Monoculture

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Monoculture

Monocultural potato field

In agriculture, monoculture is the practice of growing one crop species in a field at a time. Monocultures increase ease and efficiency in planting, managing, and harvesting crops short-term, often with the help of machinery. However, monocultures are more susceptible to diseases or pest outbreaks long-term due to localized reductions in biodiversity and nutrient depletion. Crop diversity can be added both in time, as with a crop rotation or sequence, or in space, with a polyculture or intercropping.

Monocultures appear in contexts outside of agriculture and food production. Grass lawns are a common form of residential monocultures. Several monocultures, including single-species forest plantations, have become increasingly abundant throughout the tropics following market globalization, impacting local communities.

Genetic monocultures refer to crops that have little to no genetic variation. This is achieved using cultivars, made through processes of propagation and selective breeding, and can make populations susceptible to disease.

Agroecological practices, silvo-pastoral systems, and mixed-species plantations are common alternatives to monoculture that help preserve biodiversity while maintaining productivity.

Agriculture

Agricultural monocultures refer to the practice of planting one crop species in a field. Monoculture is widely used in intensive farming and in organic farming. In crop monocultures, each plant in a field has the same standardized planting, maintenance, and harvesting requirements resulting in greater yields and lower costs. When a crop is matched to its well-managed environment, a monoculture can produce higher yields than a polyculture. Modern practices such as monoculture planting and the use of synthesized fertilizers have reduced the amount of additional land needed to produce food, called land sparing.

Diversity of crops in space and time; monocultures and polycultures, and rotations of both.

			Diversity in time
			Low	Higher
				Cyclic	Dynamic (non-cyclic)
Diversity in space	Low	Monoculture, one species in a field	Continuous monoculture, monocropping	Crop rotation (rotation of monocultures)	Sequence of monocultures
	Higher	Polyculture, two or more species intermingled in a field (intercropping)	Continuous polyculture	Rotation of polycultures	Sequence of polycultures

Note that the distinction between monoculture and polyculture is not the same as between monocropping and intercropping. The first two describe diversity in space, as does intercropping. Monocropping and crop rotation describe diversity over time.

Environmental impacts

Monocultures of perennials, such as African palm oil, sugarcane, tea and pines, can change soil chemistry leading to soil acidification, degradation, and soil-borne diseases, ultimately having a negative impact on agricultural productivity and sustainability. The use of unregulated irrigation practices on popular monocultures, such as soy, can also lead to erosion and water loss. As soil health declines, use of synthetic fertilizers on monocultural fields increases, often having negative implications on human health via chemical run-off.

In addition to soil depletion, monocultures can cause significant reductions in biodiversity due to unavailability of resources, native species displacement, and loss of genetic variation. Following large-scale oil palm plantations in Latin America, research has revealed extensive declines in mammal, bird, amphibian, and pollinator diversity, particularly in Colombia and Brazil.

Due to insufficient biodiversity and population balance, monocultures are associated with higher rates of disease and pest outbreaks. In response, pesticides are widely applied to agricultural fields, further harming insect and pollinator diversity and human health. Increasing rotations of crop monocultures or using alternatives agricultural practices can help mitigate the risk of disease and attack.

Social impacts

Environmental consequences of monocultural farming have notable social impacts, commonly concentrated to the reduction of small-scale farmers and pesticide-related health issues.Monoculture is contradictive to several primitive, more sustainable farming practices utilized by small-scale farmers. Following pest outbreaks, over 600 million liters of pesticides are sprayed annually, contaminating nearby small-scale farming and causing communal health decline. Research has revealed increased prevalence of pesticide-related disorders, diseases, and cancers affecting the human neurological, gastrointestinal, skin, and respiratory systems.

Agro-extractivism

Agro-extractivism is a form of extractivism in which foreign territorial, political, and economical dominance over agriculture is motivated by the large-scale production and exportation of agricultural commodities, often in the form of monocultures.

Several monocultures in the Global South, such as sugar and coffee, were first planted in the 1800s following European colonization. These plantations used slave labor, setting a precedent for agriculture being a field dominated by foreign entities in the rest of Latin America and the Caribbean. This social framework has shaped the oppression of Black people and smaller-scale farmers in the face of present-day land acquisition for monocultural use.

The large-scale establishment of monocultures in the tropics has led to hindrance of local small-scale farms and indigenous land rights in the forms of reduced food sovereignty, food security, land and water access, and hunting. Land privatization and pressure for monocultural expansion by larger companies takes different forms: silent evictions, violence, and reverse leasing arrangements. Introduction to global trade makes small-scale farmers vulnerable to international demand, prices, and variations in climate affecting crop production. Farmers who make contracts or take out loans with large corporations can face debt and loss of land if they fail to meet certain crop yields or profit.

Monocultures are an aspect of agro-extractivism on account of high percentages of the produced crop being exported for processing and marketing by large transnational corporations, often in developed countries. For instance, following the North American Free Trade Agreement (NAFTA), agave production increased three-fold in Mexico from 1995 to 2019 due to foreign consumption, specifically by the United States. Pararguay sees similar demands with soy crops, exporting the majority of production without nutrients returning to native soil. More than 46 million hectares of soy has been planted across South America while over half a million hectares of land are being deforested annually to make land for cultivation. Some international companies relevant in the field of agro-extractivist monocultures are Syngenta and Bayer (biotech), Los Grobo, CRESUD, El Tejar, and Maggi (landowners), and Cargill, ADM, and Bunge (grain and seed providers).

Forestry

In forestry, monoculture refers to plantations of one species of tree. In many areas of the world, forest monocultures are planted as an efficient way to produce and harvest timber. Because timber harvest from monoculture forests is often an export-driven industry, these plantations can be a form of extractivism. Following deforestation, monoculture afforestation has become increasingly popular due to the necessity for ecosystem services, such as mitigating the effects of climate change via carbon sequestration and gas regulation. Eucalyptus, pines, and acacias are examples of popular monocultures being utilized in the tropics and the Global South following rainforest deforestation.

Environmental impacts

While forest monocultures are efficient ways of producing timber, studies show single-species forests reduce biodiversity, causing declines in forest productivity and native tree, animal, and insect populations over time. The loss of biodiversity in forest monocultures is associated with lower forest resistance to pathogens, attack by insects, and adverse environmental conditions, such as an acceleration of pedolysis.

Social impacts

Monoculture plantations have been shown to have substantial social impacts on local communities. Forest monocultures have motivated migrations across Latin America due to localized water cycle interference, declining soil health, and changes in resource availability. While industrial agriculture can increase employment opportunities, studies show forest plantations often have limited employment opportunities, with most workers coming from outside of the community. Profits made from monoculture plantations historically follow a "boom and bust" trend, temporarily benefitting the community in increased income, revenue, and quality of life until resources are exhausted, with profits rarely distributed back into the deforested land.

Environmental changes caused by monoculture forests are particularly felt among indigenous communities given their reliance and connection to the land while additionally becoming subject to land privatization. These lands are frequently acquired through land grabbing and dispossession by large companies in global trade, ultimately reducing rural land, cutting off access to locals, and changing agricultural and community dynamics.

Residential monoculture

Lawn monoculture in the United States was historically influenced by English gardens and manor-house landscapes, but its inception into the American landscape is fairly recent. Aesthetics drove the evolution of the residential green areas, with turfgrass becoming a popular addition to many American homes. Turfgrass is a nonnative species and requires high levels of maintenance. At the local level, governments and organizations, such as Homeowner Associations, have pressured the maintenance of lawn aesthetics and influenced real estate value. Disagreements in residential maintenance of weeds and lawns have resulted in civil cases or direct aggression against neighbors.

High levels of maintenance required for turfgrass created a growing demand for chemical management, i.e. pesticides, herbicides, insecticides. A 1999 study showed that in a sample of urban streams, at least one type of pesticide was found in 99% of the streams. A major risk associated with lawn pesticide use is the exposure to chemicals within the home through the air, clothing, and furniture, which can be more detrimental to children than to the average adult.

Genetic monocultures

While often referring to the production of the same crop species in a field (space), monoculture can also refer to the planting of a single cultivar across a larger regional area, such that there are numerous plants in the area with an identical genetic makeup to each other. When all plants in a region are genetically similar, a disease to which they have no resistance can destroy entire populations of crops. As of 2009 the wheat leaf rust fungus caused much concern internationally, having already severely affected wheat crops in Uganda and Kenya, and having started to spread in Asia as well. Given the very genetically similar strains of much of the world's wheat crops following the Green Revolution, the impacts of such diseases threaten agricultural production worldwide.

Historic examples of genetic monocultures

Great Famine of Ireland

In Ireland, exclusive use of one variety of potato, the "lumper", led to the Great Famine of 1845–1849. Lumpers provided inexpensive food to feed the Irish masses. Potatoes were propagated vegetatively with little to no genetic variation. When Phytophthora infestans arrived in Ireland from the Americas in 1845, the lumper had no resistance to the disease, leading to the nearly complete failure of the potato crop across Ireland.

Bananas

Until the 1950s, the Gros Michel cultivar of banana represented almost all bananas consumed in the United States because of their taste, small seeds, and efficiency to produce. Their small seeds, while more appealing than the large ones in other Asian cultivars, were not suitable for planting, meaning all new banana plants had to be grown from the cut suckers of another plant. As a result of this asexual form of planting, all bananas grown had identical genetic makeups which gave them no traits for resistance to Fusarium wilt, a fungal disease that spread quickly throughout the Caribbean where they were being grown. By the beginning of the 1960s, growers had to switch to growing the Cavendish banana, a cultivar grown in a similar way. This cultivar is under similar disease stress since all the bananas are clones of each other and could easily succumb as the Gros Michel did.

Cattle

Aerial view of deforested area prepared for monoculture or cattle ranching, near Porto Velho in Rondônia, Brazil, in 2020

Genetic monoculture can also refer to a single breed of farm animal being raised in large-scale concentrated animal feeding operations (CAFOs). Many livestock production systems rely on just a small number of highly specialized breeds. Focusing heavily on a single trait (output) may come at the expense of other desirable traits – such as fertility, resistance to disease, vigor, and mothering instincts. In the early 1990s, a few Holstein calves were observed to grow poorly and died in the first 6 months of life. They were all found to be homozygous for a mutation in the gene that caused bovine leukocyte adhesion deficiency. This mutation was found at a high frequency in Holstein populations worldwide. (15% among bulls in the US, 10% in Germany, and 16% in Japan.) Researchers studying the pedigrees of affected and carrier animals tracked the source of the mutation to a single bull that was widely used in livestock production. In 1990 there were approximately 4 million Holstein cattle in the US, making the affected population around 600,000 animals.

Benefits of genetic diversity

Increasing genetic diversity through the introduction of organisms with varying genes can make agricultural and livestock systems more sustainable. By utilizing crops with varying genetic traits for disease and pest resistance, chances of disease outbreak decrease due to the likelihood of neighboring plants having strain-resistant genes. This can aid in increasing crop productivity while decreasing pesticide usage.

Alternatives to monoculture

Alternatives to monoculture include the consultation of agroecology, silvo-pastoral systems, and mixed-species plantations.

Agroecology

Agroecology consults the entire food system, considering how agricultural inputs and outputs affect social, environmental, and economic systems. Despite the recent dominance of GMO monoculture crop rotations of soy, corn, and cotton across the deforested Amazon, many Afrodescendant-run farms in Brazil continue to use traditional practices of agroecology that have the capacity to sustain the local community, environment, and economy. Ecosystem-specific ecological damage done by monocultural practices and byproducts, including the use of biocides and soil degradation, can be irreparable. However, the increasing modern prevalence of regenerative farming reinstates crop rotation and natural nutrient cycling to repair biodiversity and improve soil productivity.

Silvopasture

Silvopasture is a traditional practice that incorporates the use of various trees and forage in pastures to increase land and livestock productivity. Incorporating other plants in pastures, such as tree legumes, has been shown to enhance pollinator activity, benefitting local biodiversity and food security. Silvopastoral systems provide greater pasture species richness and grazing feed, increasing economic and environmental outcomes on various size scales.

Mixed-species plantations

In several studies, well-managed mixed-species plantations have been shown to produce greater economic outcomes than monocultures with regard to timber sales. Mixed-species forests are also associated with greater carbon sequestration and biodiversity, presenting a possible mitigation tactic against the climate crisis and current global carbon levels. However, mixed-species plantations are less common under the misconception of being more expensive and harder to manage.

Diversity index

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Diversity_index

A diversity index is a method of measuring how many different types (e.g. species) there are in a dataset (e.g. a community). Diversity indices are statistical representations of different aspects of biodiversity (e.g. richness, evenness, and dominance), which are useful simplifications for comparing different communities or sites.

When diversity indices are used in ecology, the types of interest are usually species, but they can also be other categories, such as genera, families, functional types, or haplotypes. The entities of interest are usually individual organisms (e.g. plants or animals), and the measure of abundance can be, for example, number of individuals, biomass or coverage. In demography, the entities of interest can be people, and the types of interest various demographic groups. In information science, the entities can be characters and the types of the different letters of the alphabet. The most commonly used diversity indices are simple transformations of the effective number of types (also known as 'true diversity'), but each diversity index can also be interpreted in its own right as a measure corresponding to some real phenomenon (but a different one for each diversity index).

Many indices only account for categorical diversity between subjects or entities. Such indices, however do not account for the total variation (diversity) that can be held between subjects or entities which occurs only when both categorical and qualitative diversity are calculated.

Diversity indices described in this article include:

• Richness, simply a count of the number of types in a dataset.
• Shannon index, which also takes into account the proportional abundance of each class under a weighted geometric mean.
  • The Rényi entropy, which adds the ability to freely vary the kind of weighted mean used.
• Simpson index, which too takes into account the proportional abundance of each class under a weighted arithmetic mean
• Berger–Parker index, which gives the proportional abundance of the most abundant type.
• Effective number of species (true diversity), which allows for freely varying the kind of weighted mean used, and has a intuitive meaning.

Some more sophisticated indices also account for the phylogenetic relatedness among the types. These are called phylo-divergence indices, and are not yet described in this article.

Effective number of species or Hill numbers

True diversity, or the effective number of types, refers to the number of equally abundant types needed for the average proportional abundance of the types to equal that observed in the dataset of interest (where all types may not be equally abundant). The true diversity in a dataset is calculated by first taking the weighted generalized mean M_q−1 of the proportional abundances of the types in the dataset, and then taking the reciprocal of this. The equation is:

${\displaystyle {}^{q}D=\frac{1}{M_{q-1}}=\frac{1}{\sqrt[q-1]{\sum _{i=1}^R{p}_i{p}_i^{q-1}}}={\left(\sum _{i=1}^R{p}_i^q\right)}^{1/(1-q)}}$

The denominator M_q−1 equals the average proportional abundance of the types in the dataset as calculated with the weighted generalized mean with exponent q − 1. In the equation, R is richness (the total number of types in the dataset), and the proportional abundance of the ith type is p_i. The proportional abundances themselves are used as the nominal weights. The numbers ${\displaystyle {}^{q}D}$ are called Hill numbers of order q or effective number of species.

When q = 1, the above equation is undefined. However, the mathematical limit as q approaches 1 is well defined and the corresponding diversity is calculated with the following equation:

${\displaystyle {}^{1}D=\frac{1}{\prod _{i=1}^R{p}_i^{p_i}}=\exp \left(-\sum _{i=1}^R{p}_i\ln (p_i)\right)}$

which is the exponential of the Shannon entropy calculated with natural logarithms (see above). In other domains, this statistic is also known as the perplexity.

The general equation of diversity is often written in the form

${\displaystyle {}^{q}D={\left(\sum _{i=1}^R{p}_i^q\right)}^{1/(1-q)}}$

and the term inside the parentheses is called the basic sum. Some popular diversity indices correspond to the basic sum as calculated with different values of q.

Sensitivity of the diversity value to rare vs. abundant species

The value of q is often referred to as the order of the diversity. It defines the sensitivity of the true diversity to rare vs. abundant species by modifying how the weighted mean of the species' proportional abundances is calculated. With some values of the parameter q, the value of the generalized mean M_q−1 assumes familiar kinds of weighted means as special cases. In particular,

• q = 0 corresponds to the weighted harmonic mean,
• q = 1 to the weighted geometric mean, and
• q = 2 to the weighted arithmetic mean.
• As q approaches infinity, the weighted generalized mean with exponent q − 1 approaches the maximum p_i value, which is the proportional abundance of the most abundant species in the dataset.

Generally, increasing the value of q increases the effective weight given to the most abundant species. This leads to obtaining a larger M_q−1 value and a smaller true diversity (^qD) value with increasing q.

When q = 1, the weighted geometric mean of the p_i values is used, and each species is exactly weighted by its proportional abundance (in the weighted geometric mean, the weights are the exponents). When q > 1, the weight given to abundant species is exaggerated, and when q < 1, the weight given to rare species is. At q = 0, the species weights exactly cancel out the species proportional abundances, such that the weighted mean of the p_i values equals 1 / R even when all species are not equally abundant. At q = 0, the effective number of species, ⁰D, hence equals the actual number of species R. In the context of diversity, q is generally limited to non-negative values. This is because negative values of q would give rare species so much more weight than abundant ones that ^qD would exceed R.

Richness

Main article: Species richness

Richness R simply quantifies how many different types the dataset of interest contains. For example, species richness (usually noted S) is simply the number of species, e.g. at a particular site. Richness is a simple measure, so it has been a popular diversity index in ecology, where abundance data are often not available. If true diversity is calculated with q = 0, the effective number of types (⁰D) equals the actual number of types, which is identical to Richness (R).

Shannon index

The Shannon index has been a popular diversity index in the ecological literature, where it is also known as Shannon's diversity index, Shannon–Wiener index, and (erroneously) Shannon–Weaver index. The measure was originally proposed by Claude Shannon in 1948 to quantify the entropy (hence Shannon entropy, related to Shannon information content) in strings of text. The idea is that the more letters there are, and the closer their proportional abundances in the string of interest, the more difficult it is to correctly predict which letter will be the next one in the string. The Shannon entropy quantifies the uncertainty (entropy or degree of surprise) associated with this prediction. It is most often calculated as follows:

${\displaystyle H^{\prime }=-\sum _{i=1}^R{p}_i\ln (p_i)}$

where p_i is the proportion of characters belonging to the ith type of letter in the string of interest. In ecology, p_i is often the proportion of individuals belonging to the ith species in the dataset of interest. Then the Shannon entropy quantifies the uncertainty in predicting the species identity of an individual that is taken at random from the dataset.

Although the equation is here written with natural logarithms, the base of the logarithm used when calculating the Shannon entropy can be chosen freely. Shannon himself discussed logarithm bases 2, 10 and e, and these have since become the most popular bases in applications that use the Shannon entropy. Each log base corresponds to a different measurement unit, which has been called binary digits (bits), decimal digits (decits), and natural digits (nats) for the bases 2, 10 and e, respectively. Comparing Shannon entropy values that were originally calculated with different log bases requires converting them to the same log base: change from the base a to base b is obtained with multiplication by log_b(a).

The Shannon index (H') is related to the weighted geometric mean of the proportional abundances of the types. Specifically, it equals the logarithm of true diversity as calculated with q = 1:

${\displaystyle H^{\prime }=-\sum _{i=1}^R{p}_i\ln (p_i)=-\sum _{i=1}^R\ln \left(p_i^{p_i}\right)}$

This can also be written

${\displaystyle H^{\prime }=-\left(\ln \left(p_1^{p_1}\right)+\ln \left(p_2^{p_2}\right)+\ln \left(p_3^{p_3}\right)+\cdots +\ln \left(p_R^{p_R}\right)\right)}$

which equals

${\displaystyle H^{\prime }=-\ln \left(p_1^{p_1}{p}_2^{p_2}{p}_3^{p_3}\cdots p_R^{p_R}\right)=\ln \left(\frac{1}{p_1^{p_1}{p}_2^{p_2}{p}_3^{p_3}\cdots p_R^{p_R}}\right)=\ln \left(\frac{1}{\prod _{i=1}^R{p}_i^{p_i}}\right)}$

Since the sum of the p_i values equals 1 by definition, the denominator equals the weighted geometric mean of the p_i values, with the p_i values themselves being used as the weights (exponents in the equation). The term within the parentheses hence equals true diversity ¹D, and H' equals ln(¹D).

When all types in the dataset of interest are equally common, all p_i values equal 1 / R, and the Shannon index hence takes the value ln(R). The more unequal the abundances of the types, the larger the weighted geometric mean of the p_i values, and the smaller the corresponding Shannon entropy. If practically all abundance is concentrated to one type, and the other types are very rare (even if there are many of them), Shannon entropy approaches zero. When there is only one type in the dataset, Shannon entropy exactly equals zero (there is no uncertainty in predicting the type of the next randomly chosen entity).

In machine learning the Shannon index is also called as Information gain.

Rényi entropy

The Rényi entropy is a generalization of the Shannon entropy to other values of q than 1. It can be expressed:

${\displaystyle {}^{q}H=\frac{1}{1-q}\ln \left(\sum _{i=1}^R{p}_i^q\right)}$

which equals

${\displaystyle {}^{q}H=\ln \left(\frac{1}{\sqrt[q-1]{\sum _{i=1}^R{p}_i{p}_i^{q-1}}}\right)=\ln ({}^{q}D)}$

This means that taking the logarithm of true diversity based on any value of q gives the Rényi entropy corresponding to the same value of q.

Simpson index

The Simpson index was introduced in 1949 by Edward H. Simpson to measure the degree of concentration when individuals are classified into types. The same index was rediscovered by Orris C. Herfindahl in 1950. The square root of the index had already been introduced in 1945 by the economist Albert O. Hirschman. As a result, the same measure is usually known as the Simpson index in ecology, and as the Herfindahl index or the Herfindahl–Hirschman index (HHI) in economics.

The measure equals the probability that two entities taken at random from the dataset of interest represent the same type. It equals:

${\displaystyle \lambda =\sum _{i=1}^R{p}_i^2,}$

where R is richness (the total number of types in the dataset). This equation is also equal to the weighted arithmetic mean of the proportional abundances p_i of the types of interest, with the proportional abundances themselves being used as the weights. Proportional abundances are by definition constrained to values between zero and one, but it is a weighted arithmetic mean, hence λ ≥ 1/R, which is reached when all types are equally abundant.

By comparing the equation used to calculate λ with the equations used to calculate true diversity, it can be seen that 1/λ equals ²D, i.e., true diversity as calculated with q = 2. The original Simpson's index hence equals the corresponding basic sum.

The interpretation of λ as the probability that two entities taken at random from the dataset of interest represent the same type assumes that the first entity is replaced to the dataset before taking the second entity. If the dataset is very large, sampling without replacement gives approximately the same result, but in small datasets, the difference can be substantial. If the dataset is small, and sampling without replacement is assumed, the probability of obtaining the same type with both random draws is:

${\displaystyle \ell =\frac{\sum _{i=1}^R{n}_i(n_i-1)}{N(N-1)}}$

where n_i is the number of entities belonging to the ith type and N is the total number of entities in the dataset. This form of the Simpson index is also known as the Hunter–Gaston index in microbiology.

Since the mean proportional abundance of the types increases with decreasing number of types and increasing abundance of the most abundant type, λ obtains small values in datasets of high diversity and large values in datasets of low diversity. This is counterintuitive behavior for a diversity index, so often, such transformations of λ that increase with increasing diversity have been used instead. The most popular of such indices have been the inverse Simpson index (1/λ) and the Gini–Simpson index (1 − λ). Both of these have also been called the Simpson index in the ecological literature, so care is needed to avoid accidentally comparing the different indices as if they were the same.

Inverse Simpson index

The inverse Simpson index equals:

${\displaystyle \frac{1}{\lambda }=\frac{1}{\sum _{i=1}^R{p}_i^2}={}^{2}D}$

This simply equals true diversity of order 2, i.e. the effective number of types that is obtained when the weighted arithmetic mean is used to quantify average proportional abundance of types in the dataset of interest.

The index is also used as a measure of the effective number of parties.

Gini–Simpson index

The Gini-Simpson Index is also called Gini impurity, or Gini's diversity index in the field of Machine Learning. The original Simpson index λ equals the probability that two entities taken at random from the dataset of interest (with replacement) represent the same type. Its transformation 1 − λ, therefore, equals the probability that the two entities represent different types. This measure is also known in ecology as the probability of interspecific encounter (PIE) and the Gini–Simpson index. It can be expressed as a transformation of the true diversity of order 2:

${\displaystyle 1-\lambda =1-\sum _{i=1}^R{p}_i^2=1-\frac{1}{{}^{2}D}}$

The Gibbs–Martin index of sociology, psychology, and management studies, which is also known as the Blau index, is the same measure as the Gini–Simpson index.

The quantity is also known as the expected heterozygosity in population genetics.

Berger–Parker index

The Berger–Parker index, named after Wolfgang H. Berger and Frances Lawrence Parker, equals the maximum p_i value in the dataset, i.e., the proportional abundance of the most abundant type. This corresponds to the weighted generalized mean of the p_i values when q approaches infinity, and hence equals the inverse of the true diversity of order infinity (1/^∞D).

Dangling modifier

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Dangling_modifier

A dangling modifier (also known as a dangling participle, illogical participle or hanging participle) is a type of ambiguous grammatical construct whereby a grammatical modifier could be misinterpreted as being associated with a word other than the one intended. A dangling modifier has no subject and is usually a participle. A writer may use a dangling modifier intending to modify a subject while word order may imply that the modifier describes an object, or vice versa.

An example of a dangling modifier appears in the sentence "Turning the corner, a handsome school building appeared". The modifying clause Turning the corner describes the behavior of the narrator, but the narrator is only implicit in the sentence. The sentence could be misread as the turning action attaching either to the handsome school building or to nothing at all. As another example, in the sentence "At the age of eight, my family finally bought a dog", the modifier At the age of eight is dangling. It is intended to specify the narrator's age when the family bought the dog, but the narrator is again only implicitly a part of the sentence. It could be read as the family was eight years old when it bought the dog.

Dangling-modifier clauses

As an adjunct, a modifier clause is normally at the beginning or the end of a sentence and usually attached to the subject of the main clause. However, when the subject is missing or the clause attaches itself to another object in a sentence, the clause is seemingly "hanging" on nothing or on an inappropriate noun. It thus "dangles", as in these sentences:

Ambiguous: Walking down Main Street (clause), the trees were beautiful (object). (Subject is unclear / implicit)

Unambiguous: Walking down Main Street (clause), I (subject) admired the beautiful trees (object).

Ambiguous: Reaching the station, the sun came out. (Subject is unclear - who reached the station?)

Unambiguous: As Priscilla reached the station, the sun came out.

In the first sentence, the adjunct clause may at first appear to modify "the trees", the subject of the sentence. However, it actually modifies the speaker of the sentence, who is not explicitly mentioned. In the second sentence, the adjunct may at first appear to modify "the sun", the subject of the sentence. Presumably, there is another, human subject who did reach the station as the sun was coming out, but this subject is not mentioned in the text. In both cases, whether the intended meaning is obscured or not may depend on context - if the previous sentences clearly established a subject, then it may be obvious who was walking down Main Street or reaching the station. But if left alone, they may be unclear if the reader takes the subject as an unknown observer; or misleading if a reader somehow believed the trees were walking down the street or the sun traveled to the station.

Many style guides of the 20th century consider dangling participles ungrammatical and incorrect. Strunk and White's The Elements of Style states that "A participle phrase at the beginning of a sentence must refer to the grammatical subject". The 1966 book Modern American Usage: A Guide, started by Wilson Follett and finished by others, agrees: "A participle at the head of a sentence automatically affixes itself to the subject of the following verb – in effect a requirement that the writer either make his [grammatical] subject consistent with the participle or discard the participle for some other construction". However, this prohibition has been questioned; more descriptivist authors consider that a dangling participle is only problematic when there is actual ambiguity. One of Follett's examples is "Leaping to the saddle, his horse bolted", but a reader is unlikely to be genuinely confused and think that the horse was leaping into a saddle rather than an implicit rider; The Economist questioned whether the "clumsy examples" of the style guides proved much. Many respected and successful writers have used dangling participles without confusion; one example is Virginia Woolf whose work includes many such phrases, such as "Lying awake, the floor creaked" (in Mrs Dalloway) or "Sitting up late at night it seems strange not to have more control" (in The Waves). Shakespeare's Richard II includes a dangling modifier as well.

Absolute constructions

Dangling participles are similar to clauses in absolute constructions, but absolute constructions are considered uncontroversial and grammatical. The difference is that a participle phrase in an absolute construction is not semantically attached to any single element in the sentence. A participle phrase is intended to modify a particular noun or pronoun, but in a dangling participle, it is instead erroneously attached to a different noun or to nothing; whereas in an absolute clause, is not intended to modify any noun at all, and thus modifying nothing is the intended use. An example of an absolute construction is:

The weather being beautiful, we plan to go to the beach today.

Non-participial modifiers

Non-participial modifiers that dangle can also be troublesome:

After years of being lost under a pile of dust, Walter P. Stanley, III, left, found all the old records of the Bangor Lions Club.

The above sentence from a photo caption in a newspaper suggests that it is the subject of the sentence, Walter Stanley, who was buried under a pile of dust, and not the records. It is the prepositional phrase "after years of being lost under a pile of dust" which dangles.

In the film Mary Poppins, Mr. Dawes Sr. dies of laughter after hearing the following joke:

"I know a man with a wooden leg called Smith". "What was the name of his other leg?"

In the case of this joke, the placement of the participial phrase "called Smith" implies that it is the leg that is named Smith, rather than the man. ("Called Smith" is a participial phrase, as "called" is a past participle.)

Another famous example of this humorous effect is by Groucho Marx as Captain Jeffrey T. Spaulding in the 1930 film Animal Crackers:

One morning I shot an elephant in my pajamas. How he got into my pajamas I'll never know.

Though under the most plausible interpretation of the first sentence, Captain Spaulding would have been wearing the pajamas, the line plays on the grammatical possibility that the elephant was instead.

Certain formulations can be genuinely ambiguous as to whether the subject, the direct object, or something else is the proper affix for the participle; for example, in "Having just arrived in town, the train struck Bill", did the narrator, the train, or Bill just arrive in the town?

Modifiers reflecting the mood or attitude of the speaker

Participial modifiers can sometimes be intended to describe the attitude or mood of the speaker, even when the speaker is not part of the sentence. Some such modifiers are standard and are not considered dangling modifiers: "Speaking of [topic]", and "Trusting that this will put things into perspective", for example, are commonly used to transition from one topic to a related one or for adding a conclusion to a speech. An example of a contested use would be "Frankly, he is lying to you"; such usage is not uncommon by writers, but strictly speaking that sentence would be in violation of older style guide prohibitions as it is the speaker being frank, not "he" in such a sentence.

Usage of "hopefully"

Main article: Hopefully

Since about the 1960s, controversy has arisen over the proper usage of the adverb hopefully. Some grammarians object to constructions such as "Hopefully, the sun will be shining tomorrow". Their complaint is that the term "hopefully" is understood as the manner in which the sun will shine if read literally, with the suggested modification "I hope the sun will shine tomorrow" if it is the speaker that is full of hope. "Hopefully" used in this way is a disjunct (cf. "admittedly", "mercifully", "oddly"). Disjuncts (also called sentence adverbs) are useful in colloquial speech for the concision they permit.

No other word in English expresses that thought. In a single word we can say it is regrettable that (regrettably) or it is fortunate that (fortunately) or it is lucky that (luckily), and it would be comforting if there were such a word as hopably or, as suggested by Follett, hopingly, but there isn't. [...] In this instance nothing is to be lost – the word would not be destroyed in its primary meaning – and a useful, nay necessary term is to be gained.

What had been expressed in lengthy adverbial constructions, such as "it is regrettable that ..". or "it is fortunate that .."., had of course always been shortened to the adverbs "regrettably" or "fortunately". Bill Bryson says, "those writers who scrupulously avoid 'hopefully' in such constructions do not hesitate to use at least a dozen other words – 'apparently', 'presumably', 'happily', 'sadly', 'mercifully', 'thankfully', and so on – in precisely the same way".

Merriam-Webster gives a usage note on its entry for "hopefully"; the editors point out that the disjunct sense of the word dates to the early 18th century and has been in widespread use since at least the 1930s. Objection to this sense of the word, they state, became widespread only in the 1960s. The Merriam Webster editors maintain that this usage is "entirely standard".

There are similar complications with the term "doubtless" or "doubtlessly". "Alex doubtlessly ran out of gas" either means Alex was doubtless when he ran out of gas, or the speaker is doubtless in declaring that Alex ran out of gas.

Perturbation (astronomy)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Perturbation_(astronomy) 

The perturbing forces of the Sun on the Moon at two places in its orbit. The blue arrows represent the direction and magnitude of the gravitational force on the Earth. Applying this to both the Earth's and the Moon's position does not disturb the positions relative to each other. When it is subtracted from the force on the Moon (black arrows), what is left is the perturbing force (red arrows) on the Moon relative to the Earth. Because the perturbing force is different in direction and magnitude on opposite sides of the orbit, it produces a change in the shape of the orbit.

In astronomy, perturbation is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other massive body. The other forces can include a third (fourth, fifth, etc.) body, resistance, as from an atmosphere, and the off-center attraction of an oblate or otherwise misshapen body.

Introduction

The study of perturbations began with the first attempts to predict planetary motions in the sky. In ancient times the causes were unknown. Isaac Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations, recognizing the complex difficulties of their calculation. Many of the great mathematicians since then have given attention to the various problems involved; throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for marine navigation.

The complex motions of gravitational perturbations can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is a conic section, and can be described in geometrical terms. This is called a two-body problem, or an unperturbed Keplerian orbit. The differences between that and the actual motion of the body are perturbations due to the additional gravitational effects of the remaining body or bodies. If there is only one other significant body then the perturbed motion is a three-body problem; if there are multiple other bodies it is an n‑body problem. A general analytical solution (a mathematical expression to predict the positions and motions at any future time) exists for the two-body problem; when more than two bodies are considered analytic solutions exist only for special cases. Even the two-body problem becomes insoluble if one of the bodies is irregular in shape.

Mercury's orbital longitude and latitude, as perturbed by Venus, Jupiter, and all of the planets of the Solar System, at intervals of 2.5 days. Mercury would remain centered on the crosshairs if there were no perturbations.

Most systems that involve multiple gravitational attractions present one primary body which is dominant in its effects (for example, a star, in the case of the star and its planet, or a planet, in the case of the planet and its satellite). The gravitational effects of the other bodies can be treated as perturbations of the hypothetical unperturbed motion of the planet or satellite around its primary body.

Mathematical analysis

General perturbations

In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically, usually by series expansions. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects. Historically, general perturbations were investigated first. The classical methods are known as variation of the elements, variation of parameters or variation of the constants of integration. In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations.

General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body. In the Solar System, this is usually the case; Jupiter, the second largest body, has a mass of about ⁠1/ 1000 ⁠ that of the Sun.

General perturbation methods are preferred for some types of problems, as the source of certain observed motions are readily found. This is not necessarily so for special perturbations; the motions would be predicted with similar accuracy, but no information on the configurations of the perturbing bodies (for instance, an orbital resonance) which caused them would be available.

Special perturbations

In methods of special perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion. In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements.

Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small. Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs. Special perturbations are also used for modeling an orbit with computers.

Cowell's formulation

Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body ${\displaystyle i}$ (red), and this is numerically integrated starting from the initial position (the epoch of osculation).

Cowell's formulation (so named for Philip H. Cowell, who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet) is perhaps the simplest of the special perturbation methods. In a system of ${\displaystyle n}$ mutually interacting bodies, this method mathematically solves for the Newtonian forces on body ${\displaystyle i}$ by summing the individual interactions from the other ${\displaystyle j}$ bodies:

${\displaystyle {\ddot{\mathbf{r}}}_i=\sum _{\underset{j\ne i}{j=1}}^{n}G{m}_j\frac{({\mathbf{r}}_j-{\mathbf{r}}_i)}{\Vert {\mathbf{r}}_j-{\mathbf{r}}_i{\Vert }^3}}$

where ${\displaystyle {\ddot{\mathbf{r}}}_i}$ is the acceleration vector of body ${\displaystyle i}$, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle m_j}$ is the mass of body ${\displaystyle j}$, ${\displaystyle {\mathbf{r}}_i}$ and ${\displaystyle {\mathbf{r}}_j}$ are the position vectors of objects ${\displaystyle i}$ and ${\displaystyle j}$ respectively, and ${\displaystyle r_{ij}\equiv \Vert {\mathbf{r}}_j-{\mathbf{r}}_i\Vert }$ is the distance from object ${\displaystyle i}$ to object ${\displaystyle j}$, all vectors being referred to the barycenter of the system. This equation is resolved into components in ${\displaystyle x,}$ ${\displaystyle y,}$ and ${\displaystyle z,}$ and these are integrated numerically to form the new velocity and position vectors. This process is repeated as many times as necessary. The advantage of Cowell's method is ease of application and programming. A disadvantage is that when perturbations become large in magnitude (as when an object makes a close approach to another) the errors of the method also become large. However, for many problems in celestial mechanics, this is never the case. Another disadvantage is that in systems with a dominant central body, such as the Sun, it is necessary to carry many significant digits in the arithmetic because of the large difference in the forces of the central body and the perturbing bodies, although with high precision numbers built into modern computers this is not as much of a limitation as it once was.

Encke's method

Encke's method. Greatly exaggerated here, the small difference δr (blue) between the osculating, unperturbed orbit (black) and the perturbed orbit (red), is numerically integrated starting from the initial position (the epoch of osculation).

Encke's method begins with the osculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time. Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification. Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously.

Letting ${\displaystyle \mathit{\rho }}$ be the radius vector of the osculating orbit, ${\displaystyle \mathbf{r}}$ the radius vector of the perturbed orbit, and ${\displaystyle \delta \mathbf{r}}$ the variation from the osculating orbit,

${\displaystyle \delta \mathbf{r}=\mathbf{r}-\mathit{\rho }}$, and the equation of motion of ${\displaystyle \delta \mathbf{r}}$ is simply

1

${\displaystyle \delta \ddot{\mathbf{r}}=\ddot{\mathbf{r}}-\ddot{\mathit{\rho }}}$.

2

${\displaystyle \ddot{\mathbf{r}}}$ and ${\displaystyle \ddot{\mathit{\rho }}}$ are just the equations of motion of ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathit{\rho },}$

${\displaystyle \ddot{\mathbf{r}}={\mathbf{a}}_{\text{per}}-\frac{\mu }{r^3}\mathbf{r}}$ for the perturbed orbit and

3

${\displaystyle \ddot{\mathit{\rho }}=-\frac{\mu }{{\rho }^3}\mathit{\rho }}$ for the unperturbed orbit,

4

where ${\displaystyle \mu =G(M+m)}$ is the gravitational parameter with ${\displaystyle M}$ and ${\displaystyle m}$ the masses of the central body and the perturbed body, ${\displaystyle {\mathbf{a}}_{\text{per}}}$ is the perturbing acceleration, and ${\displaystyle r}$ and ${\displaystyle \rho }$ are the magnitudes of ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathit{\rho }}$.

Substituting from equations (3) and (4) into equation (2),

${\displaystyle \delta \ddot{\mathbf{r}}={\mathbf{a}}_{\text{per}}+\mu \left(\frac{\mathit{\rho }}{{\rho }^3}-\frac{\mathbf{r}}{r^3}\right),}$

5

which, in theory, could be integrated twice to find ${\displaystyle \delta \mathbf{r}}$. Since the osculating orbit is easily calculated by two-body methods, ${\displaystyle \mathit{\rho }}$ and ${\displaystyle \delta \mathbf{r}}$ are accounted for and ${\displaystyle \mathbf{r}}$ can be solved. In practice, the quantity in the brackets, ${\displaystyle \frac{\mathit{\rho }}{{\rho }^3}-\frac{\mathbf{r}}{r^3}}$, is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra significant digits. Encke's method was more widely used before the advent of modern computers, when much orbit computation was performed on mechanical calculating machines.

Periodic nature

Gravity Simulator plot of the changing orbital eccentricity of Mercury, Venus, Earth, and Mars over the next 50,000 years. The zero-point on this plot is the year 2007.

In the Solar System, many of the disturbances of one planet by another are periodic, consisting of small impulses each time a planet passes another in its orbit. This causes the bodies to follow motions that are periodic or quasi-periodic – such as the Moon in its strongly perturbed orbit, which is the subject of lunar theory. This periodic nature led to the discovery of Neptune in 1846 as a result of its perturbations of the orbit of Uranus.

On-going mutual perturbations of the planets cause long-term quasi-periodic variations in their orbital elements, most apparent when two planets' orbital periods are nearly in sync. For instance, five orbits of Jupiter (59.31 years) is nearly equal to two of Saturn (58.91 years). This causes large perturbations of both, with a period of 918 years, the time required for the small difference in their positions at conjunction to make one complete circle, first discovered by Laplace. Venus currently has the orbit with the least eccentricity, i.e. it is the closest to circular, of all the planetary orbits. In 25,000 years' time, Earth will have a more circular (less eccentric) orbit than Venus. It has been shown that long-term periodic disturbances within the Solar System can become chaotic over very long time scales; under some circumstances one or more planets can cross the orbit of another, leading to collisions.

The orbits of many of the minor bodies of the Solar System, such as comets, are often heavily perturbed, particularly by the gravitational fields of the gas giants. While many of these perturbations are periodic, others are not, and these in particular may represent aspects of chaotic motion. For example, in April 1996, Jupiter's gravitational influence caused the period of Comet Hale–Bopp's orbit to decrease from 4,206 to 2,380 years, a change that will not revert on any periodic basis.