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Biology Essay, Research Paper
Term paper: Principles of Ecology 310L
New Ecological Penetrations:
The Application of Fractal Geometry to Ecology
7 December 1995
New penetrations into the natural universe are merely a few of the consequences from the usage of fractal geometry.
Examples from population and landscape ecology are used to exemplify the utility of fractal
geometry to the field of ecology. The coming of the computing machine age played an of import function in the
development and credence of fractal geometry as a valid new subject. New penetrations gained from
the application of fractal geometry to ecology include: understanding the importance of spacial and
temporal graduated tables ; the relationship between landscape construction and motion tracts ; an increased
apprehension of landscape constructions ; and the ability to more accurately exemplary landscapes and
ecosystems. Using fractal dimensions allows ecologists to map carnal tracts without making an
unwieldy flood of information. Computer simulations of landscapes provide utile theoretical accounts for
deriving new penetrations into the coexistence of species. Although many ecologists have found fractal
geometry to be an highly utile tool, non all concur. With all the new penetrations gained through the
appropriate application of fractal geometry to natural scientific disciplines, it is clear that fractal geometry a
utile and valid tool.
New penetration into the natural universe is merely one of the consequences of the increasing popularity and usage of
fractal geometry in the last decennary. What are fractals and what are they good for? Scientists in a
assortment of subjects have been seeking to reply this inquiry for the last two decennaries. Physicists,
chemists, mathematicians, life scientists, computing machine scientists, and medical research workers are merely a few of
the scientists that have found utilizations for fractals and fractal geometry.
Ecologists have found fractal geometry to be an highly utile tool for depicting ecological
systems. Many population, community, ecosystem, and landscape ecologists use fractal geometry as
a tool to assist specify and explicate the systems in the universe around us. As with any scientific field, there
has been some discord in ecology about the appropriate degree of survey. For illustration, some
being ecologists think that anything larger than a individual being obscures the world with excessively
much item. On the other manus, some ecosystem ecologists believe that looking at anything less than
an full ecosystem will non give meaningful consequences. In world, both positions are right.
Ecologists must take all degrees of organisation into history to acquire the most out of a survey. Fractal
geometry is a tool that bridges the & # 8220 ; spread & # 8221 ; between different Fieldss of ecology and provides a common
Fractal geometry has provided new penetration into many Fieldss of ecology. Examples from population
and landscape ecology will be used to exemplify the utility of fractal geometry to the field of
ecology. Some population ecologists use fractal geometry to correlate the landscape construction with
motion tracts of populations or beings, which greatly influences population and
community ecology. Landscape ecologists tend to utilize fractal geometry to specify, depict, and
theoretical account the scale-dependent heterogeneousness of the landscape construction.
Before researching applications of fractal geometry in ecology, we must foremost specify fractal geometry.
The exact definition of a fractal is hard to trap down. Even the adult male who conceived of and
developed fractals had a difficult clip specifying them ( Voss 1988 ) . Mandelbrot & # 8217 ; s foremost published
definition of a fractal was in 1977, when he wrote, & # 8220 ; A fractal is a set for which the
Hausdorff-Besicovitch dimension purely exceeds the topographical dimension & # 8221 ; ( Mandelbrot 1977 ) .
He subsequently expressed sorrow for holding defined the word at all ( Mandelbrot 1982 ) . Other efforts to
gaining control the kernel of a fractal include the undermentioned quotation marks:
& # 8220 ; Different people use the word fractal in different ways, but all agree that fractal objects
contain constructions nested within one another like Chinese boxes or Russian dolls. & # 8221 ; ( Kadanoff
& # 8220 ; A fractal is a form made of parts similar to the whole in some way. & # 8221 ; ( Mandelbrot 1982 )
Fractals are & # 8230 ; & # 8221 ; geometric signifiers whose irregular inside informations recur at different scales. & # 8221 ; ( Horgan
Fractals are & # 8230 ; & # 8221 ; curves and surfaces that live in an unusual kingdom between the first and
2nd, or between the 2nd and 3rd dimensions. & # 8221 ; ( Thomsen 1982 )
One manner to specify the elusive fractal is to look at its features. A cardinal feature of
fractals is that they are statistically self-similar ; it will look like itself at any graduated table. A statistically
self-similar graduated table does non hold to look precisely like the original, but must look similar. An illustration of
self-similarity is a caput of Brassica oleracea italica. Imagine keeping a caput of Brassica oleracea italica. Now break off a big floweret ;
it looks similar to the whole caput. If you continue interrupting off smaller and smaller flowerets, you & # 8217 ; ll see
that each floweret is similar to the larger 1s and to the original. There is, nevertheless, a bound to how little
you can travel before you lose the self- similarity.
Another placing feature of fractals is they normally have a non- whole number dimension. The fractal
dimension of an object is a step of space-filling ability and allows one to compare and categorise
fractals ( Garcia 1991 ) . A consecutive line, for illustration, has the Euclidean dimension of 1 ; a plane has the
dimension of 2. A really jaggy line, nevertheless, takes up more infinite than a consecutive line but less infinite
so a solid plane, so it has a dimension between 1 and 2. For illustration, 1.56 is a fractal dimension.
Most fractal dimensions in nature are about 0.2 to 0.3 greater than the Euclidian dimension ( Voss
1988 ) .
Euclidian geometry and Newtonian natural philosophies have been profoundly frozen traditions in the scientific
universe for 100s of old ages. Even though mathematicians every bit early as 1875 were puting the
foundations that Mandelbrot used in his work, early mathematicians resisted the constructs of fractal
geometry ( Garcia 1991 ) . If a construct did non suit within the boundaries of the recognized theories, it
was dismissed as an exclusion. Much of the early work in fractal geometry by mathematicians met
this destiny. Even though early scientists could see the abnormality of natural objects in the universe around
them, they resisted the construct of fractals as a tool to depict the natural universe. They tried to coerce
the natural universe to suit the theoretical account presented by Euclidean geometry and Newtonian natural philosophies. Yet we
all know that & # 8220 ; clouds are non domains, mountains are non cones, coastlines are non circles, and bark is
non smooth, nor does lightning go in a consecutive line & # 8221 ; ( Mandelbrot 1982 ) .
The coming of the computing machine age, with its sophisticated artworks, played an of import function in the
development and credence of fractal geometry as a valid new subject in the last two decennaries.
Computer-generated images clearly show the relevancy of fractal geometry to nature ( Scheuring and
Riedi 1994 ) . A computer- generated coastline or mountain scope demonstrates this relevancy. Once
mathematicians and scientists were able to see graphical representations of fractal objects, they
could see that the mathematical theory behind them was non capricious but really describes natural
objects reasonably good. When explained and illustrated to most scientists and non-scientists likewise, fractal
geometry and fractals make sense on an intuitive degree.
Examples of fractal geometry in nature are coastlines, clouds, works roots, snowflakes, lightning, and
mountain scopes. Fractal geometry has been used by many scientific disciplines in the last two decennaries ; natural philosophies,
chemical science, weather forecasting, geology, mathematics, medical specialty, and biological science are merely a few.
Understanding how landscape ecology influences population ecology has allowed population
ecologists to derive new penetrations into their field. A dominant subject of landscape ecology is that the
constellation of spacial mosaics influences a broad array of ecological phenomena ( Turner 1989 ) .
Fractal geometry can be used to explicate connexions between populations and the landscape
construction. Interpreting spacial and temporal graduated tables and motion tracts are two countries of
population ecology that have benefited from the application of fractal geometry.
Different tools are required in population ecology because the declaration or graduated table with which field
informations should be gathered is attuned to the survey being ( Wiens et al. 1993 ) . Insect motions, like
works root growing, follow a uninterrupted way that may be punctuated by Michigans but the tools required
to mensurate this uninterrupted tract are really different. Plant motion is measured by detecting
root growing through exposure, insect motion by tracking insects with flag arrangement, and
carnal motion by utilizing tracking devices on larger animate beings ( Gautestad and Mysterud 1993,
Shibusawa 1994, Wiens et Al. 1993 ) .
Spatial and temporal graduated table are of import when mensurating the place scope of a population and when
tracking carnal motion ( Gautestad and Mysterud 1993, Wiens et Al. 1993 ) . Animal waies have
local, temporal, and scale-specific fluctuations in tortuousness ( Gautestad and Mysterud 1993 ) that are
best described by fractal geometry. The function of insect motion besides required usage of the proper
spacial or temporal graduated table. If excessively long of a clip interval is used to map the insect & # 8217 ; s advancement, the
sections will be excessively long and the elaboratenesss of the insect & # 8217 ; s motions will be lost. The usage of really
short intervals may make unreal interruptions in behavioural moves and might increase the sampling attempt
required until it is unwieldy ( Wiens et al. 1993 ) .
Movement tracts are one of the chief features influenced by the landscape. Motion
tracts are influenced by the flora spots and spot boundaries ( Wiens et al. 1993 ) . Root
warp in a growth works is similar to an carnal tract being changed by the landscape
construction. Waies of carnal motion have fractal facets.
In a continuously changing landscape, it is hard to specify the country of a coinage & # 8217 ; s home ground ( Palmer
1992 ) . Application of fractal geometry has given new penetrations into carnal motion tracts. For
illustration, carnal motion determines the place scope. Because carnal motion is greatly
influenced by the fractal facet of the landscape, place scope is straight influenced by the landscape
construction ( Gautestad and Mysterud 1993 ) . Animal motion is non random but greatly influenced by
the landscape of the place scope of the animate being ( Gautestad and Mysterud 1993 ) . Structural
complexness of the environment consequences in Byzantine animate being tracts ( Gautestad and Mysterud 1993 ) ,
which in bend lead to ragged place scope boundaries.
Gautestad and Mysterud ( 1993 ) found that place scope can be more accurately described by its
fractal belongingss than by the traditional area-related estimates. Since limit of place
scope is a hard undertaking and place scope can & # 8217 ; t be described in traditional units like square metres or
square kilometres, they used fractal belongingss to better depict the place scope country as a composite
country use form ( Gautestad and Mysterud 1993 ) . Fractals work good to depict place scope
because as the sample of location observation additions, the overall form of Thursday
e place secret plans
takes the signifier of a statistical fractal ( Gautestad and Mysterud 1993 ) .
Fractal dimensions are used to stand for the tracts of beetling motion because the fractal
dimension of insect motion tracts may supply penetrations non available from absolute steps
of pathway constellations ( Wiens et al. 1993 ) . Using fractal dimensions allowed ecologists to map
the tract without making an unwieldy flood of information ( Wiens et al. 1993 ) .
Insect behaviour such as forage, coupling, population distribution, predator- quarry interactions or
community composing may be mechanisticly determined by the nature of the landscape. The spacial
heterogeneousness in environmental characteristics or patchiness of a landscape will find how organisms
can travel about ( Wiens et al. 1993 ) . As a beetle or an other insect walks along the land, it does
non travel in a consecutive line. The beetle might walk along in a peculiar way looking for something
to eat. It might go on in one way until it comes across a shrub or bush. It might travel around the
shrub, or it might turn around and head back the manner it came. Its way seems to be random but is
truly dictated by the construction of the landscape ( Wiens et al. 1993 ) .
Another betterment in population ecology through the usage of fractal geometry is the mold of
works root growing. Roots, which besides may look random, do non turn indiscriminately. Reproducing the
fractal forms of root systems has greatly improved root growing theoretical accounts ( Shibusawa 1994 ) .
Landscape ecologists have used fractal geometry extensively to derive new penetrations into their field.
Landscape ecology explores the effects of the constellation of different sorts of environments on the
distribution and motion of beings ( Palmer 1992 ) . Emphasis is on the flow or motion of
being, cistrons, energy, and resources within complex agreements of ecosystems ( Milne 1988 ) .
Landscapes exhibit non-Euclidean denseness and perimeter-to-area relationships and are therefore
suitably described by fractals ( Milne 1988 ) . New penetrations on graduated table, increased apprehension of
landscape constructions, and better landscape construction patterning are merely some of the additions from
using fractal geometry.
Troubles in describing and patterning spatially distributed ecosystems and landscapes include the
natural spacial variableness of ecologically of import parametric quantities such as biomass, productiveness, dirt and
hydrological features. Natural variableness is non changeless and depends to a great extent on spacial graduated table.
Spatial heterogeneousness of a system at any graduated table will forestall the usage of simple point theoretical accounts
( Vedyushkin 1993 ) .
Most landscapes exhibit forms intermediate between complete spacial independency and complete
spacial dependance. Until the reaching of fractal geometry it was hard to pattern this intermediate
degree of spacial dependance ( Palmer 1992, Milne 1988 ) .
Landscapes present beings with heterogeneousness happening at a myriad of length graduated tables.
Understanding and foretelling the effects of heterogeneousness may be enhanced when
scale-dependent heterogeneousness is quantified utilizing fractal geometry ( Milne 1988 ) . Landscape
ecologists normally assume that environmental heterogeneousness can be described by the form, figure,
and distribution on homogenous landscape elements or spots. Heterogeneity can change as a
map of spacial graduated table in landscapes. An illustration of this is a checker board. At a really little graduated table,
a checker board is homogenous because one would remain in one square. At a somewhat larger graduated table,
the checker board would look to be heterogenous since one would traverse the boundaries of the
ruddy and black squares. At an even larger graduated table, one would return to homogeneousness because of the
form of ruddy and black squares ( Palmer 1992 ) .
An increased apprehension of the landscape structures consequences from utilizing the fractal attack in the
field of distant detection of forest flora. Specific advantages include the ability to pull out
information about spacial construction from remotely sensed informations and to utilize it in favoritism of these
informations ; the compaction of this information to few values ; the ability to construe fractal dimension
values in footings of factors, which determine concrete spacial construction ; and sufficient hardiness of
fractal features ( Vedyushkin 1993 ) .
Computer simulations of landscapes provide utile theoretical accounts for deriving new penetrations into the
coexistence of species. Fake landscapes allow ecologists to research some of the effects
of the geometrical constellation of environmental variableness for species coexistence and profusion
( Palmer 1992 ) . A statistically self-similar landscape is an abstraction but it allows an ecologist to
theoretical account fluctuation in spacial dependance ( Palmer 1992 ) . Spatial variableness in the environment is an
of import determiner of coexistence of rivals ( Palmer 1992 ) . Spatial variableness can be
modeled by changing the landscape & # 8217 ; s fractal dimension.
The consequences of this computing machine simulation of species in a landscape show that an addition in the fractal
dimension increases the figure of species per microsite and increases species habitat comprehensiveness.
Other consequences show that environmental variableness allows the coexistence of species, lessenings beta
diverseness, and increases landscape undersaturation ( Palmer 1992 ) . Increasing the fractal dimension of
the landscape allows more species to be in a peculiar country and in the landscape as a whole ;
nevertheless, highly high fractal dimensions cause fewer species to coexist on the landscape graduated table
( Palmer 1992 ) .
Although many ecologists have found fractal geometry to be an highly utile tool, non all concur.
Even scientists who have used fractal geometry in their research point out some of its defects.
For illustration, Scheuring and Riedi ( 1994 ) province that & # 8220 ; the failing of fractal and multifractal
methods in ecological surveies is the fact that existent objects or their abstract projections ( e.g. ,
flora maps ) contain many different sorts of points, while fractal theory assumes that the natural
( or abstract ) objects are represented by points of the same kind. & # 8221 ;
Many scientists agree with Mandelbrot when he said that fractal geometry is the geometry of nature
( Voss 1988 ) , while other scientists think fractal geometry has no topographic point outside a computing machine simulation
( Shenker 1994 ) . In 1987, Simberloff et Al. argued that fractal geometry is useless for ecology
because ecological forms are non fractals. In a paper called & # 8220 ; Fractal Geometry Is Not the
Geometry of Nature, & # 8221 ; Shenker says that Mandelbrot & # 8217 ; s theory of fractal geometry is invalid in the
spacial kingdom because natural objects are non self-similar ( 1994 ) . Further, Shenker states that
Mandelbrot & # 8217 ; s theory is based on want and has no scientific footing at all. He conceded nevertheless that
fractal geometry may work in the temporal part ( Shenker 1994 ) . The unfavorable judgment that fractal
geometry is merely applicable to precisely self-similar objects is addressed by Palmer ( 1982 ) . Palmer
( 1982 ) points out that Mandelbrot & # 8217 ; s early definition ( Mandelbrot 1977 ) does non advert
self-similarity and therefore allows objects that exhibit any kind of fluctuation or abnormality on all
spacial graduated tables of involvement to be considered fractals.
Harmonizing to Shenker, fractals are eternal geometric procedures, and non geometrical signifiers ( 1994 ) ,
and are hence useless in depicting natural objects. This position is kindred to stating that we can & # 8217 ; T usage
Newtonian natural philosophies to pattern the way of a missile because the missile & # 8217 ; s exact mass and speed
are impossible to cognize at the same clip. Mass and speed, like fractals, are abstractions that allow
us to understand and pull strings the natural and physical universe. Even though they are & # 8220 ; merely & # 8221 ;
abstractions, they work rather good.
The value of critics such as Shenker and Simberloff is that they force scientists to clearly understand
their thoughts and premises about fractal geometry, but the critics go excessively far in demanding preciseness
in an imprecise universe.
With all the new penetrations and new cognition that have been gained through the appropriate
application of fractal geometry to natural scientific disciplines, it is clear that is a utile and valid tool.
The new penetrations gained from the application of fractal geometry to ecology include: understanding
the importance of spacial and temporal graduated tables ; the relationship between landscape construction and
motion tracts ; an increased apprehension of landscape constructions ; and the ability to more
accurately theoretical account landscapes and ecosystems.
One of the most valuable facets of fractal geometry, nevertheless, is the manner that it bridges the spread
between ecologists of differing Fieldss. By supplying a common linguistic communication, fractal geometry allows
ecologists to pass on and portion thoughts and constructs.
As the information and computing machine age advancement, with better and faster computing machines, fractal geometry
will go an even more of import tool for ecologists and life scientists. Some future applications of
fractal geometry to ecology include clime mold, conditions anticipation, land direction, and the
creative activity of unreal home grounds.
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