
Results ................................................................................................................. 61
Discussion ........................................................................................................... 69
Conclusions ................................................................................................................... 70



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4 REVIEW: COMPLEXITY AND COMMUNITY ...................................................... 73
Prelude to the Three-Point Interaction ......................................................................... 73
Predator-Prey Interactions ........................................................................................... 73
Parts of Predation ................................................................................................ 73
Functional Response ........................................................................................... 75
Optim al Foraging ................................................................................................ 77
Prey Size A ffected by Predation Technique ....................................................... 82
Im pact of Arena on Predator-Prey Interactions ........................................................... 84
Behavioral Effects ............................................................................................... 87
M echanistic Effects ............................................................................................. 90
Changes in density ....................................................................................... 90
Changes in form ........................................................................................... 96
Sum m ary .................................................................................................................... 102

5 PREDATION ON A COMPLEX SURFACE ........................................................... 105
Predicting the Obvious ............................................................................................... 113
Introduction ....................................................................................................... 113
M aterials and M ethods ...................................................................................... 115
Results ............................................................................................................... 117
D iscussion ......................................................................................................... 119
Predator Size in a Com plex Environm ent .................................................................. 120
Introduction ....................................................................................................... 120
M aterials and M ethods ...................................................................................... 121
Results ............................................................................................................... 122
D iscussion ......................................................................................................... 125
Prey Size in a Com plex Environm ent ........................................................................ 135
Introduction ....................................................................................................... 135
M aterials and M ethods ...................................................................................... 136
Results ............................................................................................................... 137
D iscussion ......................................................................................................... 141
Conclusions ................................................................................................................ 151

6 REVIEW: PREDATION AND COMMUNITY ........................................................ 155
H ow Do You M easure Com m unity Shape? .............................................................. 156
Richness and Diversity ..................................................................................... 157
Size V ersus Frequency ...................................................................................... 159
Clum ps .............................................................................................................. 165
Predator Effect on Com m unity Shape ....................................................................... 169
D irect Effects .................................................................................................... 170
Cascade Effects ................................................................................................. 177
Predation as a Constant Force ........................................................................... 180
Arena Effect on Com m unity Shape ..................................................................... t ...... 188
Am ount of Structure ......................................................................................... 189
V ariety of Structure .......................................................................................... 192
Form of Structure .............................................................................................. 195
Sum m ary .................................................................................................................... 205


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7 COMMUNITY ON A COMPLEX SURFACE ......................................................... 211
Effect of Plant Species on Com m unity Shape ........................................................... 212
M aterials and M ethods ...................................................................................... 212
Results ............................................................................................................... 215
D iscussion ......................................................................................................... 237
Effect of Plant Form on Com m unity Shape ............................................................... 240
M aterials and M ethods ...................................................................................... 240
R esults ............................................................................................................... 241
D iscussion ......................................................................................................... 264
Conclusions ................................................................................................................ 270

8 SU M M A RY AN D CON CLU SION S ........................................................................ 276
M easurem ent of Form ................................................................................................ 277
Form as an Interactive Surface .................................................................................. 281
Form as a Com m unity Tem plate ............................................................................... 291
A lternative Explanations ............................................................................................ 307
Future Research ......................................................................................................... 316

APPENDIX MATLAB LANGUAGE COMPUTER PROGRAMS ............................ 319

REFEREN CES ............................................................................................................... 321

BIO G R APH ICA L SK ETCH .......................................................................................... 348



























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Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

PREDATION AND COMMUNITY ON A COMPLEX SURFACE: TOWARD A FRACTAL ECOLOGY By

Robert Glen Lowen

August 2001


Chairman: Dale H. Habeck
Major Department: Entomology and Nernatology

The technique of box-counting as a method for measuring fractal dimensions is reviewed. A set of three axioms are developed that standardize methods and allow for multi-scale evaluation of complexity. A program was written that performed boxcounting on images of aquatic plants. Results were highly correlated with mean subjective evaluation of complexity. Box-counting plots of plant images were used to determine indices of prey delectability and accessibility. Detectability was defined as being proportional to the mean prey size over the mean size of plant surface at the scale of the prey. Accessibility was defined as being proportional to the mean size of plant surface at the scale of the predator over the mean size of plant surface at the scale of the prey. Laboratory experiments of fish predation on mosquito larvae were found to be highly correlated to both indices.





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Field samples of aquatic arthropods were taken from single species patches of

Myriophyllum spicatum L., Hydrilla verticillata (L.f.), and Vallisneria americana Michx. Bundles of plastic aquarium plants resembling each of these species were placed at all three sites and also sampled for aquatic arthropods. Species richness tended to follow mean plant complexity. Individual abundance did not follow mean complexity but showed scale specific peaks. Adding a mismatched plastic form to a site resulted in scale specific changes to the fauna. The data suggest that M. spicatum supports a full evenly distributed fauna, while V americana has a depauperate uneven fauna. Hydrilla verticillata supports an abundant fauna biased toward a few species in a particular size range. A predation shadow was defined as the point of transition from a small-scale linear region to a large-scale complex region. Box-counting plots show that predation shadows correspond to major structural features of the plants. Peaks in individual abundance or biomass corresponded with these regions. Distribution of species by mean body mass appears to be clumped with the number of clumps showing mean plant complexity. The data do not support the distribution of resources as a causal factor of community structure. The data are consistent with predation as a causal factor of community structure.
















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CHAPTER I
INTRODUCTION

Of all the unanswered questions in science, the simplest ones are often the hardest to answer. In ecology, these questions usually take the form of "Why are there so many/so few of species (x) around here?" Occasionally, we can provide a simple answer to such a question. There are so many Asian Tiger mosquitoes because they breed inside the numerous discarded car tires. There are so few manatees because we hit them with boat propellers. Such answers are true but are also oversimplified, often to the point of being misleading. Usually, we have little or no idea why a given species occurs at the population level that it does.

In perhaps no other field of endeavor does this shortcoming have a greater impact than in the field of classical biological control. Biological control is usually defined as the control of a noxious species (plant or animal) through the actions of natural predators (DeBach and Rosen 199 1). Classical biological control comes into play when the pest species is not native to the area but has become established and now causes a problem. The theory behind this is that the species is a problem in the new area because it has left its natural enemies behind and thus flourishes without them. If a species is declared a target of a biological control program, researchers will search its native ranges for any species that acts as a natural enemy of the target. Theoretically, such a natural enemy could serve as a biological control agent if released into the area where the target species is a pest.





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When biological control is successful, it can be phenomenally successful. There are numerous examples of target species permanently reduced to non pest levels resulting in enormous savings of money in yearly pest control expenditures (Pimental 1965, van Lenteren 1980, DeBach and Rosen 1991). In the face of such obvious benefit and success, one may wonder why being unable to predict population levels of a species can be said to have a profound effect on biological control efforts. While applauding the numerous successes and not denigrating current research efforts, the fact remains that most intentional introductions fail for one reason or another. Summarizing data on the introduction of parasites and predators, van Lenteren (1980) found that only about one fourth of targeted pests were successfully controlled and that roughly the same ratio of agents could be said to be established at the site of release. In other words, three quarters of all targets failed to be controlled and three quarters of all introduced agents failed to establish. More recently, DeBach and Rosen (199 1) summarized all data and found slightly better numbers. Of 416 species of insect pests targeted for biological control, 164 were at least partially controlled. Of the 4,226 species of natural enemies released, at least 1,251 could be said to have become established. The numbers were even better when the targeted pests were plant species. Of 125 species of weeds targeted for biological control, 49 were effectively controlled. Of 701 importations of control agents, 398 became successfully established in the wild.

Given that the food of the biological control agent is so abundant that we

determine it to be a pest, why is it that so few species become established upon release into the wild? If we examine the procedure used to choose potential agents, we see that great effort goes into ensuring that the agent preferentially feeds on the target. Since the





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pest species is in such great abundance, it becomes obvious that the introduced agent has no serious competitors for its preferred food. Great effort is made to ensure that the agent comes from climatic conditions similar to the area of release. Quarantine procedures are used to screen out harmful parasites and pathogens. Enough individuals of the agent are released into a localized enough spot so as to ensure the probability of continued reproduction. Why then do introductions fail?

One can never discount the possibility that the new area is lacking some

biological factor that was present in the native range but missing in the introduced area. But just as one cannot discount a missing biological factor, neither can one assert its presence beforehand. Careful initial breeding and host range testing within the country of origin should be enough to catch any critical factors. Perhaps the introduced organism is having trouble finding its food in the new area. In the case of herbivorous agents being released against target plant species, this does not seem likely. Releases of herbivorous agents are made directly onto large patches of the pest plant and often they are restricted by cages (DeBach and Rosen 199 1). Failure to find the target might be a factor in the case of predatory or parasitic agents being released against animal targets. It is possible that the target species is finding enough shelter amidst the structure of its surroundings so that even though it is numerous enough to be a pest it is not numerous enough to be readily found by the biological control agent. Another possible explanation for failure to establish is that the structure of the surroundings provides so little protection that predators are decimating the agents.

For the purposes of this dissertation, let us call the structure of the surroundings the arena and define arena as the identifiable area in which at least one prey and one





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predator species interact, at least potentially so. By this definition the arena can be as complex as a parasitic wasp searching the entire surface of a plant or as simple as water striders (Gerridae) hunting only at the water surface. This definition also isolates the complexity of the arena from the complexity of the trophic interactions occurring there. In other words, simple interaction could occur in complex places and complex interactions can occur in simple places. Ladybird beetles (Coccinellidae) hunting aphids on a complex plant located within a greenhouse would be an example of the former. While the plant may be structurally complex, there is a relatively straightforward relationship between predator and prey. An example of the latter situation could be represented by the water striders on the water surface. While the water surface is essentially two-dimensional and simple, the water striders must pick out what is prey while avoiding strikes from fish after the same prey and distinguishing the artificial fly cast on to the water to lure the fish.

If complexity of the arena influences the interaction between predator and prey, and if we can quantify the complexity of the arena, then we should be able to predict what the relative probability of surviving predation would be for a given species introduced into that arena. This suggests that predation, acting over an arena of given complexity, could structure the prey community by providing differential levels of enemy free space. Assuming that reproduction or immigration rates are not overpowering, those species afforded with maximal protection from that arena should achieve the greatest population levels and leave a depauperate fauna of species that are relatively exposed.

It was with these ideas in mind that the present study was begun. It was hoped that by understanding the structural complexity of plants, we could begin to predict the





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vulnerability of an organism to predation before time and effort were invested in the study of its biology. The resulting reduction in time spent searching for and testing organisms doomed to failure would result in increased success rates for biological control programs. The choice of aquatic plants and organisms was based purely on convenience and a pre-existing familiarity of the organisms involved.













CHAPTER 2
REVIEW: COMPLEXITY AND COMMUNITY Before we continue, a few definitions are in order. A predator is an animal that over the course of its life kills and eats numerous other animals. A parasitoid is an animal that feeds on one individual animal and generally kills it. A parasite is an animal that feeds upon one or more animals without killing them. A scavenger is an animal that feeds on dead animals killed by some other source. A prey item is any animal that has the potential of being killed and eaten by a predator. A host is any animal that has the potential of being fed upon by a parasitoid or a parasite. For the purposes of this dissertation, the term predator includes parasitoids, but not parasites or scavengers. Prey includes the hosts of parasitoids but not parasite hosts or the food of scavengers. While it is acknowledged that plants may serve in any of the roles mentioned above, they are not considered here as active participants.

Do not be misled by previous sentence and assume that since plants are not considered here as active participants that they are not considered at all. The entire premise of this study is that the structural complexity of plants (along with inanimate objects) forms the arena that helps control the outcome of the drama between predator and prey. So this review begins with a consideration of how to measure the complexity of the arena. Specifically, the concept of dimension as a measure of complexity is developed. With this concept in place, this review continues on to consider the predator and prey as actors on this stage.



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Arm

Clouds are not spheres, mountains are not cones, coastlines are not circles,
and bark is not smooth, nor does lightning travel in a straight line.
Mandelbrot (1983)

Infinite Coastline

Lewis Fry Richardson was an eccentric scientist who was interested in the

measurement of complexity. His specialty lay in asking deceptively simple questions such as "Does the wind possess a velocity?" and then using simple experiments to analyze them. He was reported to have studied the question of turbulence by dumping a sack of white turnips into the Cape Cod Canal. At one point he became interested in the roots of conflict between nations. He sought to examine the theory that the length of the border between two nations was proportional to the level of hostility they had previously exhibited to each other. He started gathering the data he needed by consulting the encyclopedias produced by the different countries and this is where he ran into a complication. Each country seemed to have a unique value for the length of common borders. Clearly something was amiss as even the poorest country could well afford an accurate survey of its national borders. How could two countries come up with different answers to the question of how long their common border was?

He began to make his own measurements using maps and a set of dividers. He set the dividers to the desired scale and carefully walked the dividers across the map, counting the number of steps needed to travel various borders and coastlines. An accurate estimate of the length of the border was obtained by multiplying the number of steps by the scale the dividers were set to. He then sought to increase the precision of the measurement by reducing the scale and taking fmer steps with the dividers. When he did so, he found that the dividers captured more of the detail of the boundary and resulted in





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a longer estimate of the border's length. In fact, the estimated length continued to increase and never converged to a single value no matter how fine he made the precision. Theoretically, if one could measure the boundaries with an infinitely fine precision then the boundaries would appear to be infinitely long.

Richardson had discovered that the apparent length of a boundary based on a natural feature such as a coastline, river, or mountain range, had no true length. The apparent length of such an object was strictly relative to the scale used to measure it. Furthermore, if the apparent length of such a boundary is plotted as a function of the scale used to measure it, the result is a straight line if plotted on a log-log graph (Fig. 1).




C
4
C






1 1.5 2 2.5 3 3.5
Log 10 (Length of Side in Kin)
AUSTRALIAN COAST 0 SOUTH- AFRICAN COAST
GBVIAN LAND3-F1RJNTIB 1900 X WEST COAST OF BRITAINJ
LANIC-FRONTIER OF PRTUGAL


Figure 1. Length of national boundaries as a function of scale of measurement (Richardson 1961).



The exact relationship can be expressed as follows (using notation from Mandelbrot 1983):

AI-D
L





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Where F -Dintervals of length, are needed to approximate a boundary of length Lo. The constant value Sis termed the unit length and represents the value of, when, is equal to Lo. Richardson did not proceed further with this work and all of these measurements and graphs were found among his papers after he died. The work was finally published posthumously (Richardson 196 1) in an obscure Yearbook where it seemed doomed to be ignored.

Mathematics of Noise

Benoit B. Mandelbrot came across Richardson's paper and it crystallized many of the ideas he had been considering previously. Mandelbrot's forte was geometry and for several years he had been interested in long series of numbers representing the behavior of seemingly random events. One of his early works focused on the stock market and considered changes in commodity prices (Mandelbrot 1963). If someone wished to measure the total price change of a given commodity, then they would have to define when they took their measurements. Year-end prices were easy to obtain but missed much of the fluctuation. Daily prices required more vigilance yet they missed much of the rise and fall of prices that occurred over the course of the day. Before his work, brokers considered commodity prices to be driven by large-scale forces determining long-term trends. Small daily changes were considered to be independent of long-term trends and to be essentially random in nature. Mandelbrot discovered that the variability in prices was time independent and symmetrical across scale if one considered it logarithmically. In other words, the ratio of the number of price changes of size x to the number of price changes of size 1 Ox was always the same no matter what time span was





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considered and no mater how big x was. To put it mathematically, if the number of changes of size x could be written as Nx then the following formula would hold true; N. -NIO.
NIOX N10OX

The distribution of commodity prices was self-similar, such that part of the graph looked just as complicated as the whole graph.

The idea of symmetry across scale was still fresh in his mind when he began to work on the problem of predicting errors in the transmission of computer information across phone lines. Static is an unavoidable feature of this kind of communication and while one can boost the signal strength to drown out the noise, there will always be an occasional burst of static large enough to cause an error in the flow of information. The distribution of these errors appeared to be random yet clustered. Mandelbrot (1965) realized that the distribution of these errors were analogous to the distribution of commodity price changes he had studied previously. If one examined a day where errors had occurred, then one could obtain a ratio of the number of hours with effor-firee transmission to the number of hours that contained at least one error. If one then looked at one of those hours that contained an error, you would get exactly the same ratio by comparing the number of minutes with error-free transmission to the number of minutes that contained at least one error. And again, if one then looked at one of those minutes that contained an error, one would get exactly the same ratio by comparing the number of seconds with effor-free transmission to number of seconds that contained at least one error. The distribution was self-similar and independent of any time scale, exactly like the distribution of commodity price changes.








When Mandelbrot came across the paper by Richardson (196 1) it all began to click into place. The abstract changes in the prices of cotton, the real-time electronic static interrupting a data stream, and the physical ruggedness of an actual coastline all exhibited the same qualities. A value that was thought to be real (total change in price, amount of effor-free time, and length of coastline) was entirely relative to the scale of measurement. The complexity of the measured phenomena remained constant across a broad range of scales. This realization led to the proposal that the exponent D derived from Richardson's measurement of natural boundaries was in fact a dimension, as loosely defined by mathematical convention, although D can and often does hold a non-integer value. As defined, dimension became an expression of complexity. The very nature of D being a fractional dimension led Mandelbrot (1967) to coin the term Fractal Dimension

(Df).

Euclidean World and its Monsters

The idea that dimensions can be an expression of complexity is a little difficult to grasp. It might best be shown by first examining some of the classic objects in Euclidean geometry and their integer value dimensions. It is easy to think of a cube or sphere as being three-dimensional (D = 3) and a filled square or circle as being two-dimensional (D = 2). Without too much difficulty it would be easy to convince someone that a cube is more complicated than a square (3 > 2), and that both are more complicated than a straight line (D = 1). By final extension, all three objects would be more complicated than a single point (D = 0). The dimension of the object thus becomes a numerical value of its complexity.

Obviously, these perfect objects from Euclidean geometry do not occur in the real world. There are neither perfect circles nor straight lines in nature, but this is how we





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have always modeled natural objects and events. We discuss point sources of pollution and square kilometers of a species range. We measure application rates for pesticides as so many liters per hectare. We measure the circumference of a tree trunk and calculate the volume of wood. We do these things even though we know there are no points, squares or circles in nature. We do these things because Euclid has provided us with a usefid set of models with which to approximate the real world. While we never actually see these perfect shapes, it is comforting to know that in the theoretical world of pure mathematics, these objects are real and easy to analyze, yielding predictable results.

Be wary of what comfort you draw from this. Pure mathematics has also

spawned objects that are just as real but have been impossible to analyze. Such objects bend logic to incredulity providing answers that we are not prepared to hear. These objects are the monsters of mathematics and they are also real.

One such monster is the Cantor set, first published by Georg Cantor in 1883 (in Peitgen et al. 1992). The best way to understand the Cantor set is to envision its construction (Fig. 2). Construction begins by imagining a line along the unit interval [0,I]. One then removes the open interval (1/3,2/3) leaving behind two closed intervals [0,1/3] and [2/3, 1] of length 1/3 each. The next step is to take these remaining intervals and remove an open interval third from what is left of each. This results in four closed intervals of length 1/9 each. This process is repeated so that after n iterations there are 2" intervals of length 1/3". This removal-procedure taking place an infinite number of times completes the set. The set of points remaining is the Cantor set. This strange set of points has an infinite number of members enclosed within a finite space and occupying no length. There is not the tiniest of intervals along the Cantor set that does not contain





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empty space. If we wished to place the Cantor set in the sequence of complexity we discussed above we would have to say that it is less than a line (D < 1) but more than a point (D > 0). In other words, the Cantor set has a fractal dimension (0 < Df < 1).













Figure 2. The first few steps towards creation of the Cantor set.



The next monster we need to examine is the Sierpinski gasket. Waclaw

Sierpinski was a mathematician who published his now famous gasket in 1916 (in Peitgen et A 1992). Construction of the Sierpinski gasket is similar to that of the Cantor set in that we begin with an object, perform a deletion function to it, and then repeat the process on the remaining parts an infinite number of times. However, instead of a line we begin with a filled in equilateral triangle. We divide the triangle into four equilateral triangles by drawing lines between the midpoints of the three sides. We then delete the middle triangle, leaving three triangles each 1/4 the size of the original. This process is repeated on the remaining triangles so that after n iterations there are 3" triangles, each of which is 1/4" the size of the original. The gasket itself is created when this deletion process has been completed an infinite number of times. What is left is an object that has a perimeter but no area, being composed of an infinite number of holes (Fig. 3). The Sierpinski gasket is clearly more complicated than a line (D > 1) since the lines used to





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define the original triangle are still in place, but it is also clearly less than a completely filled triangle (D < 2). The Sierpinski gasket also has a fractal dimension (I < Df< 2).

























Figure 3. Sierpinski gasket approximation.



The last monster I wish to discuss is the Koch curve, first described by Helge von Koch in 1904 (in Peitgen et al. 1992). Unlike the previous two examples, adding to an object instead of deleting creates the Koch curve. One starts with a straight line and then divides it into three equal sections (Fig. 4). The middle section is replaced with an equilateral triangle missing its base. The resulting shape now has 4 line segments each of which is 1/3 the length of the original. As before, this process is repeated so that after n iterations there are 4nl line segments, each of which is 1/3' the size of the original. The Koch curve is achieved when this process has been completed an infinite number of





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times. If one starts out with an equilateral triangle instead of a line, then this process creates a Koch island or Koch snowflake as it is sometimes called (Fig. 5). A Koch curve is not a curve in the sense of it being composed of smoothly bent lines. Rather, the Koch curve is all comers and cannot be differentiated. In other words, nowhere along its length is there a point that has a unique tangent. Furthermore, like the coastlines measured by Richardson (1961), a Koch curve is infinitely long. If we were to consider the Koch snowflake, then we need to rationalize an infinitely long border enclosing a finite space. While clearly more complicated than a line (D > 1), the Koch curve is a long way from filling the plane and so is also less than a filled circle (D < 2). The Koch curve also has a fractal dimension (I < Df < 2).




0








2 3








4

Figure 4. Four iterations towards the construction of the Koch curve.





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Figure 5. Approximation of the Koch snowflake.



Power Laws and a Multitude of Dimensions

The Cantor set, the Sierpinski gasket and the Koch curve are three examples of the many objects in geometry that seem to fall between the dimensions of Euclid. They defied rulers and tape measures and could not be counted. But slowly and in piecemeal fashion, mathematicians began to tame them through refinement of the concept of dimension. Or rather, mathematicians came up with numerous definitions of dimension. It is beyond the scope of this work to fully explore all the different possible dimensions. Discussion is limited to those immediately applicable to the problems at hand. Suffice it to say that each of these dimensions measures some subtly different component and how it scales relative to another aspect of the object being measured.

The roots to understanding dimension lie in realizing that most (but not all) are derived from the exponents of power law relationships. A power law is a relationship where the behavior of one variable behaves as a power of another. They take the generalized form of the following formula;

Y =,?Xe





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where Nis a constant and e is the exponent from which a dimension can be determined. This last point might best be shown by considering the power law formulae for the area of a circle and square.

Area Br'

Area (squm) = s 2

For the circle, the area is dependent on a constant (B) multiplied by the radius

raised to a power of 2. For the square the constant is I and the side length is raised to the power of 2. The area of these ob ects is a two-dimensional feature and the exponent is 2. The perimeter of these objects is one-dimensional and the formulae for perimeter involve an exponent equal to 1.

Perimeter (cirri,) = 213?

Perimeter (,qu.,) = 4s'

Similarly, the formulae for volume involve different constants, and r and s raised to the power of 3.

1 wish to point out that there is not one true dimension for any object. Rather the same object can have numerous dimensions depending on which aspects are being measured. The above examples of the circle and square are relatively simple, yet they can simultaneously be Dimension one or Dimension two depending on whether one is measuring the perimeter or the area. Complex objects or groups of objects can have even more subtle distinctions and so great care must be taken to ensure that you are measuring the feature you are most interested in. In the case of the mathematical monsters discussed above, there are no simple formulae for perimeter or area and so their dimension is not intuitively obvious.





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Measure of the Monsters

The theoretical groundwork for determining the dimension of a mathematical monster lay in the separate works of Carath~odory, Hausdorff and Besicovitch (in Mandeibrot 1983). Suppose you wished to evaluate the area of a planar shape S, a classical approach to doing so would be to cover the set with a collection of small squares. One could then approximate the area of S by multiplying the number of squares needed by the area of one square. Carath,9odory reasoned that one was not always able to use known coordinates so he substituted discs for squares. He also avoided making the assumption that S is planar by using spheres, which are equivalent to discs in two-dimensional space. As one decreased the size of the spheres, the estimated area of a standard object (i.e. where D is an integer) would asymptotically approach its true value. Hausdorff realized that one did not need to know the dimension of the object beforehand. The dimension could be determined from the relationship between length and volume as measured by the spheres. If length is infinite and volume is zero, then the shape could only be two-dimensional. Besicovitch extended this argument to include dimensions of non-integer value. The resulting dimension is termed the HausdorffBesicovitch dimension (DH).

The mathematics is complicated but the DH can be defined from the limit


d 0. log rJ

where N(r) is the minimum number of spheres of radius r needed to cover the object and d is equal to J-DH (Mandelbrot 1983). Rearranging this formula we get the following;





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~j'log1



This formula is equivalent to the following power law N(r) = ?r.-d

or

N(r)-= Ar.IDH

where 8 is a constant. We can determine d as the negative value of the slope of a log-log plot of the minimum number of spheres of radius r plotted as a function of r. Since this slope is already negative, a negative of a negative results in a positive value for d. The Hausdorff-Besicovitch dimension (DH) can then be determined as 14d Or, by using the second version of this formula we can say that DH is equal to 1 the slope of the log-log plot.

The problem is that there is no simple way to determine what the minimum number of spheres would be. To overcome this difficulty, mathematicians have developed the similarity dimension (Ds) as an estimate of DH. While not a perfect match with DH, D is easy to calculate for most mathematical objects. Like most dimensions, D, is derived from a power law relationship. For D,, the power law lies in the relationship between the number of pieces an object can be divided into (n), and the reduction factor

(S).


-sD

or





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D, log n 10
S
g(l)

If we take a straight line and divide it into 3 pieces (n = 3) we will have reduced the size of each piece to 1/3 that of the original (s = 1/3). This means that for a straight line D,, = I since the following is true

3 = I / (1/3)'

or

I = log 3/ log (1/(1/3)) or

I = log 3 / log 3 Similarly, if we take a square and divide it into 9 pieces (n = 9) we will have

reduced the length of each side to 1/3 that of the original square (s = 1/3). This means that for a square Ds = 2 since the following is true

9 = I / (1/3)2

or

2 = log 9 / log (1/(1/3)) or

2 = log 3 2/ log 3 This is easily extended to the fractal objects we have already considered if we remember how they were constructed. During construction of the Cantor set, each iteration resulted in twice as many line segments (n = 2) each of which were 1/3 the length of the original (s = 1/3). Thus for the Cantor set we arrive at the following value for D,





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D, = log 2 / log (1/(1/3)). 0.6309

Similarly, during construction of the Sierpinski gasket, each iteration resulted in 3 times as many triangles each of which had sides 1/2 the length of the original (n = 3, s--112)

D, = log 3 / log (1/(1/2)). 1.5850

and in constructing the Koch curve each iteration resulted in 4 times as many line segments each of which had sides 1/3 the length of the original (n = 4, s = 1/3) Ds = log 4 / log (1/(1/3)). 1.2619

The end result is that we now have a simple way of estimating the dimension of any mathematical object as long as we have some idea of how it was created. The monsters remain bizarre, but they are now understandable and we can compare the complexity of one versus the other. One dark comer of pure mathematics has been illuminated, but how does this relate to measuring the length of a coastline? Measure of Nature

Mandelbrot (1983) maintained that a plethora of natural objects and phenomena were fractal in nature. Theoretically then, natural objects should have non-integer DH values. If we attempt to measure this, we again run into the problem of a priori determination of the minimum number of spheres needed to cover the object. Without Us knowledge we cannot determine DH. Neither can we easily determine Ds since natural objects do not have readily observable factors of reduction and replication.

Similarity dimension is based on the principle of self-similarity. If we examine the monsters in close detail, we note that each of the objects is composed of parts that resemble the whole. The Cantor set is composed of two parts that are an exact match to





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the whole set except that each is 1/3 the size of the whole. Each of these parts is itself composed of two copies 1/9 the size of the whole. Similarly, the Koch curve and Sierpinski gasket are composed of reduced copies of themselves.

Natural objects such as coastlines are not self-similar in the sense of being

composed of exact copies of themselves. However, as evidenced by Richardson's work with dividers, natural objects can maintain a constant level of complexity across a wide range of scales. This results in the previously described phenomena of a log linear increase in the apparent length of an object in response to a log decrease in scale of measurement. This so-called Richardson effect is what led Mandelbrot to coin the phrase fractal dimension in the first place. So that while a coastline cannot be said to be exactly self-similar, it can be described as statistically self-similar. The object resembles itself not in being an exact copy but in being just as complicated at all scales.

The problem with using dividers to estimate the dimension of an object (sensu Mandelbrot) is that the resultant counts are sensitive to the initial placement of the dividers. Also, the dividers cannot take into account a path that crosses itself or an object composed of disjoint parts. Neither can it measure the holes in an object. A more userfriendly method of measuring the dimension was needed.

The roots of a simpler method lay in the definition of another dimension, the socalled Minkowski-Bouligand dimension (Dm). This dimension is similar to DH but is determined in a different manner (Schroeder 199 1). To find the Dm of a curve, we let the center of a small circle with radius r follow along the curve. The area F(r) that this circle sweeps out as it follows the curve is termed the Minkowski content or the Minkowski sausage. If we divide this area by 2r and allow r to approach zero, we begin to





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asymptotically approach the length of the curve. If this curve is linear then this division of the Minkowski sausage approaches a finite value. If the curve is fractal in nature, then the estimated length never levels off but continues to increase. In fact, the rate of increase can be used to define Dm since the ratio of F(r)12r is proportional to rIDM. If we wish to write out the entire formula, it would take the following form:




DM=~~log(r)j+


It becomes immediately clear that the formula for Dm is close to that for the DH. Instead of a count of minimal number of discs (N(r)) needed to cover an object, we use the area of the sausage (F(r)). This formula is no easier to use than the Hausdorff-Besicovitch dimension, but it intuitively led to a more easily determined dimension.

Mandelbrot (1983) pointed out that trying to measure Dm of an object like the

coastline of Britain was like somebody laying an end-to-end line of rubber tires along the coast. You could then determine Dm by straightening the line of tires. The diameter of one tire multiplied by the number of tires needed results in the estimated length of the line and the diameter of a tire multiplied by the length gives us the area. It is then possible to determine Dm from the slope of a log-log plot of the area versus the diameter of the tire. By further extension, if we imagine that we are using squares instead of circles, then we can approximate Dm by overlaying the object with a grid and counting the number of boxes that contain a piece of the edge of the object. This technique is called mosaic amalgamation by Kaye (1989) but is more commonly referred to as the





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box-counting or grid overlay method. The following formula can be used to obtain a dimension that is commonly called the box-counting dimension (DB).



D8 = lmlog N(s)1



Note the similarity to the formula for DH. The main difference being that DB uses squares on a grid of side length s instead of circles of radius r. More importantly, instead of trying to determine the minimum number of circles that will completely cover an object (N(r)), DB uses the relatively simple count of the number of squares that intersect the object.

Fractal Geometry of Nature

Mandeibrot (1983) declared that nature is fractal and with the tools of boxcounting and dividers step-counting, a flurry of papers were published to show that indeed, nature was fractal. So far, fractals have piqued great interest among ecologists but few studies have tried to apply them. Most studies do not go beyond simple measuring and demonstration that fractals do occur in nature.

One area where fractals have been investigated more fully is the branching pattern of fungi cultures and plant roots with regards to the exploration and exploitation of the environment. Ritz and Crawford (1990) demonstrated that fungal colonies were fractal. They found that young cultures had a low dimension, which steadily rose as the culture aged. They hypothesized that a low dimension would be more effective in exploring the immediate environment but that to exploit a resource the fungi needed a more complicated pattern, i.e., higher dimension. Bolton and Boddy (1993) pursued this idea fur-ther by rearing different species of fungi on media of different nutrient quality.





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Although the different fungal species had differing dimensions, they did exhibit the same trend with regard to nutrient level. High nutrient levels resulted in high dimension values for the fungal mycelia, while low nutrient levels led to lower dimension. Similar relationships have been found for the dimension of plant roots and nutrient level (Eghball et al 1993; Lynch and van Beem 1993; Berntson 1994). Fitter and Stickland (1992) provided contradictory evidence measured from the roots of two species of grasses and two dicots. Using step counting, they found that the dimension of the roots increased with age but showed no relationship to nutrient level.

Plant structures other than the roots have been found to be fractal and this has led to useful applications in forest ecology. Taylor (1988) measured tree rings and found a fractal relationship between their variance and mean. He was able to detect changes in growth regime, although he was unable to correlate these with a particular cause. The crown of trees has been measured as fractal and found to be directly related to site quality and inversely related to the self-thinning tendency of mature trees (Zeide and Gresham 1991; Osawa 1995).

On a larger scale, much promise has been shown in the measurement of patches within the discipline of landscape ecology. Using satellite and aerial photographs, researchers have begun to understand and measure the impact of man on landscape-sized patterns. For example, Krummell et al (in Milne 1988) found that dimension would be low at smaller scales. They postulated that the square shape of agricultural plots caused this. But general principles in landscape ecology have been few and far between. Wickham and Norton (1994) found that agriculture increased the dimension of wetlands.





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Instead of imposing the linear shape of plots, agriculture fragmented the wetlands leading them to reflect larger landscape patterns.



Animal Community

Even though there have been few papers that deal with the animal community as it exists on a fractal surface, there have been many papers that look at the arena and how it influences animal communities. This section summarizes the literature that attempts to find the causal relationship between structure and community. Passive Physical Response

Plant complexity can have a natural sorting action on the animal community that requires no other interaction. All other things being equal, large erect plants are easier to find and colonize than small plants and should therefore support larger population of invertebrates. Lawton (1986) reviewed published records of the insect fauna on British plants. He found that if one compares different plants with similar size ranges, the more complex one will have more species of insect on it. An alternative explanation lies in his definition of complexity, which included diversity of structural characteristics as well as the tendency to occupy space.

Diversity in plant structure can create new microhabitats resulting in an increased number of species. Tallamy and Denno (1979) demonstrated this. They examined two grass species for the structure of the sap-feeding invertebrate community on them. Distichilis spicata stems reach heights of 50 cm with stiff cuims that tend to stay erect. Subsequently, this species forms a thick thatch layer. Spartina alternifolia stems reach heights of 10-40 cm. with leaves that tend to be more divergent. Older leaves lie right on the marsh surface and rapidly decay. The thick thatch layer results in D. spicata having a





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richer and more diverse fauna than S. alternifolia. If the thatch is removed, richness drops significantly but diversity and evenness fluctuate.

There does appear to be an impact on the arthropod community from the presence of a diverse understory. Stoner and Lewis (1985) examined the macro-crustacean community in seagrass beds, Thalassia testudinum, with an understory of calcareous algae, Halimeda opuntia (L.). Manipulative experiments involved the removal of one or both plant species. These resulted in no particular impact on arthropod numbers when compared as total number versus plant surface area, but there was a decided impact on particular species that were assumnedly adapted to particular structures. Their conclusions suggest that total faunal abundance was a function of habitat surface area, but that the faunal diversity was a function of qualitative aspects of the area's surface.

Another possible passive impact on arthropods arises from the effect of plants on the wind and water currents around them. While, there does not appear to be any studies on the effect of plants on wind speed and subsequent colonization rates, the impact of plants on water currents and colonization rates has been clearly documented. Gregg and Rose (1985) sank trays of plants into unvegetated streams. The plant's impact on water velocity seemed to be the determining factor in what the resultant invertebrate fauna would be. Arthropod guilds were about equally represented in number regardless of type of plant cover, but the guilds were composed of different species. Unfortunately, no size comparisons were made. Dean and Connell (I1987b) found similar results using plastic algaea" mats. They saw this as a sampling artifact in that their idea of increasing complexity included larger size as well as form. Bigger mats contained a greater number of invertebrates resulting in increased numbers of species. Gibbons (1988) showed that





28

sedimentation rates on artificial "algae" were profoundly affected by the size and shape of the fake plants and that diversity increases may be due to these effects. Kern and Taghon (1986) put forward a contrary view. They found that passive recruitment alone could not account for the abundance of harpacticoid copepods since the frequency distribution of some copepods late in the year was opposite to that found on the same plants early in the year.



It is possible for animals to actively choose the plant forms they inhabit aside from passively settling at the whim of physical factors. Stoner (1980) found that when three species of gaminaridean amphipods were offered a choice between three species of seagrass, the clump of seagrass with the highest biomass was chosen. If biomass was equal, then the clump with the highest surface area was always chosen. If the biomass was close and all surface areas were equal, then no preference was shown. However, Stoner also found invertebrates did select for the densest plants (i.e., highest biomass per unit area).

Hacker and Steneck (1990) also examined the size abundance patterns of

amphipods on algae. They found that the number and size of spaces between the fronds had a positive impact on larger amphipods but little effect on the smaller ones. Highly branched and thin filamentous algae supported larger populations of amphipods. This was especially apparent for the smaller amphipods. They compared these findings to laboratory experiments that utilized the same algae as well as artificial versions. These experiments excluded predation or food value, yet produced similar patterns as those found in the wild populations. They determined that the distribution of these amphipods on different algae types was the result of active choice rather than just a response to





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differential predation rates. They did acknowledge that predation might be an ultimate factor in the habitat selection.

Fish also show active choice towards particular plant structures. Anderson (1994) found that the number of kelp perch, Brachyistiusfrenatus (Gill), showed a strong correlation with the canopy cover provided by the giant kelp, Macrocystis pyrifera (L.). The correlation was strongest for juveniles, which theoretically had more to fear from predators. Adult fish were more even in their distribution. Furthermore, if the canopy cover broke down, then the fish congregated in lower plant structures such as the fronds.

Positive taxis cannot be considered a universal cause of species distributions on plants. Norton and Benson (1983) found that in the wild, all amphipods on brown seaweed, Sargassum muticum, were more abundant on the distal well-illuminated portion of the plants, which maintained higher densities of diatoms. While this distribution seems adaptive, laboratory experiments showed that some amphipods were not attracted towards either S. muticum or diatoms. So, while all of these species have a similar distribution, this could not be attributed to a universal behavioral response of the amphipods. Dean and Connell (1987a,b) found that four common invertebrates showed positive taxis to increased biomass of algae but they made no preferential selections between algae species of the same biomass. Russo (1987) found that epiphytic amphipods showed no preference for any black nylon bottle brushes regardless of their complexities or mass.

Trophic/Energetic Response

The idea that the structure of the arena can have an impact on the trophic or energetic responses of animals has been expressed before. It has been suggested that increased structure leads to a greater number of distinct resources, i.e., niches, which





30


results in more species being able to coexist in a given area (MacArthur 1972; Pielou 1975; Whittaker 1975; and Pianka 1978). Smith (1972) postulated that increased habitat complexity allows for more ways for feeding strategies to differ. In a more indirect sense, it has been reasoned that increased complexity would allow for increased attachment sites for the food of the associated fauna (Abele 1974; and Hicks 1980).

Actual experiments on these concepts have proven to be sparse. August (1983) used principle component analysis to study the effect of vertical variation in habitat physiognomy (complexity) and horizontal variation in habitat form (heterogeneity) on various aspects of the small mammal community in a Venezuelan forest. Species richness showed a positive correlation with complexity but not with heterogeneity. This was likely due to guild expansion rather than addition of new guilds. Diversity, abundance, biomass, and evenness showed little correlation with either complexity or heterogeneity.

Lawton (1986) reviewed the literature on the impact of plant architecture on insect diversity. He focused on phytophagous insects, and his idea of complexity encompassed size of plant through space as well as variety of plant structures. Comparing different plants with geographic ranges of similar extent, the more complex plant will have more species of insects on it. There are two possible reasons for this; size per se and resource diversity. Both explanations are similar to theories employed in island biogeography. Size per se has already been discussed, i.e., larger plants are more visible and more likely to be colonized. But the literature also suggests that resource diversity will impact the animal community. Plants with a greater variety of resources will support a greater variety of herbivores. While some of the variety comes from non-





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trophic: reasons (e.g., microclimate variations, refuge sites, over-wintering sites, opposition sites, etc.), a plant's tendency to occupy space does affect the trophic; interactions of the animal community. Bigger and older plants can have a different assemblage of insects as compared to younger plants. For instance, wood-boring insects would not colonize a seedling. Plants do not get this increase in diversity if they are annuals.

Lawton (1986) goes on to promote the idea that the energetic relations of an insect interacting with the fractal dimension of plant surfaces as an explanation for the relative abundance of insect species. If plant surfaces are fractal, then small insects outnumber large insects not only because they take up less space but also because there is absolutely more living space at the smaller scale. It would be meaningless to say how much physical space an insect has available to it without stating the size of the insect.

The energetics part of Lawton's argument comes from the power laws of animal metabolism as put forward by Peters (1983). Because of the way an animal's metabolism changes with its body mass (W), the number of animals (N) able to be supported by a given unit of energy is proportional to W-0-" or (L 3)-0,75 if one wishes to approximate with an animals length (L). This relationship means that if resources limit the number of individual animals living in an area, then there should be more little animals than big animals because each little animal uses less of these limiting resources. The power laws suggest that an order of magnitude decrease in animal length should result in density almost 178 times higher ((0. 13)-0.71 = 177.8). Lawton suggests that if the plant has fractal dimension, then the area perceived by a smaller animal would be greater than that perceived by the larger animal. By way of example, he shows that if the plant surface





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had Df = 1.5, then an order of magnitude reduction in scale of measurement would result in the apparent distance between two points on that plant increasing by a factor of 3.16.
2
The perceived area would increase by a factor of 3.16 or roughly one order of magnitude. The expected density of an animal 3 mm long should therefore be longer than that of an animal 3 cm long by a factor of 10 based on perceived area and an additional factor of about 178 based on metabolic demands. Data from Janzen (1973) and Morse et al. (1985) seem to support this.

What Lawton fails to discuss is exactly why the energy resources would be distributed in a fractal manner. Yes, the surface of the plant clearly shows the Richardson effect and appears to have more available surface when viewed by a smaller animal, but there is only so much resource material inherent in the plant. At any given instant, the volume of the plant is fixed and does not change regardless of the apparent size of the surface. The situation is analogous to the example of the Koch snowflake in which an infinite boundary surrounds a fixed volume. The energy resources in a plant are part of the fixed volume and not the subjective boundary. One would have to show that an animal was space limited in its access to resources in order for a fractal surface to impact on an animal's abundance in this manner. This is clearly not the case. So if an animal's abundance correlates to a plant's fractal dimension, then the answer as to why this occurs lies elsewhere.

Co=tition Response

While there are no studies that compare an area's complexity with the competition level between animal species, there are numerous studies that compare the level of competition between animals of particular sizes. In a landmark study Hutchinson (1959) compared linear measurements of some body part from sympatric species and animals.





33


What he found was that the ratio of the larger species to the smaller one had a mean value of approximately 1.3. Many ecologists leapt on this value and hailed it as an ecological constant. Soon numerous studies began to appear in the literature finding the 1.3 size ratio in a variety of communities. Roth (198 1) pointed out that while many of these studies showed a similarity to Hutchinson's ratio, there was an almost universal lack of statistical validation to the perceived similarities. The studies varied in their definitions of sympatric and in the meaning of the ratio itself. The ratio has been called the mean, modal and optimum value for species coexistence. It has also been called the maximum ratio for successful coexistence as well as the proper sum of differences along all n axes of niche space. It could be argued that such diversity of approaches all find a value near

1.3 was evidence for a universal underlying principle but Roth (1981) discounts this as evidence. She points out that the sum of published studies is not a random sample since any study that failed to find a pattern would be less likely to be published.

Roth (1981) goes on to reexamine the work of Schoener (1965) to show the lack of fit with Hutchinson's ratio. Schooner gathered 4 10 ratios of bill sizes of sympatric bird species. When plotted, the distribution had a mean and mode that was considerably less than 1.3. In fact, the overall distribution most closely resembled a discrete approximation of an exponential distribution. This suggests that the ratio of bill sizes between sympatric birds was no different from a random value. Roth does go on to caution against dismissing the possibility of pattern existing in natural populations but she does stress the need for statistical verification of observed patterns.

Even if the value of 1.3 has no special significance, there is evidence that

interspecific competition can structure the body sizes of species in a community. Bowers





34


and Brown (1982) examined the coexistence of granivorous desert rodents in three different North American deserts. They rejected the null hypothesis that the species body sizes were no different than random assemblages. Species of similar body size (body mass ratio < 1.5) coexisted locally and overlapped in geographic distribution less often than would have resulted from chance. Price (1983) developed a mathematical model that helped to explain this. The size of the rodent determined the size of the patch the rodent would use so that animals of different size classes could coexist by utilizing different sized patches of food.

Working on a much smaller yet a more numerous scale, Walter and Norton (1984) extracted and measured 20,000 oribatid mites of 85 different species. No minimal size difference was noted except for congeners. This implied that they do not compete, but there was a strong pattern between congeners implying that these do compete. Given that there was not much variation in the possible diet, the authors were uncertain what could lead to this type of pattern. They were certain that some sort of biological factor was involved, but the data seemed to suggest that exploitative competition was not it. Tonkyn and Cole (1986) found that comparative size ratios of competing species had limited value. They found that if one drew randomly from any distribution of animals and plotted a graph of the number of species versus their relative size ratios, what you get is a graph that is monotonically declining and concave up. If there were some special ratio between adjoining species, this graph would appear with peaks. Dean & Connell (1987c) examined 50 pairs of species of aquatic invertebrates for potential competitive interactions. None of the comparisons showed a significant negative correlation. They admit that seasonal changes in overall numbers could affect these results, but an





35

additional test showed that the presence of a potential competitor had no significant effect on the substrate choice of an introduced amphipod. Nelson (1979) found no evidence for significant interspecific competition in amphipod fauna found is seagrass beds. Coen et al. (198 1) found some evidence for competition for shelter between two caridean shrimps, at least within a laboratory setting.

What these negative reports seem to have in common is that they all examine communities that are on rich or abundant resources. Kohler (1992) found that competition could be a significant structuring factor in an area where resources are limited. Two periphyton grazers, a caddisfly larva, Glossosoma nignor Banks, and a mayfly naiad, Baetis tricaudatus Dodds, were examined in the lab and in the wild. They did not affect each other's survival, but they did have significant impact on each other's growth. In addition, exclusion of the caddisfly led to significant changes in the size structure of the other invertebrates

Succession in Time or Space

A lot of the observed successional changes in an animal community are simply a question of luck of colonization. Whichever animal species finds and exploits an area first has a decided edge in winning out. Robinson and Dickerson (1987) showed this to be the case, at least on a small scale. They took small jars and colonized them with algae and other microorganisms at specific rates and sequences. Four different stable communities arose depending on which species were added first and at what rate they were added. A follow up study (Robinson and Edgemon 1988) examined this in more detail, i.e., more species and stricter measurements of timing. They found that the invasion rate was most influential. The order of invasion had almost no effect but the timing between the invasions was highly significant. In other words, second place can





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still be successful as long as you're not too early or too late. Not that post colonization factors of the arena cannot also influence communities. Chironomid colonization of wading pools was found to be dependent on substrate particle size and organic matter content and that this effect occurs because of differential survival of the eggs (Francis and Kane 1995). But the lottery of colonization, while modified, is still considered the main structuring feature.

In this view of succession, age of the arena and colonization rates are what

structure the animal community. But plant structure clearly has an impact on the animal communities and a progressive change in plant structures is almost the definition of succession. Dean and Connell (1987ab, c) envisioned a steadily changing animal community responding to the steady changes in the structure of the plant community. In their investigation of the fauna on marine algal clumps, invertebrate species richness and abundance steadily increased with successional stage in response to changes in algal structure even though the greatest variety of forms occurred at middle successional stages. This is a successional pattern where bare rock is first colonized by low biomass levels of flat and smooth species of algae, followed by intermediate biomass levels of numerous species of varying shape, and finally, dominated by a higher and taller species with few lower branches and more complex tops. Evenness of the invertebrate fauna remained constant throughout succession, while temporal variation declined and spatial variation remained constant. This is somewhat supported by Beckett et al. (1992) who found that the total abundance of aquatic invertebrates was positively correlated with deteriorating condition of the aquatic macrophyte Potamogeton nodosus Poiret. The authors suggest that this may be a response to plant age rather than condition. This seems





37

slightly antagonistic to the findings of Fowler (1985) who reported that the faunal composition and overall species richness was essentially the same for birch seedlings, saplings and trees. Possibly, this could indicate an important difference in the fauna of aquatic versus terrestrial habitats.

Faster change such as the seasonal growth pattern of plants can also impact the

final animal community. Hargeby (1990) compared the invertebrate communities on two species of macroalgae growing in the same water body and having roughly the same plant form. One species dies off every winter and is dominated by fast colonizing chironomids. The other species forms more permanent patches and becomes dominated by slower colonizers like Gammarus spp. In other words, the yearly die off leaves the invertebrate community in a permanently early successional state.

Other authors have found successional impacts to be more of a step-function response to structure rather than a gradual shift in response to time. Hurd and Fagan (1992) found a good example of this. They examined the spider assemblages along a gradient of four temperate successional communities. Diversity, richness and evenness exhibited a dichotomy between herbaceous and woody communities rather than a progressive change. Diversity, richness and evenness were all higher in younger fields, and clearly showed a step-function response to successional stage.

The interpretation of animal responses to successional changes is further muddied by the idea that stability across time and space is scale dependant. Ogden and Ebersole (198 1) found that artificial reef fish communities were variable if examined over short time periods or small areas. But if a study is expanded to cover large reefs or greater time spans, then a stable structure of species presence and abundance becomes apparent.





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Murdoch et aL (1972) compared the plant species and structural diversity in old-field habitats with the species diversity of homopteran insects. Any correlation was weak if one considered I-m2 patches but strong if one considered the whole field.



SUMMM

So what can be said about the interaction between the arena and the animal community? Undeniably, the natural world has a variety of possible shapes. For a variety of possible reasons, animals have been shown to interact with those shapes in a manner that seems to dictate the frequency of animal size classes. Any fundamental laws controlling this interaction have proven to be elusive. Absolute statements regarding passive size sorting, animal choice, trophic responses, competition, or succession as size determining factors have proven to be case specific and are often contradictory.

The implication is that these are modifying factors instead of basic underlying principles and that some other factor is in action that is strong enough to override the effects of these other factors. There is no need to assume one underlying principle. Nature is complicated enough to present us with multiple structuring principles, any or all of which could influence species abundance. Nonetheless, any factor that acts in a near universal manner needs to be considered as a potential organizing force. Predation has a near universal impact. Predation comes to nearly every animal species, at some point in its life, and should be examined as a potential basic controlling factor of the animal community structure. The problem is that the interaction between predation and arena is poorly understood.

Box-counting techniques have been purported to quantify the complexity of the environment and so suggest themselves as a potential technique to evaluate the





39

relationships between arena and the animal community. However, if predation is the structuring force in an animal community then naive measurements of the arena will not elicit much information on the structure of the animal community. First, we must understand how the structure of the arena affects the interaction between the predator and prey. Only once these mechanisms are known, can we begin to understand how they might shape a community of animals. So, if the techniques of fractal geometry can provide us with a fi-amework for understanding the structure of the arena, we should be able to make objective measurements and predictions.













CHAPTER 3
PERCEPTION OF COMPLEX SURFACES



Box-counting has emerged as the most common method of measuring the

dimension of objects. People who study chaotic dynamics have used it to measure the dimension of strange attractors. Ecologists have used it to measure the outline of habitat patches. Hydrologists have used it to measure water basins. And economists have used it to measure the fluctuation of prices over time. Yet no published guide exists on how to actually apply this technique. The result is haphazard application of different methods leading to often misleading or outright contradictory results. This has led to a backlash against fractal geometry as a tool in the natural sciences.



Rules for Box-Counting

"Then you should say what you mean," the March Hare went on. "I do,"
Alice hastily replied; "at least-at least I mean what I say-that's the
same thing, you know." "Not the same thing a bit!" said the Hatter. "Why,
you might as well say that 'I see what I eat' is the same thing as 'I eat
what I see'!" Alice's Adventures in Wonderland, Lewis Carroll

Box-counting can be a sensitive measure of dimension but one must be careful on how it is applied. Subtle nuances in meaning for the words 'complexity' and 'dimension' have led to strange results. Nobody would trust the numbers provided by someone who uses a yardstick without understanding what 'straight' or 'length' really means, yet numerous papers have been published that utilize box-counting without defining their technique or what surface is being measured.


40





41

Apparent Dimensio

Let us define an object's inherent complexity as the tendency for it to occupy

space-time. In this view, a lake is more complex than a puddle because it occupies more space over a greater length of time. Furthermore, let us define an object's ecological complexity as the tendency to occupy space-time of that part of an object that interacts with another object or observer. So a seagull flying over a wave-tossed lake perceives one level of complexity, which is different from that of a fish, swimming in the lake, that is different again from that perceived by a benthic insect crawling along the bottom of that same lake. All three animals perceive a different level of complexity but the inherent complexity of the lake has not changed. What has changed is the level of interaction, or the surface that is being interacted with. How can box-counting dimension measure this complexity?

Consider the dimensions of classical objects of Euclidean geometry (DE) and how a box-counting plot would appear if such perfect objects did occur. If we took a grid of squares and overlaid a perfect point (DE = 0), the point would always fall in one and only one square. Remember that we determine DB by using the following formulae introduced in the previous section,



Dg = lim logN(S)
s-+O
lo
S

We note that if N(s) always equals one, then log N(s) always equals zero and thus for a perfect point DB = 0. Next, imagine overlaying a grid on a perfect line (DE = I)- If we cover this line with N(s) = x grids, then s = l/x and therefore I/s = x. No matter how small we make s, DB will always equal log x/log x, i.e., DB = 1. Similarly, imagine





42


overlaying a grid on a perfect square (DE = 2). If the side of the grid (s) were equal to the side of the square, then it would only take one grid to cover the square. If we reduce s to 1/2, then it would take 4 grids to cover the square. If we reduce s to 1/3, then it would take 9 grids to cover the square. No matter how small we make s, N(s) = I /s2, and therefore DB = log (I/S2)/ log (1/s) = 2. Using similar arguments for a cube (DE = 3), we find that N(s) = I/S3 and that DB = 3. So, we see that for geometrically perfect (i.e. Euclidean) objects DB = DE.

Note that the plots can be expressed in units of length and area rather than the
2
count of boxes. We can convert N(s) into area by multiplying N(s) by s and convert N(s) into units of length by multiplying N(s) by s. But these two values are not the same. Careful judgment must be made to ensure you are measuring the interaction you think you are measuring. Consider a disc drawn on a piece of paper. From the point of view of anything else on that paper, all interactions with the disc occur at the edges and the disc is indistinguishable from a circle. It is only when we rise above the surface of the paper that we can perceive the middle of the object and measure the area. So that while the disc has an inherent complexity of DB = DE = 2, anything interacting with that disc would find that it had an ecological complexity of DB = 1, which is less than the Euclidean dimension (DE = 2). The dimension used to create an object will be termed the Inherent Dimension (DI), which is a fixed value for any given object or distribution. The ecological dimension can vary depending on how the object appears to a given observer so that it is here termed the Apparent Dimension (DA). Box-counting dimension can be used to estimate either D, or DA. Any dimension derived from the formula used to create the object is D, (e.g., area of a circle = BrD, D = 2). Any technique for measuring the





43

dimension of an object based on how that object is perceived is in fact measuring DA (e.g., circumference of a circle = 2B?), D = 1). DA and DI may be fractal or Euclidean or one of them may be fractal, while the other is Euclidean. This is not to say that Di cannot be determined from DA, only that they are related but different aspects of the object in question.

Minimum Size of Existence

So, with the cautionary explanation of the differences between inherent and ecological complexity out of the way, let us return to our thought experiments and consider what a box-counting plot would look like. Envision our previously considered perfect point. Euclidean geometry tells us that this object has a dimension of 0. It would be infinitely small and occupy only one point in space. Clearly, no such object exists in nature and it would be impossible to draw one. But imagine that we had such a point drawn on a piece of paper and we began to overlay it with our grid of squares. No matter how small we made the grid size (s), the point would always be covered by one and only one square. Our estimate of the size of the point would always be equal to the scale of measurement (s). There is of course, a small chance that the point would lay directly on the boundary between two grids, but if we assume our grid to be made up of perfect line segments, then this possibility approaches zero. A log-log plot of the estimated size of this point versus the scale of measurement (s) would have a slope of I and would intercept the x and y axis at the point [0,0]. Since this slope of I is equal to (1-DB), we see that the box-counting dimension is 0, and this accurately reflects the Euclidean dimension of a perfect point. Note, that for a perfect point DB = DE = DI = DA = 0-





44

The box-counting plot of a perfect point can be considered as a minimum line of existence. If an object exists, then it will intersect a minimum of one box. It is impossible to intersect less than one box. It is therefore impossible for an object to be measured as smaller than the scale of measurement. Only points above and to the left of the minimum line of existence can be measured. It is impossible to measure a point that lies below and to the right of the minimum line of existence (Fig. 6). This leads us to our first axiom about measuring objects with the grid-overlay method.

Axiom 1. No object can exist and be measured as being smaller than the minimum scale of measurement.


Minimum Line of Existence

lo


C
a 1


0.1
0.1 10
scale


Figure 6. Box-counting plot of a perfect point.


We might be able to infer the size of an object smaller than the minimum scale of measurement, but we cannot measure it as such. A corollary to this axiom is that all objects smaller than the minimum scale of measurement are indistinguishable from points and that the size of a point will always be overestimated. Also, note that this axiom applies to all measurement systems, not just the grid-overlay method of box-counting.





45


All Thin2s Are Relative

Let us now extend our thought experiment and consider a perfect line. By Euclidean definition a perfect line has dimension equal to I, is perfectly straight, and stretches to infinity. If we had an infinitely large grid, we could overlay the line and count out an infinite number of squares that the line intersects. Changing the scale of the grid (s) results in there still being an infinite number of squares. This approach does not provide us with graphable data, and so we must consider this problem from a different angle. Imagine that we are flying along our perfect line in a rocket ship that is traveling at a constant velocity (v). If we travel for a set period of time (t), then we will encounter a set number of grid squares. Multiply that number of squares by s and we get an estimate of the distance we traveled. If we cut the scale of measurement in half, then we will encounter twice as many squares on our journey and the estimate of the length of travel is unaffected. In fact, no matter how we change s, the apparent distance traveled is constant. While changing velocity and the time period does affect the distance traveled, this distance is always a constant with respect to s. A log-log plot of estimated distance versus scale has a slope of 0 and therefore DB = 1. Note that for a perfect line DB = DE = Di DA = I Similar logic will show that a perfect plane has a slope of -1, and therefore DB 2. Note, for a perfect plane DB = DE = D, = DA = 2.

But what about measuring real objects? The curious thing about real objects is that they are not equally complex at all levels of scale. The point, line and plane are unusual objects in that they involve infinities. The point is infinitely small, while the line and plane stretch out to infinity. Box-counting plots for these objects have constant slopes at all scales from 0 to 4. Such is not the case for objects of finite size. Let us





46

consider a perfect square with each side length equal to one hypothetical unit. The boxcounting plot of this square is shown in Figure 7. Moving to the left of the plot (decreasing scale), we see that the line is asymptotically approaching an estimated perimeter of four and that the slope is becoming 0. For reasons that will be discussed later, this estimate of perimeter is always an overestimate except at s = 0. We conclude from the slope that the box-counting dimension of a perfect square is I since DB is defined as s 4 0. The slope of this graph is never 0 except at s = 0. At scales larger than 0, the slope becomes increasingly different from 0. At first, the slope is slightly positive but as s approaches 1, the slope levels off and becomes negative. The implication is that at s = 0, the circle is clearly a line (DB =1). As s increases, the slope is slightly positive indicating that the circle is finite compared to a perfect line (DB<1). With increasing scale, the slope becomes negative, reflecting its two dimensional nature (DB>I). When s exceeds the size of the circle, the slope is positive again and the circle begins to resemble a point (DB=O).

A circle appears to have a fractional dimension at larger scales. This apparent

fractal nature of linear objects measured under large scales has been noticed before. It is termed a "fractal rabbit" and has previously been dismissed as an artifact of the technique (Kaye 1989). But it becomes an intriguing idea if this phenomenon is more am a mere artifact. What if this apparent complexity is in fact an accurate representation of our perception of the object? A square is intuitively more complex than a straight line. Added information is required to bend that straight line in on itself to form the square. For years, people imagined that the world was fiat. If we increase our scale of observation by moving higher into the air, we begin to perceive the curvature until finally





47

the world appears to be a sphere. Our perception of the complexity of objects is scale dependent.




Box-counting plot Unit Square 10

OF
to
0 OW 00

E




0.1 1 10
Scale


Figure 7. Box-counting plot of a perfect square with side length of I unit minimum line of existence; - estimated perimeter of square).



At the largest scales of observation, the apparent complexity of the square levels off and the box-counting plot of a square asymptotically approaches the minimum line of existence (Fig. 7). The interpretation of this behavior is that at small scales one does not capture the angles of the square in the analysis and the square appears little different from a straight line. As the scale increases, more of the angles are captured in each box and the apparent complexity increases as we begin to appreciate the two dimensional nature of a square. As the scale continues to increase, it becomes easier to consider the square as a single point in space and the apparent complexity drops and continues to drop until the object becomes indistinguishable from a perfect point. In fact, this behavior was





48

alluded to as a corollary of axiom one. If no object can be measured as being smaller than the scale of measurement, then all objects appear to be points if the scale of measurement is larger than the object. Mandelbrot (1983) used the analogy of a ball of string to point out that the dimension of a real object can change depending on how far away one is from the object. Viewed at from a great distance, our ball of string appears to be no more than a point (DE = 0). Moving closer, we see the ball has width, height, and depth (DE = 3). Even closer, we see that the ball has texture, and eventually we would see that the ball is composed of lines of string (DE = 1), which can appear to be small columns (DE = 3). The same object has a varying DE depending on scale of observation, in this case defined as distance from the object. If we were persistent, we could count N(s) at scales as small as the individual fibers and since DB is defined at the limit as s approaches zero, we might conclude that a ball of string is no more complex than a strand of hair. This approach would lose all information about complexity at larger scales.

The solution would be to use the Apparent Dimension (DA) to describe

complexity at these larger scales. We had previously defined DA Of an object as being relative to the surface of interaction. We can also define it as being relative to the scale of interaction across that surface. In other words, whereas DB was defined at the limit as s approaches zero, DA can be defined using the derivative with respect to s.

DA =I- d(length)
d(scale)

Defining DA as being relative to scale of observation results in ecological

complexity having no intrinsic meaning without reference to scale or range of scales. This is important enough to be called axiom 2.





49

Axiom 2. The ecological complexity of an object and its measure (DA) are strictly relative to the scale of observation.

Since no other dimension is defined as being scale specific, any D shown with

reference to any scale greater than zero can be assumed to be DA and the subscript A may be dropped for convenience sake (e.g., D[ 1 to 5mm] = DA I to 5 mm]). Rule of Averaging

Now that we have determined DA to be measurable at any scale, we run into the question as to whether to use minimum N(s) or mean N(s) to plot our graphs. A convenient feature about objects that extend or contract to infinity is that these two values are always equal. But for objects with finite size, N(s) is not only a:ftmction of s but can also vary based on the orientation of the object relative to the grid. By definition, the minimum value of N(s) needs to be used to plot the dimension but there is no known way to predict minimum N(s) for an object with a complex shape. As previously discussed, box-counting dimension was developed as an alternative to Hausdorff dimension because it was difficult to determine the minimum number of circles needed to cover an object. The implication is that box-counting uses an average number of squares since it was developed as an estimate of the undeterminable minimum number of circles needed to cover an object. Minimum N(s) can be extremely difficult to find for a complex shape, and so most researchers use mean N(s) without worrying about potential differences. Most of the time, mean N(s) will produce the same results as minimum N(s) except at scales near where two different scaling regions meet. Minimum N(s) produces a sharp transition between scaling regions while mean N(s) results in an asymptotic shift between regions.





50

This behavior might best be shown by considering box-counting plots of simple objects using both minimum N(s) and mean N(s). Figure 8 illustrates the box-counting plot of two points separated by some unit distance x =1. The longest distance within a grid cell is a diagonal cross-section. As long as this cross-section is smaller than the distance x, there is no way that the two points can fall in the same grid and N(s) always equals 2 and the estimated size of the two points is 2s. As soon as that cross-section exceeds x, then minimum N(s) will always be I and the estimated size of the two points will be Is. If we use mean N(s), then there is still a chance that the two points will fall into two separate grid cells. This probability decreases with increasing s and so the estimated size of the two points asymptotically approaches the minimum line of existence.


Box-counting Plot of two points

10
.0 0




0 I
0.1 1 10
Scale


Figure 8. Box-counting plot of two points set a unit distance aparL The solid line represents the minimum determined distance and the dotted line represents the mean determined distance.


Another thing to consider when working with minimum N(s) is that s needs to be equal to I/n the size of the object being measured, where n is an integer. Reconsider the





51

box-counting plot of a unit square, but this time include the distance determined from minimum N(s) as well as mean N(s) (Fig. 9). We see that the minimum N(s) plot resembles a series of octaves of regions with apparently increasing length. This happens because if the size of the object being measured is not a multiple of s sized grids, then one whole grid square is used to account for the fraction of length left over. Thus if we estimate length by multiplying minimum N(s) by s, we steadily overestimate length unless the object is a whole number multiple of s. Note that mean N(s) produces a smooth continuous line.




Box-counting plot Unit Square

10- Or
.0.0
.00


E
IL



0.1 1 10
Scale


Figure 9. Box-counting plot of a unit square. The solid line represents the minimum determined distance and the dashed line represents the mean determined distance.


Whether to use minimum N(s) or mean N(s) should be decided with great care and with careful consideration of exactly what it is that you wish to measure. If one is interested in the exact structure of an ob ect and knowing at exactly what point of scale





52

that the complexity begins to change, then one should use min N(s). If on the other hand, one wishes to study interactions across a complex surface and how probabilities are affected by changes in structure, then one should use mean N(s). To date, the only evidence to support this assertion is anecdotal. Imagine watching an object disappear into the distance. The object does not suddenly shift from a three dimensional object to a zero dimensional object. Rather, there is a gradual transition in size until the object begins to be too small to be seen. An observer would probably lose the object for a moment, as the probability of detecting it becomes low, possibly regaining it, only to eventually lose it completely. Minnaert (1954), who pointed out that colors begin to blend and merge as they are viewed from further away, expressed a sense of this averaging of perception. Thus Minnaert points out that dandelions on a lawn will appear to be a green-yellow mixture and that apple blossoms will appear dirty white when viewed at great distances. The averaging of perception also seems to be related to the ecological approach to perception founded by Gibson (pg. 25-26 in Goldstein 1989). Without going too deep into the topic of mind and perception, the ecological approach states that we do not perceive static images, but rather we perceive objects dynamically through time and movement. While both Minnaert and Gibson were talking about vision, this concept should extend to any interaction.

Axiom 3. The probable outcome of an interaction is determined as a continuous averaging fiction across scale.

In other words, the probability of an outcome is the integration of its ecological complexity across all scales of interaction. This statement appears to hold true whether the interaction is the mean N(s) value from a grid-overlay, a visual inspection of an





53

object, or a physical interaction across a complex surface. This can be shown by imagining a tall tree trunk growing straight. At a far distance we can only examine this tree at a large scale and it appears smooth edged. If we stick our nose right on the bark, we restrict our interaction to small scales and we can only see the complex surface. If we step back far enough to see the extent of the trunk and yet still notice the bark, we see that it is both smooth and rough with a continuous transition between them. We can say that it is smooth at this one large scale or that it is rough at this one small scale but we cannot pick a particular scale where rough becomes smooth. Theoretically, one could devise a scenario wherein a perfectly smooth surface had rough features below a certain resolution limit, but in the real world, such distinct boundaries do not exist. Now imagine that we are bouncing a ball off of the tree trunk. Providing that it is a large ball, the rebound will be predictable. On the other hand, a small ball would catch the irregularities of the bark and fly off in an unpredictable direction. Gradually increasing the size of the ball will gradually increase the predictability, but there will never be a sudden shift in probability. Thus, we see that DA, our visual inspection of the tree, and the probable outcome of physically rebounding objects off of it all show continuous shifts between different zones of complexity



TestinLy Real-World Validity

The three axioms developed in the preceding section have profound implications for our understanding of interactions. If valid, they imply that we will be able to a priori determine the probable outcome of an event providing that the event is based on the shape of the object. In the instance of this dissertation, the event we are interested in is the perception of a plant's complexity by an animal observer. The rest of this section will





54

present an experiment designed to show that box-counting, when looked at with an appreciation of the three axioms already discussed, can be used as a fair and objective measure of the human perception of complexity. Introduction

By definition, box-counting provides a number that is a dimension (DII). What people have assumed is that DBis, immediately translatable as complexity. But the term complexity is highly subjective even if we remain within an anthropomorphic point of view. Human beings, at least, have little difficulty in comparing two objects and deciding which is more complicated. This does not mean that all people are in complete agreement. All observers may unanimously separate the extremes of a series but neighboring objects in a series can be judged quite differently. While proponents of boxcounting argue that it provides us with a means of transcending the subjectivity, no one has tested the validity of the technique as a measure of subjective impressions.

This experiment will attempt to show that the perception of complexity is

measurable by box-counting and that the view of the environment is quantifiable in an objective manner. While some may argue that human beings might not be a good model species for making universal claims about animal perception, they do have the distinct advantage of being able to tell us their opinion. To this end, I provided a set of images and ask people to evaluate the complexity of the individual images. I then measured the images using box-counting techniques and compared the subjective human response with the calculated values.

Materials and Methods

The choice was made to work with aquatic plants based solely on the author's preexisting familiarity with the flora and fauna of aquatic environments. The twelve





55

species listed in Table 1 were chosen because they are all common Florida plants representing a diverse array of forms, yet all growing fully submersed and rooted to the substrate.


Table 1: List of plant species used to create images.
Hydrocharitaceae Egeria densa Planch.
Hydrilla verticillata (L.f.)
Vallisneria americana Michx. Najadaceae Najas guadalupensis (Spreng.) Magnus
Najas marina L.
Potamogetonaceae Potamogeton illinoensis Morong Potamogeton pusillus L.
Ruppiaceae Ruppia maritima L.
Ceratophyllaceae Ceratophyllum demersum L.
Haloragaceae Myriophyllum spicatum L.
Lentibulariaceae Utricularia inflata Walt.
Scrophulariaceae Bacopa caroliniana (Walt.) Robins.


Due to the inherent difficulty in three-dimensional imaging, two-dimensional approximations were used. Pressed and dried specimens of the above species were obtained from the Division of Plant Industry Herbarium, Florida Department of Agriculture and Consumer Services, Doyle Connor Building, in Gainesville, Florida. The herbarium pages were directly scanned using a Desk Scan II and were saved as PCX files. The image type was as black and white drawings with a resolution of 150 dpi x 150 dpi. The resulting images were then imported into Corel Paint and cleaned up so that all pixels within the outline of the plant were given values of 256 and all pixels outside the outline of the plant were given values of zero. The final images were printed at 100% scaling and were individually mounted on poster board for ease of handling.





56

Each plant was thus represented by a single image saved in two formats. There was a hard copy for visual evaluation and subjective rating as well as an electronic file copy for measuring by computer. Figure 10 shows the hard copy images at reduced scale.

Visual evaluation of these images was performed through the use of an informal survey. Staff and students from The University of Florida, Department of Entomology & Nematology, as well as the Department of Zoology were asked to participate. The pool of respondents was well mixed by age, race and gender but represented a biologically informed group of people. Each respondent was presented with the twelve boards containing the prepared images and asked to rate their complexity on a scale from I to 10, with 10 being the most complex. The respondents were told that they did not have to utilize all the numbers but could if they wanted to, and that ties were allowed. Respondents were questioned singly and given unlimited time to make their decisions. Complexity was not defined for them and no information as to the identification of the plants represented was provided.

Computer measurement of the images was more difficult than imagined.

Available programs that purport to measure the fractal dimension of an image were inadequate. Most of them would count all pixels within the image and not just the edges of the object being measured, thereby failing to measure the apparent dimension of the object. All of them produced a single value "D" regardless of the scale of measurement and none of them could be calibrated to give values for specific scales of measurement. New programs had to be written.





57




















A. B.











PI







C. D.

Figure 10. Final prepared images of plant forms for box-counting and survey comparisons (not to scale). A) Elodea densa, B) Hydrilla verticillata, C) Vallisneria americana, and D) Najas guadalupensis.





58





















E. F.




















G. H.

Figure 10 (Continued). E) Najas marina, F) Potamogeton illinoensis, G) Potamogeton pusilus, and H) Ruppia maritime.





59





















J.




















K. L

Figure 10 (Continued). 1) Ceratophyllum demersum, J) Myriophyllum spicatum, K) Utricularia inflate, and L) Bacopa caroliniana.





60


The end result was two Matlab@ computer programs that together provided two matched vectors. One vector is a list of counts of boxes that the edge of the object intersects, and the second vector is the scale of measurement (in mm) that resulted in that count. Program codes are included in the Appendix. These programs take the image and convert it to binary values where "1" represents a pixel on the object and "0" represents a pixel off the object. The programs then convert this image to a second image where only pixels on the edge of the object have a value of "1" and all other pixels are assigned a value of "0" (Fig. 11). Given the image resolution as input, the program calculates the size of one pixel in mm. It then divides the image into a matrix of n by n pixels and counts the number of squares that contain part of the image. The scale is matched to this count by multiplying the size of one pixel by n. The value of n is increased each iteration from 1 through 300 pixels. The number of grids containing positive values is tallied after each iteration. These tallies are appended to form a vector representing N(s) at scales ranging from 1 by 1 through 300 by 300 pixels.



oollll01ooooo ooo 111110III1oo00011 110110101100I11
000010000000000000000000111101110011111 0000100000000000000000001100101110011110
00001100000000000000000111111110001111111 0000110000000000000000011000110001110011
0001111000000000000000111111100111111110 0001111000000000000000110000100111011110
00001111000000000000011ll11i11110000 -+O 000010110000000000000110000111111110000
0001111111000000000011111IllIi ooo11000000000 0001100111000000000011000011111000000000
000111111111I000000I1111100(00000IO00 000111000111100000011001111000000000000
000001111111111000111 0IOOOOOOOOO000 000001100000111000110I10000000000000o
OOO0000001IlI1I111 1000000000000000000 0000001111000011011011000000000000000000
S000000000111 r1011110000111111111100000 0000000001111001011110000111111111100000
0000000000001111111000011111111111111000 0000000000001101111000011100000000111000
0000000000000111110001111111111111111100 0000000000000100110001111111111111111100
0000000000000111100111!11000000000111110 0000000000000100100111111000000000111110
0000000000001111111111100000000000001111 000000000000110011111100000000000001111
0000000000011111111100000000000000000011 0000000000011000000100000000000000000011

Figure 11. Diagrammatic representation of image preparation performed by Matlab program "yne.m".





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Results

Human evaluation of the plant images showed some variability but overall was fairly consistent (Table 2). Standard deviation was lowest on those images judged to have either low or high complexity. Responses to images judged to have intermediate complexity showed the highest standard deviations. The data is better appreciated graphically (Fig. 12).



Table 2: Response to survey on image complexity (N = 15).

Standard
Species Mean Deviation

Bacopa caroliniana 2.60 1.18
Vallisneria americana 2.80 1.26
Potamogeton illinoensis 3.67 1.40
Potamogeton pusillus 4.07 1.79
Najas guadalupensis 4.87 1.73
Egeria densa 4.87 2.07
Najas marina 5.07 1.62
Hydrilla verticillata 5.07 1.75
Ruppia maritima 5.60 2.26
Ceratophyllum demersum 7.60 1.59
Myriophyllum spicatum 8.00 1.41
Utricularia inflata 8.20 1.61


Box-counting data from the computer outputs was collected and distances of the images edges determined by multiplying the count (N(s)) by the scale of measurement

(s). Graphing distance as a function of scale on a log-log plot results in the standard boxcounting plots developed as the standard in determining fractal dimension (Dr) (Fig. 13). It is readily apparent that none of the plant images represents a truly fractal object since all of them approach a linear slope of zero (Dr= 1) at small scales. At larger scales, all of





62

the images increase in complexity at image specific rates. Slopes of these graphs were estimated and apparent dimension plotted as a function of scale (Fig. 14). The mean

response was plotted as a function of the maximum-recorded DA (Fig. 15). The resulting R2 value was low at 0.465, meaning that maximum dimension was a poor predictor of human response to complexity.




10
9
a) 8
C
._


S4
x
4)


3a 5
E



1 0



Species


Figure 12. Mean response from opinion poll on complexity of plant images. Error bars represent Vone standard deviation. B.c. = Bacopa caroliniana, V.a. = Vallisneria americana, P.i. = Potomageton illinoensis, P.p. = P. pusillus, N.g. = Najas guadalupensis, E.d. = Egeria densa, N.m = N. marina, H.v. = Hydrilla verticillata, R.m. = Ruppia maritima, C.d. = Ceratophyllum demersum, M.s. = Myriophyllum spicatum, and U.i. = Utricularia inflata..





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Perimeter Length Across Scale


100000


E 10000E _ _ _ _ _
1000100
0.10 1.00 10.00 100.00
Scale (mm)
- Egeria densa Vallisneda amnericana] A.


Perimeter Length Across Scale

100000


E 10000E
1000100
0.10 1.00 10.00 100.00
Scale (mm)
- Hydrilla verdIclata Najas guadalupensis B.

Figure 13. Estimated length of perimeter of plant images across scale. A. Egenia densa and Vallisneia americana. B. Hydrilla verticillata and Najas quadalupensis.





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Perimeter Length Across Scale

100000


E 10000
E

C
o 1000
-j

100
0.10 1.00 10.00 100.00
Scale (mm)
- Najas marina Potamogeton illinoensis C.

Perimeter Length Across Scale

100000


E 10000


1000
-J


100
0.10 1.00 10.00 100.00
Scale (mm)
- Potamogeton pusillus Ruppia maritima
D.

Figure 13 (Continued). Estimated length of perimeter of plant images across scale. C. Najas marina and Potamogeton illinoensis. B. Potamogeton pusillus and Ruppia maritime.





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Perimeter Length Across Scale

100000


10000
E


S 1000
.J


100
0.10 1.00 10.00 100.00
Scale (mm)
- Ceratophyllum demersum Myriophyllum spicatum E.


Perimeter Length Across Scale

100000


10000 ..
1

o 1000


100
0.10 1.00 10.00 100.00
Scale (mm)
- Utricularia inflata Bacopa carolineana F.

Figure 13 (Continued). Estimated length of perimeter of plant images across scale. E. Ceratophyllum demersum and Myriophyllum spicatum. F. Utircularia inflata and Bacopa carolineana.





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2.00


C
0
7R
C 1.50
E



1.00 7
-1 -0.5 0 0.5 1 1.5 2
log scale (mm)
- Egeria densa Vallisneria americana A.


2.00




S1.50
0
E
00


1.00

-1 -0.5 0 0.5 1 1.5 2
log scale (mm)

- Hydrilla verticillata Najas guadalupensis B.

Figure 14. Apparent dimension of perimeter of plant images across scale. A. Egeria densa and Vallisneria americana. B. Hydrilla verticillata and Najas quadalupensis.





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2.00


C 0
O

g 1.50
E



1.00 *
-1 -0.5 0 0.5 1 1.5 2
log scale (mm)
- Najas marina Potamogeton illinoensis C.


2.00


C 0
1.50
E



1.00
-1 -0.5 0 0.5 1 1.5 2
log scale (mm)
- Potamogeton pusillus Ruppia maritima
D.

Figure 14 (Continued). Apparent dimension of perimeter of plant images across scale. C. Najas marina and Potamogeton illinoensis. B. Potamogeton pusillus and Ruppia maritima.





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2.00


C 0
C 1.500 E



1.00
-1 -0.5 0 0.5 1 1.5 2
log scale (mm)

- Cera tophyllum dem.ersum MyHophyflum spicatum
E.


2.00


C 0
1.50- JO
E
Im


1.00
-1 -0.5 0 0.5 1 1.5 2
log scale (mm)

- Utticularia inflate -Bacopa caroline:a:na] F.

Figure 14 (Continued). E. Ceratophyflum demersum and Myriophyllum spicatum. F. Utircularia inflate and Bacopa carolineana.





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10

0 8




y *10.059x 10.374
0 R2= OA65
0 1
1.0 1.2 1.4 1.6 1.8 2.0
Maximum Dimension (Df)


Figure 15. Mean human estimate of image complexity as a function of the maximum apparent dimension.



Discusio

Even though the maximum Df represents the highest level of complexity that an object reaches, it is a poor predictor of the perception of complexity. A quick glance through the graphs displayed in Figure 14 shows why. Each plant's image has a seemingly unique pattern to the rise and fall of the complexity. Some images are steadily complex over the entire range of scales, while others are simple for the most part with only a peak in complexity. Maximum Df will not capture this distinction. If an overall evaluation of complexity is needed, then mean Df would provide more information.

Mean Df is easily approximated as I the slope of the log(distance) log(scale) plot (Fig. 13). The slope can be read directly from these graphs but it is more easily determined by converting all values for distance at scale to their log equivalents. The mean slope is then determined as the difference between the highest log distance value





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and the lowest log distance value, all divided by the difference between log scales at those distances. Plotting the mean survey value for the human response as a function of the mean Df proved to be highly significant with an R2 value of 0.958 (Fig. 16). This is highly predictive and provides strong support for the notion that people perceive overall complexity to be the average across all scales and that box-counting is a good tool for measuring this.




Human Perception of Complexity 10
y 16.497x 16.835
c 8
0 2
CL R 0.958
6

4
C
2

0
1.00 1.10 1.20 1.30 1.40 1.50 1.60
Mean Dimension (Df)


Figure 16. Mean human evaluation of complexity as a function of mean fractal dimension estimated from box-counting.



Conclusi,

Box-counting dimension appears to be a powerful estimate of an object's

complexity, but care must be used in its application. Dimension is not a method for separating objects based on their overall form. Rather, it separates objects based on one aspect of their structure, the relationship between perimeter and scale of observation.





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This is not a new finding; McLellan and Endler (1998) compared different methods for measuring and describing the shape of leaf outlines. Fractal dimension was well correlated with the ratio of perimeter to area but was not the best method for differentiating among shapes. In other words, two noticeably different objects can be equally complex and therefore have the same fractal dimension. This is noticeable by looking at the images of Ceratophyllum demersum, Myriophyllum spicatum, and Utricularia inflata (Fig. l0ijk). Each of these images is unique and easily discernable, but all were rated as having similar levels of overall complexity (Table 2 and Fig. 12), although the scale specific values of complexity differed (Fig. 14e,f).

The original thought had been that the maximum achieved complexity would

correspond to the mean human response. This had resulted from a misapplication of the second axiom of box-counting. While the complexity of the images is strictly relative to the scale of observation, this does not mean that we should attempt to evaluate observations at one particular scale. Rather, the scale of observation covered the entire range of scales from that of a single pixel to the scale where the appearance of the object begins to approach the zero line. Subsequently, people respond to the mean complexity over all these scales, which is a vindication of the third axiom. The probable outcome of the interaction (i.e., perception and evaluation of complexity) was determined to be a continuous averaging function across scale.

How then does this help us to understand the impact of the arena on predator-prey interactions? Consider the variables. We have a surface with a given dimension and complexity. We have a predator, potentially of almost any size, who must travel over and search the arena. And we have the prey, which also must travel the surface of the arena





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and can be of almost any size. In more mathematical terms, we have the box-counting function of the arena with two critical points, the scale of the predator and the scale of the prey. Theoretically, any change in predator size, prey size, or arena complexity can affect the overall interaction, but before we develop a mathematical model of their relationships, we need to review what is already in the literature.













CHAPTER 4
REVIEW: PREDATION AND SCALE

Prelude to the Three-Point Interaction

Perception, as examined in the previous chapter, is a two-point interaction. There is an object and a perceiver. Predation as examined here, is a three-point interaction; predator, prey, and arena. Two-point interactions are relatively simple and can be exactly calculated. Three-point interactions introduce the possibility of uncertainty. Insights into the subtleties of the three-point interaction can be glimpsed by trying to understand the various two-way interactions contained within the three-point problem. In this case, the potential two-way interactions are predator-prey, arena-predator, and arena-prey. This chapter reviews the literature on predation that highlights the effect that changing any of the three points has on these two-way interactions.



Predator-Prey Interactions

Parts of Predation

Predation as an act can be broken down into time spent in various activities, each of which can be affected by the other, by change in the sizes of predator and prey, or by changing the complexity of the environment. Searching, pursuit, handling, and digestion all require time investments. Search time is that time spent from when a predator decides to search for food to the time when food is found. For ambush hunters, this is the time spent waiting for prey to come by. Search time is negligible in prey rich habitats but in prey poor habitats it can represent the majority of a predator's time budget. Pursuit is


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that time between the detection of a potential prey item and its final capture, or to the point where the pursuit is cancelled in an unsuccessful hunt. Handling time is the time between the capture and final consumption of a prey item. Digestion time is usually the time between final consumption and the decision to start looking for food again. These categories are not strictly exclusive and can show considerable overlap. For instance, animals that inject venom to subdue their prey typically begin an extra-oral digestion at the same time. All of the parts of predation can be affected by changes in the three factors.

Intuitively, smaller prey items will be harder to find and will require more search time than an equivalent number of large prey items. It would seem more symmetrical if increasing predator size had the same impact as decreasing prey size, but increasing the size of the predator reduces the search time. Mittelbach (198 1 a) found that the number of prey items captured per second of search time increased linearly with a log increase in fish length. The exact relationship depended on the type of prey being hunted, but the general trend of increased feeding rate in response to increased fish length was always highly significant. Mittelbach interprets these results in light of research on the visual ability of fishes. Previous studies have found that the maximum distance from a prey item that will cause a fish to react increases with increased fish size as well as with increased prey length. Interpreted in this light, larger fish have a faster feeding rate because they are able to find more prey. Mittelbach's work provides some support for this in that the fastest feeding rates occurred in the open water experiments where there were no objects to obstruct the fish's field of view. An alternative explanation could allow for the ability of a larger predator to cover the ground faster. Ryer (1988) found





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that larger pipefish encountered more amphipods than smaller fish if relatively unobstructed. Not only were their fields of view larger but also larger fish were more likely to attack and more likely to be successful than smaller fish. The implication is that large fish were quicker than small fish. This resulted in larger pipefish having a greater consumption rate per unit body mass, than smaller fish feeding on the same prey.

Changing the relative sizes of predator and prey can dramatically affect handling time as well. The efficiency of an anthocorid predator feeding on aphids increased as the predator to prey size ratio increased (Evans 1976a). A more accurate measure of actual handling time showed that for naiads of a damselfly, Ischnura elegans (van der Lind), feeding on a cladoceran, Daphnia magna (Straus), handling time decreased linearly with increased damselfly length (Thompson 1975). Similarly, increasing the size of the prey increases the handling time resulting in a maximum size of prey for a given predator. Vince et al. (1976) found that the maximum size of prey eaten by the salt marsh killifish, Fundulus heteroclitus (L.), increased with increasing predator size so that growing killifish could consume ever-larger prey items. There is a maximum size for this fish species so that it was possible for prey items to escape predation if they managed to grow big enough. A prey item could also escape predation if it grows faster than its predator. Small instars of the big-eyed bug, Geocoris punctipes (Say), could not successfully attack large instar caterpillars of the tobacco budworm, Heliothis virescens (F.), but large instar bugs could consume large instar caterpillars (Chiravathanapong and Pitre 1980). Functional Restns
The idea of a predator's attack rate being influenced by the perceived density of prey items is almost the definition of a functional response. Since this dissertation is attempting to determine the relative protection value of plant forms, any changes to





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predation rate that incur with increasing prey density are important. A full review of the literature on functional responses is beyond the scope of this report and interested readers are referred to Holling (1961) for an excellent introduction and review of the subject. In general, changes in the relative sizes of predator and prey result in corresponding changes to the attack coefficient and handling times. This has the impact of shifting the functional response to either the left or the right of the prey density gradient. This was well documented by Thompson (1975), who examined naiads of Ischnura elegans feeding on Daphnia magna. Attack coefficient "a" increased and handling time Tb decreased linearly with increased damselfly length, resulting in a shift in the Holling type 2 functional response.

Nuances of behavior can also influence the functional response. Heimpel and Hugh-Goldstein (1994) examined the functional response of nymphal predatory pentatomids feeding on larvae of the Colorado potato beetle, Leptinotarsa decemlineata (Say). The pentatomids did not show a typical functional response. At low densities of beetle, predation rate starts out high but then drops as prey densities increase. Predation rate bottoms out and then rises again as prey densities increase. The apparent cause of this pattern is that pentatomid nymphs exhibit area specific searches after a successful kill. At high prey densities, this results in faster location of clumped prey, but at low densities, it merely wastes time. This doesn't explain the initial high success rate at low prey densities. Prey behavior can also impact on the functional response. Tostowaryk (1972) found that the attack rate of pentatomids on sawfly larvae peaked at intermediate prey densities and then declined. This was because the larvae were able to defend themselves more efficiently when in a large aggregate.





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The impacts of changes in the arena on the functional response have been less well documented. Occasionally, an arena change can change the type of functional response demonstrated instead of just shifing it. Lipcius and Hines (1986) found that predatory crabs feeding on soft-shelled clams showed a type III response in sand (sigmoid) and a type II response in mud. Topographically similar conditions produced different responses. As previously stated, this was thought to be due to probing action of crabs being hindered by sand. The probing action is a chemosensory searching action, so the impact is on the crabs' perception, and does not affect physical movement or prey capture. The result is increased search times without any change in the handling time.

Handling time does not always impact functional responses. Wiedenmann and O'Neil (199 1) compared the functional response of predatory pentatomids in simplified lab settings versus the field situation. The results differed markedly. Lab studies indicated that the pentatomids could attack 14 larvae each per day, but they only achieved

4 or 5 per day in the field. There are different limitations at work here. Search time in a petri dish is essentially zero, so that lab studies highlight the effect of handling time. In the wild, search times are so long that handling time has virtually no effect on the functional response.

Q12timal Foraging

Like the functional response, optimal foraging is concerned with predation from the predator'spoint of view. But while the functional response attempts to numerate how many successful attacks a predator will make on a given prey item, optimal foraging theory attempts to determine how much attention a predator will give to different possible prey items or feeding strategies. Loosely stated, optimal foraging theory speculates that animals feed in a manner that best maximizes food intake while minimizing energy





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expenditures. The origins of optimal foraging theory can be traced to a pair of works published sequentially in the same journal. Emlen (1966) and MacArthur and Pianka (1966) presented the idea that animals were adapted to behave in a manner that maximized their net energy gain. The idea that mathematical models could predict feeding decisions made by an animal caught the imagination of scientists and hundreds of papers have since been published on the topic. Numerous subtopics have been developed, but in general, they can be lumped into four categories (Pyke et al. 1977); 1) Optimal type of food, 2) Optimal choice of patch, 3) Optimal time spent in a patch, and 4) Optimal movement patterns and speed.

The focus of the present study is the complexity of surfaces and how variously sized predators and prey interact across them. Therefore, the concern here is with what happens to the prey when a predator is already within a patch. Inter-patch decisions can play an important part on the optimal foraging of the predator, but these decisions are dependent on, rather than causative of the small time-scale interactions that are our primary focus. It is for this reason that we restrict our attention to predators and why they feed on particular types of prey, i.e., optimal type of food.

When an animal makes the choice as to which item to eat, we might be able to predict that choice based on the amount of energy "profit" available in that item. Profitability of an item is usually defined as the net food value of an item divided by the handling time needed to consume it. Small handling time thus increases the profitability of a food item by allowing a faster rate of consumption. While it is very straightforward to declare that an animal should always feed on the best item, it becomes a more difficult question to determine how many of the lower quality items to include in the diet. In





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general, if animals are presented with a simple choice between food items of different profitability, they will choose the most profitable one (Krebs 1978).

Profitability defined by the relative value of a single individual does not include all aspects of predation costs. Obviously, some time is spent searching and chasing prey items before handling time becomes important. If we define profitability in terms of a rate of net energy intake, we then have a more dynamic measure of a food type's profitability that takes into account its relative frequency. Time spent searching is inversely proportional to the encounter rate, which is a function of the foo&s density and the speed of the searching animal. Numerous examples of animals feeding optimally have been published in the literature. One of the earlier and best-documented cases involves the bluegill, Lepomis macrochirus (Rafinesque), feeding on different sized prey. Werner and Hall (1974) examined the feeding behavior of bluegill in an aquarium, presented with Daphnia of three different size classes. If all densities were low, then the encounter rates were also low and the fish made no selection. If all three size classes were presented at high densities, then the fish overwhelmingly preferred the largest class. At intermediate levels the fish chose the two larger size classes and ignored the smallest size class. In other words, if the fish were not being kept busy feeding on the larger size classes, they would take smaller size classes in order to maintain energy flow. They were maximizing the energy intake per unit time. This aquarium study and the model developed to explain it received important validation from a study that examines the growth of bluegill in a natural lake and compared it to the energy intake predicted by the model as well as that predicted by random feeding (Mittelbach 198 1 a, 1983). The growth





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rate of the wild fish correlated with the energy intake predicted by the optimal foraging model but did not correlate with the energy intake predicted by random foraging.

A special note should be made here of the case of ambush predators where the

encounter rate is a function of the prey's speed of movement and not the predator's. Here the search time is the sit and wait time. Ambush predators too have been found to feed optimally and a classic example is that of mantids. Charnov (1976) found that mantids fed optimally according to the rational that any food item was optimum if its energy content divided by the handling time was greater than the total available energy of all possible food items divided by the total time spent waiting and handling prey.

Occasionally, a study reveals an animal that is apparently feeding in a sub-optimal manner. For example Goss-Custard (in Krebs 1978) examined the foraging strategy of the redshank, Tringa totanus (L.). He found that large polychaete worms were eaten in direct proportion to their own density but that smaller worms were not eaten in proportion to their own density but in inverse proportion to the density of the larger worms. So far this is classic optimal foraging but the redshank preferred a sub-optimal amphipod above any of the worms. It is thought, but not proven, that this apparently bad food choice was a function of mixed currencies. That is to say that while energy was one currency of optimization, there could also have been another currency such as a particular nutrient. The particular feeding pattern observed may not be optimal on energy alone but might represent an optimum mix to maximize the input of both energy and nutrients.

Switching is another phenomenon that can result in an apparently sub-optimal feeding strategy. Switching was coined by Murdoch (1969) and refers to the situation where a predator feeds on a disproportionate number of the commonest prey. If a





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different prey becomes the most common, the predator will often switch to specializing on this second prey. Murdoch and Oaten (1975) as well as Murdoch et al. (1975) did finther work on the topic. A possible explanation for switching is that a predator may become more efficient by specializing all efforts at catching a particular species of prey. Lawton et al. (1974) found that Notonecta individuals became better at attacking mayfly nymphs as they became experienced. The percentage of attacks that were successful increased with time indicating that experience may improve predator foraging behavior.

Few studies have examined how the structure of the environment affects the optimal foraging of a predator. Indeed, most studies seek to minimize structure in the testing arena in order to examine the decision process in isolation from any habitat effect. While the shape of the environment may not affect the amount of energy represented in a prey item, it can certainly affect almost all other components of optimal foraging models. One of the first studies to investigate the effect of structure on optimal foraging was Mittelbach (1983). In the laboratory experiments, bluegill were placed into aquaria with three different possible structures. The least complicated structure was a plain aquarium filled only with water. The medium complexity structure involved an aquarium with 4-5 cm of marl sediment. The most complex structure involved an aquarium with live, anchored plants. For all three situations, handling time increased exponentially with increasing prey length, but the slope of a log-linear regression increased with increasing complexity of environment, indicating that larger prey were increasingly more difficult to handle in more complex habitats. Not only handling times, but encounter rates were also affected by the complexity of the environment. Lower complexity led to higher encounter rates. This affect was greater for larger fish. This suggests that the encounter





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rate was a function of the size of the predator interacting with the complexity. Caution should be used in interpreting this paper. Each of the habitat complexity levels had a different prey species. While length of each type of prey showed the same response, inter-habitat comparisons could be biased, negating the conclusions on complexity. Additionally, complexity was not quantified, so that later papers had to arrive at the values of handling time and encounter rates by experimental analysis rather than measurement (Werner and Mittelbach 1981; Mittelbach 1983). Prey Size Affected by Predation Technique

While an optimal foraging strategy can definitely determine the size of prey item selected, a change in the manner of predation will also impact size of prey consumed. Schmitt and Coyer (1982) examined the foraging ecology of two sympatric fish in the genus Embiotoca (Embiotocidae). Both species are roughly the same size and occur together in temperate marine reefs and both share similar diet. Embiotocajacksoni Agassiz fed primarily on tubicolous amphipods and was able to separate the amphipods from surrounding medium. It fed primarily on small but numerous species and the mean weight of species in its gut contents was not different from that of a random sample of the environment. Embiotoca lateralis Agassiz was a more open water feeder and actively hunted the larger and rarer prey species. The mean weight of the individual prey items in its gut was significantly larger than that of E. lateralis of the same length.

It is more difficult to show a similar effect within individuals of the same species since the switch between potential prey items is tightly connected with optimal foraging strategies. Murdoch et al. (1975) found that guppies, Poecilia reticulates Peters, would switch feeding behaviors based on optimal foraging decisions. When the fish were offered either Drosophila flies on the water surface or tubificid worms on the aquarium





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bottom, they would feed in the manner that offered the greater rewards. In other words, a change in the fish behavior resulted in a prey species bias.

It can also be shown that a particular feeding behavior has an impact on the size frequency of prey items selected. Bluegill take more big prey than small prey. Experiments using Daphnia magna as prey items show that the number of a specific size class taken exactly matches that predicted by a model that uses the apparent size (Werner and Hall 1974). Bluegill are visual predators and always feed on the prey item that appears to be larger either because of its size or because of its proximity (O'Brien et a!. 1976). If large prey is common, then the predators will specialize in feeding on them because the large prey have a greater probability of appearing larger. As large prey becomes less common, small prey items are fed upon with higher frequency since their probability of appearing larger increases. The notion of feeding on prey "'as encountered" is affected by the apparent size of an organism.

The technique of feeding on the apparently larger prey item has profound impact on the prey species. Zaret and Kerfoot (1975) noted that predators of a waterflea, Bosmina longirostris (OF Muller), primarily used visual hunting techniques. The waterfleas were mostly transparent except for the black pigment of the eye. Ideally, large waterfleas could see food better than small waterfleas, but being large increased their vulnerability to visual predators. Transparency reduced their vulnerability, but to see food requires eye pigment. The result of being visually hunted resulted in large transparent prey with pigment only where it is needed. Presumably, a filter-feeding predator would result in populations of smaller waterfleas with more pigment.





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Hunting strategy can interact with prey behavior to have an impact on size selection. Another waterflea species, B. longispina (Leydig), reacts to a predator by swarming (Jakobsen and Johnson 1988). The larger waterfleas would move faster into the swarm resulting in smaller waterfleas being near the margins and being disproportionately fed upon by sticklebacks.



Impact of Arena on Predator-Prey Interactions

Price et al. (1980) list four different ways in which the plant structure can impact predators. Predators might have different attack rates on different plant species (Haynes and Butcher 1962, Miller 1959, Monteith 1955, Weseloh 1976) or attack rates can vary over different parts of the same plant (Askew 1961, Askew and Ruse 1974, Dowden et al. 1950, Evans 1976b, Weseloh 1976). Plants might provide structural refuges for the prey items (Arthur 1962, Ball and Dahlsten 1973, Bridwell 1918,1920, Graham and Baumhofer 1927, Levin 1973, Mitchell 1977, Pimentel 1961, Porter 1928, Wangberg 1977, Washburn and Cornell 1979) or simply interfere with enemy search movement (Bequaert 1924, Darlington 1975, Ekborn 1977, Hulspas-Jordan and van Lenteren 1978, Katanyukul and Thurston 1973, Levin 1973, Rabb and Bradley 1968, Webster 1975, Woets and van Lenteren 1976).

Changing the relative sizes of predator and prey can dramatically affect handling time. The efficiency of an anthocorid predator feeding on aphids increased as the predator to prey size ratio increased (Evans 1976a). A more accurate measure of actual handling time showed that for naiads of a damselfly, Ischnura elegans (van der Lind), feeding on a cladoceran, Daphnia magna (Straus), handling time decreased linearly with increased damselfly length (Thompson 1975). Similarly, increasing the size of the prey





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increases the handling time resulting in a maximum size of prey for a given predator. Vince et al. (1976) found that the maximum size of prey eaten by the salt marsh killifish, Fundulus heteroclitus (L.), increased with increasing predator size so that growing killifish could consume ever-larger prey items. There is a maximum size for this fish species so that it was possible for prey items to escape predation if they managed to grow big enough. A prey item could also escape predation if it grows faster than its predator. Small instars of the big-eyed bug, Geocoris punctipes (Say), could not successfully attack large instar caterpillars of the tobacco budworm, Heliothis virescens (F.), but large instar bugs could consume large instar caterpillars (Chiravathanapong and Pitre 1980).

Changes in the arena can have dramatic effects on search time. Physically preventing or hindering the movement of predators greatly increases search time. Walking speed of two predators, a coccinellid, Coleomegilla maculata (De Geer), and a lacewing, Chrysopa carnea Stephens, were seriously reduced on tobacco as compared to cotton (Elsey 1974). This was observed to be due to the glandular trichomes on tobacco hindering movement. A similar phenomenon seems to occur in predatory crabs hunting for soft-shelled clams (Lipcius and Hines 1986). The crabs' success rate rapidly increases with increasing clam density if the crabs are probing in mud. But if they are probing in sand, their success rate is reduced and lags behind increases in clam density. The speculation was that this was due to probing action of crabs being hindered by sand, leading to increased search times.

Since prey density has an effect on search time, changing the size of the arena will impact on the predator success rate. Sometimes this change in density is not immediately obvious. Need and Burbutis (1979) examined the searching efficiency of a parasitic





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wasp, Trichogramma nubildle Ertle and Davis, searching for eggs of the European corn borer, Ostrinia nubialis (Hubner). In the field, corn leaves gradually expand resulting in more area for the wasps to search. Initially, percentage parasitism increases with time, but eventually there is a point in the season when percentage parasitism begins to steadily drop. In controlled lab experiments, increasing the arena size decreased density of the eggs and as predicted, decreasing the density of eggs decreased the parasitism rate in a linear fashion.

More relevant to the theme being explored here is that it is possible to change the search time by altering the complexity of the arena. Andow and Prokrym (1990) also examined the hunting behavior of T nubilale looking for egg masses of the European corn borer. The hunting surface was a waxed paper onto which egg masses of the European corn borer had been laid. The simple hunting surface was one of these papers folded once and standing on end. A complex surface was one of these papers folded numerous times and standing on end. Parasitism rates were 2.9 times higher on the simple surfaces. With no hosts present, search time was 1.2 times longer on complex surfaces implying an effect on giving up time. Wasps found hosts on simple surfaces 2.4 times faster than on complex ones. Keep in mind that the original pieces of waxed paper had the same dimension. In our terminology, the two papers had the same Euclidean area, but the multiple folded piece of paper had a greater apparent dimension.

While it is normal to think of structural effects on the basis of what their final outcome is, it is more germane to our current discussion to consider the effects on the basis of how they impact predators and prey. To this end, we can divide structural effects into two categories of behavioral and mechanistic effects. Behavioral effects are changes





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in the behavior of either the predator or the prey that are caused by changes in structure and mechanistic effects are defined as differences resulting from changes in structure without a change in the behavior of either the predator or prey. Note that these categories are not meant to be mutually exclusive and that combinations can occur. Behavioral Effects

It is easy to assume that there are more animals in complex areas because the added cover provides a refuge from predation. But we must be careful to separate between the effects of refuge on the differential survival of the prey and some aspect of the structure causing the prey to choose to be in the area. Prey items could be attracted to particular structural characteristics for a variety of reasons. Rejmankova et al. (1987) attributed the distribution of larvae of the mosquito Anopheles albimanus Wiedemann to opposition choices made by the adult. They used cluster analysis to define 16 different larval habitats and found that the mosquito was most common in habitats with emergent graminoids. The presence of filamentous algae or small floating plants was detrimental to them. Habitats that were complex on a large scale (e.g., mangrove roots) were not favorable to the larvae.

Most behavioral reactions probably are predator avoidance responses. The

avoidance of habitats that were complex on a large scale by adult mosquitoes might be considered an adaptive response considering that these areas are readily accessible by fish. Additionally, the presence of a predator may exaggerate a behavior already present. Lynch and Johnson (1989) found bluegill sought artificial pipes for shelter regardless of predators present or not present, but that a predator being present did result in an increase in the rate of shelter seeking. In this situation, intimidation by predators is visual so it was not surprising that turbid water resulted in less shelter seeking.





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It is also not surprising that behavioral responses to changes in complexity have been found to be species specific. Edgar and Robertson (1992) found that when leaves or epiphytes were removed from stems in Australian seagrass beds, some species of mobile epifauna increased and some species decreased. Most species reacted to either the reduction of leaves or epiphytes, but not to both. Decreasing the seagrass density had the same effect on the same species. Cage exclusion in the wild showed that the cause was from active choice and not predator mediated.

Similarly, predators can be attracted to particular structures. It has been argued that areas of greater complexity attract predators because they provide an increased abundance of attachment sites for the associated fauna's food (Abele 1974; Hicks 1980). There is some evidence that this attraction occurs, since mixed vegetable crops attracted and sustained higher populations of predatory anthocorids (Hemiptera) than monocultures (Letoumneau 1990). Prey was not the attractant though, since densities of prey items were similar. C~rcamo and Spence (1994) did not measure plant complexity, but they did find that different crops resulted in different predation pressures. They found this to be the result of the ground beetles being differentially attracted to the different crops and not because of any changes in their hunting efficiency.

Aside from choosing to be in areas of particular complexity, predators can change their behavior when presented with differing complexities. Cloarec (1990) found that the presence of the aquatic macrophyte Hydrilla verticillata caused belastomatids (Hemiptera) to switch from active hunting to ambush. This effect was independent of hunger level. There was no testing of different density levels and no alternate plant types were tested, but these results do indicate that changes in the amount of structure in the





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habitat can change predator behavior. Stoner (1979) had previously found that Pinfish became more selective with increasing macrophyte biomass, resulting in different prey being selected. Different parts of the same plant can cause similar responses. Gardner and Dixon (1985) found that a parasitic wasp searching for aphid hosts on wheat would search each leaf equally, but that the ear of the plant was searched less often. Any aphids on the ear of the plant would be somewhat protected from attack by the presence of the seeds. One explanation was that the wasp was choosing to search areas where aphids had less protection.

Just as in the prey, predator response to change in habitat complexity is also

species specific. Frazer and McGregor (1995) used dowels to mimic plant structure and examined the behavior of various coccinellid (Coleoptera) species on these surfaces. They examined movement speeds and the frequency of specific directions of movement. Tendency to move up or tendency to move to the top of an object varied in a speciesspecific manner. The searching effectiveness of coccinellids would vary between plants as well as between species if these results held for natural plants.

Care must be taken when evaluating behavior in that what appears to be a

difference in behavior might in reality be the same behavior operating at a different scale. Price (1983) developed a mathematical model to predict patch choice by "predators". He validated the model with data from seed-eating desert rodents. In other words, the size of the predateo' rodents determined the size of the seed patch that the predateo' would use. Animals of different sizes would appear to be making different behavioral decisions even thou they were operating under the same mathematical rules. Lynch and Johnson (1989) found a similar phenomenon. As cited above, they studied shelter seeking by





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bluegills in artificial pipes. They found that large fish sought large pipes and that small fish sought small pipes even though they could fit into larger pipes and still receive shelter. Apparently, the fish had some sort of ideal surrounding area to body size ratio that resulted in size selection of refuge.

Mechanittic Effects

Behavior is an important and interesting field of study with many questions yet to be answered, and it does dovetail neatly into our discussion on complexity and predation. However, the central phenomenon in consideration is interaction across a complex surface. Behavior is affected by complex surfaces, but it is generally of interest because it can prevent the interaction. Predator-prey interaction is physical and questions on size, frequency, density, and complexity need to be also considered in mechanistic terms.

Fortunately, there are already many studies published that consider predation in mechanistic terms. These can be divided into two categories. There are those studies that compare changes in the amount of structure. Then there are studies that consider changes in the form of the structure. The concepts of complexity and dimension are contained within both types of study and so both types need to be considered. Changes in density

Many studies interchange the definitions of complexity as density versus complexity as form. Certainly, the broad definition of complexity, as an object's tendency to occupy space-time, supports the obvious notion that a dense patch of vegetation is more complex than a loose patch. But the two definitions can be measured separately on a box-counting plot. Both concepts are scale specific and interrelated and both concepts can affect the predator-prey interaction. There have been a number of studies on each meaning of complexity, but sweeping generalities have proven to be





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elusive. It has proven to be easier to draw conclusion about density since density is easier to measure than dimension.

The simplest experiments on the impact of density compare the feeding

efficiencies of predators in empty versus heavily planted arenas. The most commonly reported phenomenon is that the presence of structure does not alter the fundamental dynamics but rather draws them out over time and merely delays the final outcome. This has been found to be true for European perch, Percafluviatilis L., which quickly eliminated large predatory invertebrates from areas lacking vegetation, but needed more time to have the same impact in vegetated areas (Diehl 1992). The impact is thought to represent a slowing down of the predator either by increasing the amount of area it would need to search or by physically impeding its movements. Luckinbill (1973) demonstrated this with laboratory populations of Paramecium aurelia Ehrenberg and its predator Didinium nasutum OF Muller kept in culture. Without structure, the situation was unstable. Populations would fluctuate in cycles of increasing amplitude until one or the other of the predator and prey went extinct. Structure was added in the form of methylcellulose, which slowed the frequency of contact but offered no preferential degree of movement to either species. The fundamental dynamics did not change, but were prolonged over time. Sometimes, this prolongation may be enough to allow for the survival of a prey species. Russ (1980) increased density of structure by placing models of arborescent bryozoans on a coral reef. Fish foraging efficiency was reduced and the survival of colonial ascidians was enhanced.

A problem with the all or no vegetation experiments is that we cannot note the effects of intermediate levels of structure in the environment. Most studies have found