Biology Essay Research Paper Term paperPrinciples of

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Biology Essay, Research Paper

Term paper: Principles of Ecology 310L

New Ecological Penetrations:

The Application of Fractal Geometry to Ecology

Victoria Levin

7 December 1995

Abstraction

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

linguistic communication.

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

1986 )

& # 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

1988 )

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.

Literature Cited

Garcia, L. 1991. The Fractal Explorer. Dynamic Press. Santa Cruz.

Gautestad, A. O. , Mysterud, I. 1993. Physical and biological mechanisms in animate being

motion processes. Journal of Applied Ecology. 30:523-535.

Horgan, J. 1988. Fractal Shorthand. Scientific American. 258 ( 2 ) :28.

Kadanoff, L. P. 1986. Fractals: Where & # 8217 ; s the natural philosophies? Physicss Today. 39:6-7.

Mandelbrot, B. B. 1982. The Fractal Geometry of Nature. W. H. Freeman and Company.

San Francisco.

Mandelbrot, B. B. 1977. Fractals: Form, Chance, and Dimension. W. H. Freeman. New

York.

Milne, B. 1988. Measuring the Fractal Geometry of Landscapes. Applied mathematics and

Computation. 27: 67-79.

Palmer, M.W. 1992. The coexistence of species in fractal landscapes. Am. Nat.

139:375-397.

Scheuring, I. and Riedi, R.H. 1994. Application of multifractals to the analysis of

flora form. Journal of Vegetation Science. 5: 489-496.

Shenker, O.R. 1994. Fractal Geometry is non the geometry of nature. Studies in History

and Philosophy of Science. 25:6:967-981.

Shibusawa, S. 1994. Modeling the ramification growing fractal form of the corn root

system. Plant and Soil. 165: 339-347.

Simberloff, D. , P. Betthet, V. Boy, S. H. Cousins, M.-J. Fortin, R. Goldburg, L. P.

Lefkovitch, B. Ripley, B. Scherrer, and D. Tonkyn. 1987. Novel statistical analyses in

tellurian carnal ecology: dirty informations and clean inquiries. pp. 559-572 in Developments in

Numeric Ecology. P. Legendre and L. Legendre, eds. NATO ASI Series. Vol. G14.

Springer, Berlin.

Turner, M. G. 1989. Landscape ecology ; the consequence of form on procedure. Annual Rev.

Ecological Syst. 20:171-197.

Vedyushkin, M. A. 1993. Fractal belongingss of forest spacial construction. Vegetatio. 113:

65-70.

Voss, R. F. 1988. Fractals in Nature: From Characterization to Simulation. pp. 21- 70. in

The Science of Fractal Images. H.-O. Peitgen and D. Saupe, eds. Springer- Verlag, New

York.

Wiens, J. A. , Crist, T. O. , Milne, B. 1993. On quantifying insect motions.

Environmental Entomology. 22 ( 4 ) : 709-715.

Thomsen, D. E. 1980. Making music & # 8211 ; Fractally. Science News. 117:187-190.

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