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 * 1) wikipedia:en # Indo-Aryan_migration # w?f- # 20061024205654 # 20061025033538 # NONE ###

Indo-Aryan migration refers to the theory that speakers of Indo-Aryan languages migrated into the Indian subcontinent during the 2nd millennium BCE, as opposed to being autochthonous to the region.

Based on linguistic, archaeological and cultural evidence, most scholars have argued that Indo-Aryan speakers migrated to northern India following the breakup of Proto-Indo-Iranian and the subsequent  Indo-Iranian expansion out of Central Asia. These scholars argue that, in India, the Indo-Aryans interacted with the remnants of the Indus Valley civilization, a process that gave rise to Vedic civilization.

Archaeological data indicates that there was a shift of settlements from the Indus Valley region to the east and south during the later 2nd millennium BCE, but is inconclusive with regard to a preceding immigration into India.

The linguistic facts of the situation are little disputed by the relevant scholars. However, linguistic data alone cannot determine whether this migration was peaceful or invasive. Different linguists have argued for either, or for a combination of both, on extra-linguistic grounds.

History and political background
In the earliest phase of Indo-European studies, Sanskrit was assumed to be very close to (if not identical with) the Proto-Indo-European language. Its geographical location also fitted the then-dominant Biblical model of human migration, according to which Europeans were descended from the tribe of Japhet, which was supposed to have expanded from Mount Ararat after the Flood. Iran and northern India seemed to be likely early areas of settlement for the Japhetites.

In the course of the 19th century, as the field of historical linguistics progressed, and Bible-based models of history were abandoned, it became clear that Sanskrit could no longer be given priority. In line with late 19th century ideas, an Aryan 'invasion' was made the vehicle of the language transfer. Max Muller estimated the date to be around 1500–1200 BC, a date also supported by more recent scholars.

The Indus Valley civilization, discovered in the 1920s, was unknown to 19th century scholars. The discovery of the Harappa and Mohenjo-daro sites changed the theory from an invasion of implicitly advanced Aryan people on an aboriginal population to an invasion of nomadic barbarians on an advanced urban civilization, an argument associated with the mid-20th century archaeologist Mortimer Wheeler. The decline roughly contemporaneous to the proposed migration movement was seen initially as an independent confirmation of these early suggestions (compare the causal relations between the decline of the Roman Empire and the Germanic Migration Period).

Among the archaeological signs claimed by Wheeler to support the theory of an invasion are the many unburied corpses found in the top levels of Mohenjo-daro. They were interpreted by Wheeler as victims of a conquest of the city, but Wheeler's interpretation is no longer accepted by many scholars (e.g. Bryant 2001). Wheeler himself expressed no certainty, but wrote, in a famous phrase, that "Indra stands accused".

In the later 20th century, ideas were refined, and so now migration and acculturation are seen as the methods whereby Indo-Aryan spread into northwest India around 1700 BCE. These changes are exactly in line with changes in thinking about language transfer in general, such as the migration of the Greeks into Greece (between 2100 and 1600 BCE), or the Indo-Europeanization of Western Europe (between 2200 and 1300 BCE).

Political debate
The debate over such a migration, and the accompanying influx of elements of Vedic religion from Central Asia is still politically charged and hotly debated in India. Hindutva (Hindu nationalist) organizations, especially, remain opposed to the concept, for political, religious, and scientific reasons, while many Indian Marxists and a fraction of the Dalit Movement support the theory in opposition to the Hindu nationalists. Outside India, the perceived political overtones of the theory are not as pronounced, and it is discussed in the larger framework of Indo-Iranian and Indo-European expansion.

Linguistics
Linguists have several rules of thumb they use to gauge the place of origin of a family. One is that the area of highest linguistic diversity of a language family is usually fairly close to the area of its origin ( cited in ); thus, for example, while the modern nation with the highest number of speakers of Germanic languages is the United States, the highest diversity of longstanding Germanic languages is found in northern Europe. This is due to the fact that placing the origin of a language family in the area of least heterogeneity requires postulating the fewest number of migrations, and because of the unlikelihood of several linguistic features developing in an area without leaving any representatives behind. By this criterion, India, home to only the Indo-Aryan subfamily, seems to be an exceedingly unlikely candidate for the origin of the Indo-European languages. notes that "[i]t is far more logical to imagine that the Indo-Iranian languages had moved away from the mass of other Indo-European languages rather than the converse, and to argue otherwise is to engage in the same type of absurdity as assuming that the Finno-Ugric languages originated in Hungary." Conversely, such a criterion suggests a North Indian homeland for the Dravidian languages. However, extinctions of unrecorded languages may affect this measure. Most linguists believe Indo-European to have originated somewhere around the Black Sea : a favorite candidate is the Kurgan hypothesis.

The early formation of political states also affects the distribution of languages. The Punjab was in historical times settled by Iranians, Greeks, Kushans (replacing Greeks and their language), and Hephthalites, yet Indo-Aryan languages dominate, probably due to the dominance of later Indian empires and states. Hence in regions where Persian and Indian empires dominated many languages died out. This process can be seen in the elimination of Saka and Tocharian languages through the influence of Persians, Buddhism (spreading Prakrit language), and Turks.

Dialectical variation
It has long been recognized that a binary tree model cannot capture all linguistic alignments; certain areal features cut across language groups and are better explained through a model treating linguistic change like waves rippling out through a pond. This is true of the Indo-European languages as well. Various features originated and spread while Proto-Indo-European was still a dialect continuum. These features sometimes cut across sub-families: for instance, the instrumental, dative, and ablative plurals in Germanic, Baltic and Slavic feature endings beginning with -m-, rather than the usual -*bh-, e.g. Old Church Slavic instrumental plural synŭ-mi 'with sons', despite the fact that the Germanic languages are centum languages, while Baltic and Slavic are satem languages.

There is a close relationship between the dialectical relationship of the Indo-European languages and the actual geographical arrangement of the languages in their earliest attested forms that makes an Indian origin for the family unlikely. Given the geographic barriers separating the subcontinent from the rest of Eurasia, such as the Hindu Kush mountains and the existence of the various Indo-European sub-families, an Indian Urheimat would require several successive staggered migrations (c.f. Out of India theory). However, this would destroy the close arrangement between archaic shared linguistic features and geographical arrangement noted above. This arrangement is better explained by a radial expansion of the Indo-Europeans, a corollary of which is the migration of Indo-Aryan speakers into the subcontinent.

Substrate influence
Most of the languages of North India belong to a single language family, the Indo-Aryan subgroup of the Indo-European family of languages. The languages of South India belong to a different language family, the Dravidian languages, which has not been proven to be linked with any other language family.

The presence of retroflex consonants (including L) in Vedic Sanskrit is generally taken by linguists to indicate the influence of a non-Indo-European speaking substratum population.
 * These sounds are found throughout Dravidian and Munda and are reconstructed for proto-Dravidian and proto-Munda.
 * They are neither reconstructible for proto-Indo-European nor for proto-Indo-Iranian.
 * They are also extremely rare among other Indo-European languages (they phonetically emerged in Swedish and Norwegian only in recent centuries).
 * Presence of words with Dravidian and Munda etymologies in Sanskrit (some of these etymologies have been challenged, though most have not).

Critics argue that the "substratum" influences from Dravidian and Munda could equally well be adstratum influences through mutual contact without conquest, or superstratum given the advanced nature of the precedent Mature Harappan culture.

While Dravidian languages are primarily confined to the South of India, there is a striking exception: the Brahui (which is spoken in parts of Baluchistan), the linguistic equivalent of a relict population, perhaps indicating that Dravidian languages were formerly much more widespread and were supplanted by the incoming Indo-Aryan languages.

Chronology
The Indo-Aryan migration is dated subsequent to the Mature Harappan culture and the arrival of Indo-Aryans in the Indian subcontinent dated during the Late Harappan period. Based on linguistic data, many scholars argue that the Indo-Aryan languages were introduced to India in the 2nd millennium BCE. The standard model for the entry of the Indo-European languages into India is that this first wave went over the Hindukush, forming the Gandhara grave culture or Swat culture, either into the headwaters of the Indus or the Ganges (and probably, both). The language of the Rigveda, earliest stratum of Vedic Sanskrit is assigned to about 1500-1200 BCE.

The separation of Indo-Aryans proper from Proto-Indo-Iranians has been dated to roughly 2000 BCE–1800 BCE. It is believed Indo-Aryans reached Assyria in the west and the Punjab in the east before 1500 BC: the Indo-Aryan Mitanni rulers appear from 1500, and the Gandhara grave culture emerges from 1600. This suggests that Indo-Aryan tribes would have had to be present in the area of the Bactria-Margiana Archaeological Complex (southern Turkmenistan / northern Afghanistan) from 1700 BC at the latest (incidentally corresponding with the decline of that culture).

The Swat culture is the most likely locus of the earliest presence east of the Hindukush of the bearers of Rigvedic culture, and Parpola (1999) based on this assumes an immigration to the Punjab ca. 1700-1400, but he also postulates a first wave of immigration from as early as 1900 BC, corresponding to the Cemetery H culture.

Rajesh Kochhar argues that there were three waves of Indo-Aryan immigration that occurred after the mature Harrapan phase : the Murghamu (BMAC) related people who entered Baluchistan at Pirak, Mehrgarh south cemetery etc and later merged with the post-urban Harappans during the late Harappans Jhukar phase; the Swat IV that co-founded the Harappan cemetery H phase in Punjab and the Rigvedic Indo-Aryans of Swat V that later absorbed the cemetery H people and gave rise to the PGW culture. He dates the first two to 2000-1800 BCE and the third to 1400 BCE.

Early Indo-Aryans
The earliest written evidence for an Indo-Aryan language dates to about 1500 BCE and is found in northern Syria in Hittite records regarding one of their neighbors, the Hurrian-speaking Mitanni. In a treaty with the Hittites, the king of Mitanni, after swearing by a series of Hurrian gods, swears by the gods Indara, Mitraśil, Naśatianna and Uruvanaśśil, who correspond to the Vedic gods Indra, Mitra, Nāsatya and Varuṇa. Contemporary equestrian terminology, as recorded in a horse-training manual whose author is identified as "Kikkuli the Mitannian" contains Indo-Aryan loanwords. The personal names and gods of the Mitanni aristocracy also bear traces of Indo-Aryan. In 1960, Paul Thieme demonstrated to the satisfaction of most scholars that this vocabulary was specifically Indo-Aryan, as opposed to Iranian or Indo-Iranian. Because of this association of Indo-Aryan with horsemanship and the Mitanni aristocracy, it is generally presumed that, after superimposing themselves as rulers on a native Hurrian-speaking population about the 15th-16th centuries BCE, Indo-Aryan charioteers were absorbed into the local population and adopted the Hurrian language.

Brentjes argues that there is not a single cultural element of central Asian, eastern European, or Caucasian origin in the Mitannian area and associates with an Indo-Aryan presence the peacock motif found in the Middle East from before 1600 BCE and possible as long ago as 2100 BCE.

However, received opinion rejects the possibility that the Indo-Aryans of Mitanni came from the Indian subcontinent as well as the possibility that the Indo-Aryans of the Indian subcontinent came from the territory of Mitanni, leaving migration from the north the only likely scenario.

There were also tribes (the Maiotes and Sindoi/Indoi) that spoke Indo-Aryan languages in the Ukraine. Kretschmer (1944) saw this as proof for the Pontic homeland hypothesis.

Rigveda
The Rigveda is by far the most archaic testimony of Vedic Sanskrit. It is often assumed that the Rigveda represents a pastoral or nomadic, mobile culture, still centered on the Indo-Iranian Soma cult and fire worship. With all the effort to glimpse historical information from the hymns of the Rigveda, it should not be forgotten that the purpose of these hymns is ritualistic, not historiographical or ethnographical, and any information about the way of life or the habitat of their authors is incidential and philologically extrapolated from the context.

Rigvedic society as pastoral society
The mobile nature of the Vedic religion is illustrated by the laying out of the ritual precinct as part of the ritual, rather than the existence of fixed temples. This holds for the invitation of Indra to the Soma ritual as well as for the Agnicayana, the piling-up of the fire altar. Cities or fortresses (') are mentioned in the Rigveda mainly as the abode of hostile peoples, while the Aryan tribes live in ', a term translated as "settlement, homestead, house, dwelling", but also "community, tribe, troops".

Indra in particular is described as destroyer of fortresses, e.g. RV 4.30.20ab:
 * "Indra overthrew a hundred fortresses of stone."
 * "Indra overthrew a hundred fortresses of stone."

The Rigveda does contain some phrases referring to elements of an urban civilization, other than the mere viewpoint of an invader aiming at sacking the fortresses. These references become increasingly frequent in the younger books 1 and 10, linguistically dated as contemporary to the early parts of the Atharvaveda and the mantras of the Yajurveda. Here, for example, Indra is compared to the lord of a city (purapatis) in RV 1.173.10, a ship with a hundred oars is mentioned in 1.116 and metal forts (puras ayasis) in 10.101.8. Since the Vedic books appear to have been composed over a long period of gradual change, rather than being a snapshot of society at one particular moment, these late Rigvedic books may indeed describe an urbanized amalgamation of pastoral Indo-Aryan culture with indigenous, Late Harappan elements even in the view of proponents of immigration, roughly representing the early phase of the Kuru kingdom (ca. 12th century BC). Furthermore, there were also cities in the Post-Harappan period in the Punjab region.

However, according to S.P. Gupta (1996), "ancient civilizations had both the components, the village and the city, and numerically villages were many times more than the cities. (...) if the Vedic literature reflects primarily the village life and not the urban life, it does not at all surprise us.". Gregory Possehl (1977) argued that the "extraordinary empty spaces between the Harappan settlement clusters" indicates that pastoralists may have "formed the bulk of the population during Harappan times". Agriculturalists, pastoralists as well as the city and village life may have coexisted in the same region. Such a view would imply that the only testimony surviving of Harappan times is not from the urban centers, but preserves the rituals of rural pastoralists living between the cities.

Rigvedic reference to migration
There is no explicit mention of an outward or inward migration in the Rigveda. In RV 7.6.3, Agni turned the godless and the Dasyus westward, and not southward, as would be required by some versions of the AIT. Some of the tribes that fought against Sudas on the banks of the Parusni River during the Dasarajna battle have maybe migrated to western countries in later times, as they are possibly connected with some Iranian peoples (e.g. the Pakthas, Bhalanas).

While the Avesta does mention an external homeland of the Zoroastrians, the Rigveda does not explicitly refer to an external homeland or to a migration. Later texts than the Rigveda (such as the Puranas) seem to be more centered in the Ganges region. This shift from the Punjab to the Gangetic plain continues the Rigvedic tendency of eastward expansion, but does of course not imply an origin beyond the Indus watershed.

Rigvedic Rivers and Reference of Samudra

The geography of the Rigveda seems to be centered around the land of the seven rivers. While the geography of the Rigvedic rivers is unclear in the early mandalas, the Nadistuti hymn is an important source for the geography of late Rigvedic society.

The Sarasvati River is one of the chief Rigvedic rivers. The Nadistuti hymn in the Rigveda mentions the Sarasvati between the Yamuna in the east and the Sutlej in the west, and later texts like the Mahabharata mention that the Sarasvati dried up in a desert.

Most scholars agree that at least some of the references to the Sarasvati in the Rigveda refer to the Ghaggar-Hakra River, while the Helmand is often quoted as the locus of the early Rigvedic river. Whether such a transfer of the name has taken place, either from the Helmand to the Ghaggar-Hakra, or conversely from the Ghaggar-Hakra to the Helmand, is a matter of dispute. Identification of the early Rigvedic Sarasvati with the Ghaggar-Hakra before its drying up would place the Rigveda well before 1700 BC, and thus well outside the range commonly assumed by Indo-Aryan migration theory.

A non-Indo-Aryan substratum in the river-names and place-names of the Rigvedic homeland would support an external origin of the Indo-Aryans. However, most place-names in the Rigveda and the vast majority of the river-names in the north-west of India are Indo-Aryan (Bryant 2001).

Iranian Avesta
The religious practices depicted in the Rgveda and those depicted in the Avesta, the central religious text of Zoroastrianism—the ancient Iranian faith founded by the prophet Zarathustra—have in common the deity Mitra, priests called hotr in the Rgveda and zaotr in the Avesta, and the use of a hallucinogenic compound that the Rgveda calls soma and the Avesta haoma. However, the Indo-Aryan deva, meaning 'god,' is cognate with the Iranian daeva, meaning 'demon'. Likewise, the Indo-Aryan asura, meaning 'demon,' is cognate with the Iranian ahura, meaning 'god,' suggesting that, at some point, a rivalry between Indo-Aryans and Iranians that found religious expression, as the Indologist Thomas Burrow has proposed.

Two alternative dates for Zarathustra can be found in Greek sources: 5000 years before the Trojan War, i.e. 6000 BCE, or 258 years before Alexander, i.e. the 6th century BCE, the latter of which used to provide the conventional dating but has since been traced to a fictional Greek source. Linguists such as Burrow argue that the strong similarity between the Avestan language of the Gathas—the oldest part of the Avesta—and the Vedic Sanskrit of the Rgveda pushes the dating of Zarathustra or at least the Gathas closer to the conventional Rgveda dating of 1500–1200 BCE, i.e. 1100 BCE, possibly earlier. Boyce concurs with a lower date of 1100 BCE and tentatively proposes an upper date of 1500 BCE. Gnoli dates the Gathas to around 1000 BCE, as does J.P. Mallory, with the caveat of a 400 year leeway on either side, i.e. between 1400 and 600 BCE. Therefore the date of the Avesta could also indicate the date of the Rigveda.

There is mention in the Avesta of Airyanem Vaejah, the legendary homeland of the Aryans as well as Zarathustra himself. Gnoli's interpretation of geographic references in the Avesta situates the Airyanem Vaejah in the Hindu Kush. For similar reasons, Boyce excludes places north of the Syr Darya and western Iranian places. With some reservations, Skjaervo concurs that the evidence of the Avestan texts makes it impossible to avoid the conclusion that they were composed somewhere in northeastern Iran. Michael Witzel points to the central Afghan highlands. Humbach derives Vaejah from cognates of the Vedic root "vij," suggesting the region of a fast-flowing river. Gnoli considers the lower Oxus region, south of the Aral Sea to be an outlying area in the Avestan world. However, according to Mallory and Mair, the probable homeland of Avestan is, in fact, the area south of the Aral Sea, which just happens to be the region of a fast-flowing river.

Other Hindu texts
Indologists have noted that "there is no textual evidence in the early literary traditions unambiguously showing a trace" of an Indo-Aryan migration. Texts like the Puranas and Mahabharata belong to a later period than the Rigveda, making their evidence less sufficient to be used for or against the Indo-Aryan migration theory.

According to the Yajur Veda, Yajnavalkya (one of the Vedic Seers) lived in the eastern region of Mithila. Aitareya Brahmana 33.6.1. records that Vishvamitra's sons migrated to the north, and in Shatapatha Brahmana 1:2:4:10 the Asuras were driven to the north.

Manu was said to be a king from Dravida. In the legend of the flood he stranded with his ship in Northwestern India or the Himalayas. The vedic land (e.g. Aryavarta, Brahmavarta) is located in Northern India or at the Sarasvati and Drsadvati River, according to Hindu texts. In the Mahabharata Udyoga Parva (108), the East is described as the homeland of the Vedic culture, where "the divine Creator of the universe first sang the Vedas." The legends of Ikshvaku, Sumati and other Hindu legends may have their origin in South-East Asia.

Puranas
The evidence from the Puranas is often disputed because they are a comparably late text. They are often dated from c.400 to c.1000 CE. The Rgveda dates from before 1200 BCE. Thus the Rgveda and the Puranas are separated by approximately 1600 to 2200 years, though scholars argue that some contents of the Puranas may date to an earlier period.

The Puranas record that Yayati left Prayag and conquered the region of Saptha Sindhu. His five sons Yadu, Druhyu, Puru, Anu and Turvashu became the main tribes of the Rigveda.

The Puranas also record that the Druhyus were driven out of the land of the seven rivers by Mandhatr and that their next king Ghandara settled in a north-western region which became known as Ghandara. The sons of the later Druhyu king Pracetas finally migrate to the region north of Afghanistan. This migration is recorded in several Puranas.

Vedic and Puranic genealogies
The Vedic and Puranic genealogies indicate a greater antiquity of the Vedic culture. The Puranas themselves state that these lists are incomplete. But the accuracy of these lists is disputed. In Arrian's Indica, Megasthenes is quoted as stating that the Indians counted from Shiva (Dionysos) to Chandragupta Maurya (Sandracottus) "a hundred and fifty-three kings over six thousand and forty-three years." The Brhadaranyaka Upanishad (4.6.), ca. 8th century BCE, mentions 57 links in the Guru-Parampara ("succession of teachers"). This would mean that this Guru-Parampara would go back about 1400 years, although the accuracy of this list is disputed. The list of kings in Kalhana's Rajatarangini goes back to the 19th century BCE.

Baudhayana Shrauta Sutra
Witzel (1989) quoted a passage of the Baudhayana Shrauta Sutra (BSS 18.44) as a "direct statement" of Indo-Aryan immigration. R.S. Sharma argued that this passage contains "the most explicit statement of immigration into the subcontinent". However, Witzel's translation of this passage was later criticized by Koenraad Elst, who wrote: "Far from attesting an eastward movement into India, this text actually speaks of a westward movement towards Central Asia, coupled with a symmetrical eastward movement from India's demographic centre around the Saraswati basin towards the Ganga basin." Other Indologists like Cardona, Willem Caland, C.G. Kashikar, D.S. Triveda, Toshifumi Goto and Hans Hock translated the passage like Elst. Since the BSS is a comparatively late text, its content is unsuitable as conclusive evidence regarding the hypothesis either way.

Archaeology
There is no clear evidence in the archaeological record for an intrusion of Indo-Aryan people into India. Many archaeologists argue that the available data reflects indigenous cultural developments. J. M. Kenoyer and many other archaeologists have pointed out that "current evidence does not support a pre- or proto-historic Indo-Aryan invasion of southern Asia. Instead, there was an overlap between Late Harappan and post-Harappan communities, with no biological evidence for major new populations." Furthermore, scholars like D. K. Chakrabarti have also pointed out that northwestern India always had cultural exchanges and trade contacts with Afghanistan and other western regions. According to Erdosy, cultural traits that have been associated with Vedic culture "originate in different places at different times and circulate widely" and it is therefore "impossible ... to regard the widespread distribution of certain beliefs and rituals ... as evidence of population movements.".

However, proponents of the theory believe that the Indo-Aryans were nomadic or at least peripatetic, following their herds of cows around from pasture to pasture. Consequently they had no permanent settlements; the RgVeda only mentions temporary huts These leave no archaeological record. So it is only to be expected that the migrations left no archaeological traces. The Huns are a comparable instance. No one doubts that the Huns actually invaded parts of western Europe on more than one occasion. Yet — because the Huns were nomads — they left no archaeological remains behind. The records come from other sources.

Proto-Indo-Iranians
Scholars commonly accept the identification of the Andronovo-Sintashta-Petrovka culture (ca. 2200 BC–1600 BC) as Indo-Iranian, i.e. ancestral to both Indo-Aryans and Iranians. Proto-Indo-Iranians are usually identified with the Sintashta-Petrovka culture of Russia and Kazakhstan. It is there that the earliest chariots are found. The follow-up Andronovo culture and BMAC correspond to the earliest phase of the rapid expansion that would reach into the Caucasus, the Iranian plateau, Afghanistan, and the Indian Subcontinent.

Asko Parpola (1988) has argued that the Dasas were the "carriers of the Bronze Age culture of Greater Iran" living in the BMAC and that the forts with circular walls destroyed by the Vedic Aryans of the Rigveda were actually located in the BMAC. Other scholars have argued that cultural links between the BMAC and the Indus Valley can also be explained by reciprocal cultural influences uniting the two cultures.

Other scholars have argued that the Andronovo culture cannot be associated with the Indo-Aryans of India or with the Mitannis because the Andronovo culture took shape too late and because no actual traces of their culture (e.g. warrior burials or timber-frame materials of the Andronovo culture) have been found in India or Mesopotamia. The archaeologist J. P. Mallory (1998) found it "extraordinarily difficult to make a case for expansions from this northern region to northern India" and remarked that the proposed migration routes "only gets the Indo-Iranian to Central Asia, but not as far as the seats of the Medes, Persians or Indo-Aryans". The evidence disputing this argument, is both linguistic and archaeological (for linguistic arguments, see e.g. Hans Hock in Bronkhorst & Deshpande 1999)

Indus Valley Civilization
Indo-Aryan migration into the northern Punjab is thus approximately contemporaneous to the final phase of the decline of the Indus-Valley civilization (IVC). Many scholars have argued that the historical Vedic culture is the result of an amalgamation of the immigrating Indo-Aryans with the remnants of the indigenous civilization, such as the Ochre Coloured Pottery culture. Such remnants of IVC culture is not yet present in the Rigveda, with its focus on chariot warfare and nomadic pastoralism in stark contrast with an urban civilization.

The decline of the IVC from about 1900 BC is not universally accepted to be connected with Indo-Aryan immigration. A regional cultural discontinuity occurred during the second millennium BC and many Indus Valley cities were abandoned during this period, while many new settlements began to appear in Gujarat and East Punjab and other settlements such as in the western Bahawalpur region increased in size. Shaffer and Liechtenstein stated that: "This shift by Harappan and, perhaps, other Indus Valley cultural mosaic groups, is the only archaeologically documented west-to-east movement of human populations in South Asia before the first half of the first millennium B.C..". This could have been caused by ecological factors, such as the drying up of the Ghaggar-Hakra River and increased aridity in Rajasthan and other places. The Indus River also began to flow east and floodings occurred. Jim Shaffer and other scholars have argued that these "internal cultural adjustments" could reflect "altered ecological, social and economic conditions affecting northwestern and north-central South Asia" and do not necessarily imply migrations.

At Kalibangan (at the Ghaggar river) the remains of what some writers claims to be fire altars have been unearthed. Some of their characteristics suggest that they could have been used for Vedic sacrifices. In addition the remains of a bathing place (suggestive of ceremonial bathing) have been found near the altars in Kalibangan. S.R. Rao found similar "fire altars" in Lothal which he thinks could have served no other purpose than a ritualistic one.

Horse and chariot

The spread of Indo-Aryan languages has been connected with the spread of the chariot in the first half of the second millennium BC. Elements supposedly introduced to India in the course of the migration include the Soma cult, as well as the horse and chariot.

About 1800 BCE, there is a major cultural change in the Swat Valley with the introduction of new ceramics and two new burial rites: flexed inhumation in a pit and cremation burial in an urn which, according to early Vedic literature, were both practiced in early Indo-Aryan society. The economy of the Swat culture not only includes the horse, but there are two horse burials as well as other horse-trappings.

Attempts of proponents of continuity to portray the Rigvedic culture as native to the subcontinent, such as identification of horses or chariots in Indus Valley Civilization art, have met with little or no acceptance.

Archaeogenetics and physical anthropology
Kenneth Kennedy (1984), who examined 300 skeletons from the Indus Valley civilization, concludes that the ancient Harappans “are not markedly different in their skeletal biology from the present-day inhabitants of Northwestern India and Pakistan”(p.102).

A later study finds no evidence of discontinuities in the skeletal record during and immediately after the decline of the Indus Valley Civilization. The two discontinuities that Kennedy finds in the prehistoric skeletal record do not correspond to the second millennium BCE. The first of these discontinuities occurred between 6000-4500 BCE (a separation of the Neolithic and Chalcolithic inhabitants of Mehrgarh), and the second occurred after 800 BCE (between 800-200 BCE). He concludes that "there is no evidence of demographic disruptions in the north-western sector of the subcontinent during and immediately after the decline of the Harappan culture. If Vedic Aryans were a biological entity represented by the skeletons from Timargarha, then their biological features of cranial and dental anatomy were not distinct to a marked degree from what we encountered in the ancient Harappans.” (1995: 54). Comparing the Harappan and Gandhara cultures, Kennedy remarks that: “Our multivariate approach does not define the biological identity of an ancient Aryan population, but it does indicate that the Indus Valley and Gandhara peoples shared a number of craniometric, odontometric and discrete traits that point to a high degree of biological affinity.” (1995: 49). The craniometric variables of prehistoric and living South Asians also showed an "obvious separation" from the prehistoric people of the Iranian plateau and western Asia (1995: 49).

Brian E. Hemphill and Alexander F. Christensen's study (1994) of the migration of genetic traits does not support a movement of Aryan speakers into the Indus Valley around 1500 BC. According to Hemphill's study, "Gene flow from Bactria occurs much later, and does not impact Indus Valley gene pools until the dawn of the Christian era." In a more recent study, Hemphill concludes that "the data provide no support for any model of massive migration and gene flow between the oases of Bactria and the Indus Valley. Rather, patterns of phenetic affinity best conform to a pattern of long-standing, but low-level bidirectional mutual exchange."

Alternate Theories
The opinion of the majority of professional archaeologists interviewed by Edwin Bryant in India during the 1990's was that there is no archaeological evidence to support external Indo-Aryan origins, or that the data is inconclusive. Kenoyer argued: "Although the overall socioeconomic organization changed, continuities in technology, subsistence practices, settlement organization, and some regional symbols show that the indigenous population was not displaced by invading hordes of Indo-Aryan speaking people. For many years, the ‘invasions’ or ‘migrations’ of these Indo-Aryan-speaking Vedic/Aryan tribes explained the decline of the Indus civilization and the sudden rise of urbanization in the Ganga-Yamuna valley. This was based on simplistic models of culture change and an uncritical reading of Vedic texts..."

Even though several alternate theories have been proposed, Indo-Aryan migration theory, as described in this article remains to be the most accepted. Its main contenders are the Anatolian hypothesis(1987) and Out of India theory, and there are many lesser accepted suggestions such as the Paleolithic Continuity Theory.

Anatolian hypothesis
The Anatolian hypothesis suggests that the speakers of the Proto-Indo-European language (PIE) lived in Anatolia in the Neolithic, and associates the distribution of historical Indo-European languages with the expansion during the Neolithic revolution during the 7th and 6th millennia. For this theory to be consistent with Indo-Aryan presence in India during the Mature Harappan period, the Indo-Iranians would have had to migrate east around 3000 BC, reaching the Indus Valley before 2600 BC. The Iranians could have migrating back west after 1900 BC.

Paleolithic Continuity Theory
The Paleolithic Continuity Theory suggests that the Indo-European languages somehow originated in Paleolithic times in Europe.

Out of India Theory
An "Out of India Theory" is proposed by some scholars, (e.g. S.S. Misra, David Frawley). Based mainly on archaeological evidence and references in Hindu texts, the Out of India theory proposes the idea of the Indo-European languages originating in India. However, some linguists do not consider South Asia a serious candidate for Proto-Indo-European origin, though some astronomers differ citing Hindu texts and analyzing star patterns, eclipses, and the like to justify OIT and the historicity of Hindu texts.

Bibliography and References

 * Johannes Bronkhorst and M.M. Deshpande. 1999. Aryan and Non-Aryan in South Asia. Ann Arbor, University of Michigan Press.
 * Chakrabarti, D.K. The Early use of Iron In India. Dilip K. Chakrabarti.1992. New Delhi: The Oxford University Press.
 * Chakrabarti, D.K. 1977b. India and West Asia: An Alternative Approach. Man and Environment 1:25-38.
 * Dhavalikar, M. K. 1995, "Fire Altars or Fire Pits?", in Sri Nagabhinandanam, Ed V Shivananda and M. K. Visweswara, Bangalore.
 * ,
 * ; --In Search of the Cradle of Civilization 1995. Quest Books; —Gods, Sages and Kings. 1991.Lotus Press, Twin Lakes, Wisconsin ISBN 0-910261-37-7; —The Rigveda and the History of India. 2001.; —Vedic Aryans and the Origins of Civilization (with N.S. Rajaram). Quebec: W.H. Press. 1995.
 * Fussman, G.; Kellens, J.; Francfort, H.-P.; Tremblay, X.: Aryas, Aryens et Iraniens en Asie Centrale. (2005) Institut Civilisation Indienne ISBN 2-86803-072-6
 * Gupta, S.P. 1996. The Indus Sarasvati Civilization. Delhi: Pratibha Prakashan.
 * Hemphill & Christensen: “The Oxus Civilization as a Link between East and West: A Non-Metric Analysis of Bronze Age Bactrain Biological Affinities”, paper read at the South Asia Conference, 3-5 November 1994, Madison, Wisconsin; p. 13.
 * --- 1999. "Out of India? The Linguistic Evidence." In Aryan and Non-Aryan in South Asia (1-18). Harvard Oriental Series Opera Minora 3. Ed. J. Bronkhorst and M. Deshpande. Cambridge: Department of Sanskrit and Indian Studies, Harvard University.
 * Kak, Subhash. The Astronomical Code of the Rgveda; Munshiram Manoharlal Publishers Pvt. Ltd (2000), ISBN 81-215-0986-6
 * Kennedy, Kenneth 1984. “A Reassessment of the Theories of Racial Origins of the People of the Indus Valley Civilization from Recent Anthropological Data.” In Studies in the Archaeology and Palaeoanthropology of South Asia (99-107).
 * --- 1995. “Have Aryans been identified in the prehistoric skeletal record from South Asia?”, in George Erdosy, ed.: The Indo-Aryans of Ancient South Asia, p.49-54.
 * Kenoyer, J.M. 1991a. The Indus Valley Tradition of Pakistan and Western India. Journal of World Prehistory 5:331-385.
 * Kenyoer, J.M. : (1991b) "Urban Process in the Indus Tradition: A Preliminary Model from Harappa." In Harappa Excavations 1986-1990 (29-60)
 * Klostermaier, Klaus. 1989. A Survey of Hinduism.Albany: State University of New ork Press.
 * Lal, B.B., (1984) Frontiers of the Indus Civilization.1984.
 * Lal, B.B., (1998) New Light on the Indus Civilization, Aryan Books, Delhi 1998
 * Lal, B.B. 2005. The Homeland of the Aryans. Evidence of Rigvedic Flora and Fauna & Archaeology, New Delhi, Aryan Books International.
 * Lal, B.B. 2002. The Saraswati Flows on: the Continuity of Indian Culture. New Delhi: Aryan Books International
 * Mallory, JP. 1998. A European Perspective on Indo-Europeans in Asia. In: The Bronze Age and Early Iron Age Peoples of Eastern and Central Asia. Ed. Mair. Washingion DC: Institue for the Study of Man.
 * Oppenheimer, Stephen; (2003) "The Real Eve: Modern Man's Journey out of Africa" ,
 * Pargiter, F.E. [1922] 1979. Ancient Indian Historical Tradition. New Delhi: Cosmo.
 * S.R. Rao. The Aryans in Indus Civilization.1993
 * Sethna, K.D. 1992. The Problem of Aryan Origins. New Delhi: Aditya Prakashan. ISBN 81-85179-67-0
 * Shaffer, Jim : (1984), The Indo-Aryan Invasions: Cultural Myth and Archaeological Reality, in John R Lukacs (ed.) The People of South Asia: The Biological Anthropology of India, Pakistan and Nepal, New York, Plenum Press, pp. 77-88.
 * Shaffer, Jim. 1986. Cultural Development in the Eastern Punjab. In Studies in the Archaeology of India and Pakistan (195-235). Ed. Jerome Jacobson. New Delhi: Oxford University Press.
 * Talageri, Shrikant: The Rigveda: A Historical Analysis. 2000. ISBN 81-7742-010-0 ; —Aryan Invasion Theory and Indian Nationalism. 1993.
 * Thapar, Romila. 1966. A History of India: Volume 1 (Paperback). ISBN 0-14-013835-8
 * Trautmann, Thomas. The Aryan Debate in India (2005) ISBN 0-19-566908-8
 * Mallory, JP. 1998. A European Perspective on Indo-Europeans in Asia. In: The Bronze Age and Early Iron Age Peoples of Eastern and Central Asia. Ed. Mair. Washingion DC: Institue for the Study of Man.
 * Oppenheimer, Stephen; (2003) "The Real Eve: Modern Man's Journey out of Africa" ,
 * Pargiter, F.E. [1922] 1979. Ancient Indian Historical Tradition. New Delhi: Cosmo.
 * S.R. Rao. The Aryans in Indus Civilization.1993
 * Sethna, K.D. 1992. The Problem of Aryan Origins. New Delhi: Aditya Prakashan. ISBN 81-85179-67-0
 * Shaffer, Jim : (1984), The Indo-Aryan Invasions: Cultural Myth and Archaeological Reality, in John R Lukacs (ed.) The People of South Asia: The Biological Anthropology of India, Pakistan and Nepal, New York, Plenum Press, pp. 77-88.
 * Shaffer, Jim. 1986. Cultural Development in the Eastern Punjab. In Studies in the Archaeology of India and Pakistan (195-235). Ed. Jerome Jacobson. New Delhi: Oxford University Press.
 * Talageri, Shrikant: The Rigveda: A Historical Analysis. 2000. ISBN 81-7742-010-0 ; —Aryan Invasion Theory and Indian Nationalism. 1993.
 * Thapar, Romila. 1966. A History of India: Volume 1 (Paperback). ISBN 0-14-013835-8
 * Trautmann, Thomas. The Aryan Debate in India (2005) ISBN 0-19-566908-8
 * Talageri, Shrikant: The Rigveda: A Historical Analysis. 2000. ISBN 81-7742-010-0 ; —Aryan Invasion Theory and Indian Nationalism. 1993.
 * Thapar, Romila. 1966. A History of India: Volume 1 (Paperback). ISBN 0-14-013835-8
 * Trautmann, Thomas. The Aryan Debate in India (2005) ISBN 0-19-566908-8
 * Thapar, Romila. 1966. A History of India: Volume 1 (Paperback). ISBN 0-14-013835-8
 * Trautmann, Thomas. The Aryan Debate in India (2005) ISBN 0-19-566908-8

title: User:Xiaopo/Scratchpad


text:
A Different Drum

A Different Drum was an independent record label and an online store based in Smithfield, Utah, founded in 1996. The label specialized in synthpop and related genres. A Different Drum has released hundreds of albums, singles, and compilations and was the biggest synthpop label in the U.S.

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A selection of songs from the label's artists have been featured on various versions of Konami's Dance Dance Revolution series, including the arcade version of Dance Dance Revolution X.

The label formally ceased operations in 2014.

A Different Drum published more than 350 albums, singles, and various artist compilations.

Notable artists

 * Alien#Six13
 * Alphaville
 * B! Machine
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 * Cosmicity
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 * Daybehavior
 * The Dignity of Labour
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 * Iris
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 * Real Life
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license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: A Different Drum


text:
Talk:Lotka–Volterra equations

Old discussions
Deleted a link to a computer game clan - Some guy.

Help! Can anyone give me the equations for Volterra-Lotka equations for three species.

What I'm talking about is

Fox eats Rabbit

Rabbit eats Carrot

Carrot eats sunlight (infinite supply of sunlight)

Sure, I can give lots of them. Let C, R, F denote mass of Carrots, number of Rabbits, and number of Foxes. For each of C, R, and F I will give several equations, increading in realisticness. All parameters are >= 0.

Carrot equation:

C1. C' = a - bR. This assumes the number of carrots is fixed ( they do not get killed, or propagate), a is the total growth due to sunlight, and b is how much is eaten by a rabbit.

C2. C' = aC - bR. This assumes a = p - d is the difference of the propagation rate and the 'old age' death rate for carrots. This implies unlimited exponential growth in the abscence of rabbits.

C3. C' = a ( 1 - C/maxC ) C - bR. This is called a 'logistic' equation, it assumes a 'maximum mass of carrots' called MaxC. This is appropriate in the likely event that you are dealing with a finite area for your ecology experiment. This ODE gives an 'S shaped curve', rapidly growing from 0, and slowly approaching 1. This is the best equation to use, unless you are looking for an easy equation. This ignores the effect that if carrots get low, the rabbits will have a hard time finding them. if you want to deal with that effect, see the F' equations below for ideas. In many situations, the foxes keep the rabbits from eating too many carrots.

Rabbit equations. Most Lotka-Volterra models use an absurd simple Rabbit - fox interaction term of R*F, which would indicate each fox eats a fixed proportion of all the rabbits. The only good thing about that is it leads to the simplest nonlinear term.

R1. R' = cR - d RF. This is the standard lame equation ( ignoring for the moment rabbit food).

R2. R' = R [ -e + f(C/R) ] / [ 1 + g(C/R) ] = R [ -eR + fC ] / [ R + gC ]. This is a pretty good equation for the rabbit food aspect, ignoring the foxes for the moment. Here -e indicates how fast rabbits starve if the ratio C/R goes to zero, and f/g indicates how fast rabbit population grows when there are plenty of carrots to go around. For a fixed f/g, you can fiddle with f and g to better fit whatever data you have.

R3. R' = R [ -eR + fC ] / [ R + gC ] - hF. Here we assume the foxes have no trouble finding rabbits, and each one eats h rabbits per unit time.

R4. R' = R [ -eR + fC ] / [ R + gC ] - hRF / [ kF + R ]. Here each fox eats h rabbits if there are plenty to go around, and 0 as the R/F ratio goes to zero. k is a free parameter to help fit whatever curve you thing is best.

F1     F' = F [ - m + pR ]. Traditional lame equation.

F2. F' = F [ -mF + pR ] / [ F + nR ]. Here -m is the death rate due to starvation when there are no rabbits around. p/n is the birth - (old age) death rate for well fed rabbits. p and n can be adjusted to fit whatever. For example, if you have a guess for what the R/F ratio is that keeps the Fox population constant, choose p so that m/p = stable(R/F).

equation summary: A good balance between reality and simplicity is

C' = a ( 1 - C/maxC ) C - bR R' = R [ -eR + fC ] / [ R + gC ] - hRF / [ kF + R ] F' = F [ -mF + pR ] / [ F + nR ]

Before working with this set of equations, simplify by 'scaling away parameters'. You can scale four paramiters (C, R, F, and time) which will allow you to eliminate 4 of the 11 parameters. The simplist is to measure C in units of maxC, so that maxC becomes 1. There are many ways to simplify the remaining equations. For example, letting replacing R with gR and F with (g/k)F removes the parameters in the denominators of the R' equation. Scaling time by a eliminates the a. This leads to ( the parameters will now have different values ):

C' = C ( 1 - C ) - b R R' = R [ -eR + fC ] / [ R + C ] - hRF / [ F + R ] F' = F [ -mF + pR ] / [ F + nR ]

Thus you have a three dimensional ODE with seven parameters.

The first step in the analysis is to find the (nontrivial) critical points, which are values of (C,R,F) where all right hand sides are zero.

This will require

R = C(1-C)/b, F = pR/m, [-eR + fC ] / [ R+C ] = hF / [F+R ].

subbing the second eqn into the third:

-eR + fC = [ hp / (m+p) ] [ R + C ].

subbing in the first equation:

(m+p) [ (bf-e)C + eC^2 ] = hp [ ( 1+b)C - C^2 ].

Dividing thru by C:

(m+p) [ (bf-e) + eC ] = hp [ ( 1+b) - C ]

which leads to

C = [ ( e - bf ) ( m + p ) + ( 1 + b ) hp ] / [ ( m + p ) e + hp ].

From this you can find R, and then F.

Obviously the parameters ( b,e,f,h,m,p ) will have to be such that C, R, and F are > 0. The parameter n does not affect the critical point. Interesting.

From this point on you will probably need computer assistance.

Next you classify this critical point. Find the three eigenvalues of the frechet derivative (this is a three by three matrix of partial derivatives) evaluated at the CP. If you are looking for a cyclical population situation, you probably need a complex conjugate pair with positive real part, or maybe a pure imaginary complex pair, and the third eigenvalue a positive real.

Finding the regions in (b,e,f,h,m,p) space where these two events happen would be nice, but likely involve a lot of UGLY work, maybe Monte Carlo simulation. (BTW be sure to use a GOOD random number generator. Contact me for advice. It might be a good time to find some realistic value ranges for  b, e, f, h, m, n, and p.

Ralph Kelsey --132.235.14.20 08:08, 5 Nov 2004 (UTC)

R Kelsey's contribution is much better than the article itself; both in terms of depth, as well as in clarifying the physical significance of each term. What a shame R Kelsey is not accessible, as he deserves to be encouraged to adapt his contribution to the main article.

I also regret that R Kelsey did not leave a reference. His last paragraph points out a way: I can surely anticipate the outcome, but would be glad to have a book as my companion enroute.

BTW my initial reaction is to be skeptical of the independence of the parameter n (RK writes takes note that this is 'interesting'. I have not done the work, but I can't suppress suspicion that this may mask a computational fault.)

--Philopedia 20:00, 4 January 2007 (UTC)

R Kelsey's contribution is a great example of a population model with three species. But it does not belong in this article. The Lotka-Volterra model has only a predator and a prey, per definition (otherwise it's not a Lokta-Volterra model). :-) See Population modeling for a description of more general population models. Unfortunately, the Wikipedia coverage of population models seems to be a bit of a mess at the moment. It is incomplete, often unclear and fragmented over at least three or four articles that should all be merged. Any theoretical biologists around with too much free time on their hands? The WikiProject_Ecology certainly sounds like a good initiative.

--Lvzon 23:39, 22 October 2007 (UTC)

 * You should be holding this discussion at Competitive Lotka–Volterra equations. Cuzkatzimhut (talk) 15:13, 24 June 2015 (UTC)

Are these really "the" Lotka-Volterra equations?
The Lotka-Volterra equations are not necessarily the ones given in the article now, which form the Lotka-Volterra predator-prey model. There is also the well known Lotka-Volterra competition model. They have in common that the per capita growth rates are linear functions of population densities or sizes. Which of these two models deserves the name Lotka-Volterra sec is unclear to me, and if nobody justifies the choice for the predator-prey model, I think it should be changed.

On the other hand, maybe mathematicians refer to the predator-prey model with "the Lotka-Volterra equations", while ecologists call it the Lotka-Volterra predator-prey model.

Also, the name of the article is a singular, while there are clearly two equations. OpenScience (talk) 12:13, 6 February 2010 (UTC)

Discrete version
Is there a discrete version? I would expect it to be able to produce chaotic results.--Henrygb 16:35, 17 Mar 2005 (UTC)

I know the competetive version of the equations produce chaos in 4d and up systems. I've also run across papers indicating that higher dimensional systems will also produce period doubling and limit cycles. Vortmax 22:50, 2 December 2006 (UTC)

Analytic solution
There is only approximate solution revealed in this article. But what about strong analytic solution? I know (thanks for Maple) that it is expressed in terms of $$LambertW$$ function. But it may be interesting if this solution would be referred in article not in so vaguely manner. 216.66.24.58 07:17, 20 November 2005
 * I disagree the numerical solutions are "only approximate". The "analytic" Lambert W function solution is a hardly superior implicit renaming, and it does not shed that much light, unless one understands Lambert $W$ functions well, in which case it is self-evident.
 * Absorbing the superfluous parameters into the variables as described in the text, retaining only the essential ratio $γ/α$, the solved parametric equation for constant $K$ is already provided in the article,
 * $$-y e^{-y}= - K (e^{x}/x)^{\gamma/\alpha}$$, so that   $$y=-W( - K (e^{x}/x)^{\gamma/\alpha})$$.
 * Without mastery of the nasty properties of the Lambert $W$ for negative real arguments, this is all but useless here, and not suitable for inclusion. Cuzkatzimhut (talk) 18:57, 23 June 2015 (UTC)

Volterra-Lotka equations.
I would like to know if the solutions of Volterra-Lotka equations are unique and global.

Reply: Yes, they are: the vector field is $$C^1$$.

Little problem with second EP
In the analysis of the second EP, the terminology and tools used are good for linear systems, not for nonlinear systems such as the LV system.

That's correct and I noticed it too. All the complex eigen values reveal is that the fixed point is periodic, but does not reveal if it is a center or a stable spiral.

You have to eliminate the time dependance by dividing the dy/dt equation by dx/dt equation, which by the chain rule gives a single seperable equation for dy/dx. When you seperate this and integrate you are left with an equation in x and y which you can relate to a third variable (magnitude) giving level sets of the origional equations. Since f(x) and f(y) approach 0 as x&y approach infinity, the level curve is closed. This translates into the closed orbits. Vortmax 22:46, 2 December 2006 (UTC)Matt

Confusing; needs explication
The article has the text: In this equation, δxy represents the growth of the predator population.

This is misleading in (at least) two respects.

Most simply, it seems to be saying that the Delta-x-y term represents the totality of change in y. This is clearly in error, since there is a second term as well. However, this problem is certainly just a matter of carelessness. Presumably the writer meant that the Delta-x-y term represents the rate of births of predators.

More troubling is question, why the birth rate is proportional to meetings between predator and prey. Wouldn't it be more logical to say that births among predators are proportional to meetings between male and female predators, i.e. proportional to y2? However, this would seem to be only appropriate for modelling in cases where predator-predator encounters are generally infrequent (in comparison with the period of gestation). For instance, the predators may be an endangered species. If predator-predator encounters are very frequent, we could ignore this term.

Or maybe the x-y term is meant to show the dependency on food supply. (This would make sense, else why introduce the relationships predator and prey). In this case, the xy proportionality of the term would seem to say that predators on average have some characteristic number of progeny and that this characteristic number is proportional to the amount of available food. (But since we are talking of an average birth rate for the entire predator population, shouldn't that be the amount of available food per predator, ie y/x?) However, this explanation only appears (approximately) valid for moderate ranges of populations: As I've already mentioned we ought to include a y2 factor if y is small. Similarly, we ought to model a cap on the birth rate in case of very abundant prey.

In any case, I think I've succeeded in making my point that the applicability for these equations needs to be addressed carefully and thorough if intelligent newcomers to the subject (I include myself) are to be able to learn from the article.

--Philopedia 19:32, 4 January 2007 (UTC)

The point (and also perhaps the problem) of the Lotka-Volterra model is that it is a highly simplified model. It is nice for showing that a predator-prey relationship can produce oscillating behaviour, but other than that it doesn't have much biological significance. Even less if the nullclines of the ODEs are perpendicular, as shown in many textbooks (mathematically elegant perhaps, but biologically unlikely). I think it needs to be mentioned in the article that this model is meant as a paradigm system, not a realistic model.

You are right that the delta-x-y-term shows dependency on food supply for the predators, whereby delta is the conversion-efficiency of prey into predators. This, is a simple mass-action term, like in chemical kinetics. It is the simplest way of stating that the birth-rate of predators depends on the prey density (because higher densities increase the probability of a predator catching a prey). For a more realistic model this should indeed include a saturation term.

Incidentally, attentive readers may also have noticed that the population numbers in some of the graphs fall below 1. Actually, they can become infinitely small in this model, and still no-one will go extinct. This is a general problem with these types of population models. In computer simulations it can be "solved" (= made more realistic) by adding a little random noise to the population variables, and setting negative values to zero.

--Levien 23:13, 22 October 2007 (UTC)

Elliptical diagram
What do the numbers in the key mean? What do the different solutions represent?

Repton infinity 06:22, 27 March 2007 (UTC)

 * And what does it mean for the population of baboons to be between zero and one? 209.6.22.99 (talk) 06:25, 28 March 2008 (UTC)
 * Agreed; that's what I came to this page to talk about. It really demands some sort of comment about how population going to 0 ever recovers. Vttale (talk) 12:15, 24 March 2017 (UTC)
 * Did you understand the remarks on the atto-fox problem? Do you wish to scale all numbers by millions? What are you actually proposing, specifically? This is not a critic's choice "bring me another stone" forum. Cuzkatzimhut (talk) 13:07, 24 March 2017 (UTC)

This is kind of a terrible example 134.88.191.99 (talk) 02:29, 11 March 2009 (UTC)

 * It is only slightly misleading: it insinuates ("a serious problem with this as a biological model") that the L−V model somehow dictates an "extremely small" minimum value for the prey, $x$, when that is merely a choice of the parameters made.
 * The minimum number $x$ is the solution of the standard transcendental equation $$-\delta x +\gamma \ln x =V+\alpha \ln (e\beta/\alpha)$$, where $V$ depends on the initial conditions.
 * A "safe" minimum may be achievable for suitable I.C.s and parameters——the solution for minimum $x$ being a LambertW.
 * Since the oscillations are about the fixed point of 25=γ/δ baboons and 20=α/β cheetahs, with the maximum $V$*, a sufficiently large V will entail a minimum number of baboon as close to 25 as desired. Cuzkatzimhut (talk) 11:05, 17 April 2015 (UTC)

This example isn't that useful without the parameter values. Ggmcnickle (talk) 18:06, 22 March 2017 (UTC)

 * Proceed to propose what you think and propose, specifically, beyond passive whining. Did you read the Wikimedia site figure's specs? Do you agree with them? Cuzkatzimhut (talk) 13:09, 24 March 2017 (UTC)

Example: baboon (prey) and cheetah (predator)
The initial conditions are 80 baboons and 40 cheetah, but what are the parameters α, β, γ and δ ? --141.53.37.95 10:59, 13 August 2007 (UTC)

Lol, so true. Also, if you look at the curves, none actually HIT 80 baboons, 40 cheetah ;) 134.88.191.99 (talk) 02:30, 11 March 2009 (UTC)

 * What about the black one? 98.245.92.246 (talk) 08:31, 30 April 2010 (UTC)

Proposal of Equation and citations
A. Lotka originally proposed this equation in 1910 in chemistry and extended it in 1920 to biological interactions, most definitely not as late as 1925(cited two different ways). This article could do with at lot more citations and not just some vague ref list at the end. Kae1is (talk) 00:42, 9 March 2010 (UTC)

Exact solutions
In case anyone wants to know this: The Lotka-Volterra equations are analytically solvable. Here are some papers: . At some point someone should put these information into the article --Svebert (talk) 20:41, 24 October 2011 (UTC)

 * Hi,you provide two references, for the exact solutions. The first reference has a broken link, and the second one links to an article by 'Martin J. Gander' that gives an analytical solution without derivation, and cites: A. Steiner and M. Arrigoni, \Die Losung gewisser Rauber-Beute-Systeme", Studia Biophysica vol. 123 (1988) No. 2. I cannot find this last reference by searching online. Can you please provide another reference? Ami-de-la-Terre (talk) 07:19, 12 October 2021 (UTC)

Wikipedia talk:Articles for creation/Atto-fox problem
An opinion from Wikiproject Mathematics suggests that information from the above article be merged into this article. I don't know how to do that without losing attribution, so I am leaving a note here in the hopes that an interested editor will take on this task. —Anne Delong (talk) 23:18, 11 November 2013 (UTC)
 * The Atto-Fox material is now in the history of this redirect: Atto-fox problem. —Anne Delong (talk) 02:53, 26 June 2014 (UTC)

plural!!
I can't believe this title said "Lotka–Volterra equation", with no final "s". I changed it. This is not about a type of equation of which any particular instance is a "Lotka–Volterra equation", nor about just one equation. In either of those situations, the singular title would be required under WP:MOS. It's about a particular pair of equations that constitute a system of equations. Michael Hardy (talk) 20:54, 29 September 2015 (UTC)

External links modified
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Updating Figures
I thought it would be a good idea to update the "baboon/cheetah" example to use svg figures. I can make them and propose the changes, but wanted to float the idea here first. --Eitanlees (talk) 13:10, 12 June 2018 (UTC)

 * The existing figures are poor quality, better ones would be great. I don't see any reason at all to seek preapproval.  --JBL (talk) 18:26, 12 June 2018 (UTC)

Applying Lotka–Volterra to Turchin's Ages of Discord
From "Ages of Discord: A Structural-demographic Analysis of American History" by Peter Turchin (ISBN 0996139540, 9780996139540), There is an equation similar to the equations that predict social instability.

Leconte source
Hello I agree that Leconte is a strange choice for a source here. There are several contradictory factors: Highly regarded journal from a legitimate Association; entirely incorrect journal subject area; physics is math; recentism; utter lack of cites. I think we should ignore the other factors and rm on the basis of 0 cites. It may in fact come back around to us as an appropriate source, having been cited by others in future but not today. Invasive Spices (talk) 21 March 2022 (UTC)
 * I am the contributor of the analytical solution contribution. Obviously, the cited source is recent (2022) which explains the lack of cites. About 'entirely incorrect journal subject area', I disagree, the Lotka-Volterra model may have been developed first in the ecology field, but has becomes a paradigm for nonlinear interactions in many fields of Physics, including in plasma turbulence in magnetically-confined plasmas. Ami-de-la-Terre (talk) 07:17, 25 March 2022 (UTC)
 * Invasive Spices, I am ok with removal. (I'm not convinced the other sources we have on this, at the beginning of the section, are so great either.)  Ami-de-la-Terre, it is often the case that people in one field re-invent ideas that have been known in other fields for a long time, thereby losing the intellectual history and context; this is presumably one of those cases.  --JBL (talk) 10:18, 25 March 2022 (UTC)
 * @JayBeeEll, I just think that it would be helpful for some people to know that there is actually an analytical solution, if you find another source, it would be good too. Ami-de-la-Terre (talk) 06:52, 30 March 2022 (UTC)
 * The first paragraph of the section already said this, and already contained two references, before your edits. -- JBL (talk) 11:16, 30 March 2022 (UTC)
 * Ami-de-la-Terre plasma turbulence Very interesting. Is the magnetic confinement predating upon the motive force of the plasma? (Simply imagining what the correspondence might be. I tried reading some of the papers on this and failed.) Invasive Spices (talk) 25 March 2022 (UTC)
 * @Invasive Spices, Well it is more like one large-scale component of the turbulence (a sheared flow) preys on the smaller scale turbulence which drives it.

Ami-de-la-Terre (talk) 07:00, 30 March 2022 (UTC)

Reorganised top section
I reorganised the first section to bring various important parts mentioned downstream together, and made the text more relevant to a non-mathematical audience. I have added a sub-section on the biological relevance of the model by providing some examples how the behaviour of the model relates to biology (in part by replacing previous text and figure). I also have replaced the words "baboon" and "cheetah" with "rabbit" and "fox". This is in the first place to keep to one example: the baboons didn't add anything (there is nothing baboon specific in the example), and now the predator ties in with the atto-fox. Also because the cheetah is not a main predator of the baboon. I hope this improves the quality of the article. --Pips15 (talk) 14:46, 20 April 2023 (UTC)

Proposal: Retitle as "Predator-Prey" or merge with other Lotka-Volterra equations
I find it confusing that this page is "Lotka-Volterra equations" but only considers predator-prey, while the competitive and  generalized equations have separate pages. I'd say a single page for all, or otherwise simply a retitling of this page to specific "Predator-prey" (and perhaps a clearer cross-reference in the intro) would resolve. Kyle MoJo (talk) 05:12, 17 February 2024 (UTC)

title: Talk:Lotka–Volterra equations


text:
Edgecumbe

Edgecumbe (Pūtauaki) is a town in the Bay of Plenty Region of the North Island of New Zealand, 15 km to the west of Whakatāne and 8 km south of the Bay's coast.

It is the main service town for the agricultural region surrounding the plains of the Rangitaiki River, which flows through the town.

State Highway 2 and the Tāneatua Branch railway line (disused) pass through the town.

The Edgecumbe Dairy Factory, established in 1915, employs 358 people, roughly a fifth of Edgecumbe's total population.

History
The town is named after the small village of Edgecumbe in Cornwall, United Kingdom, between Penryn and Helston. Edgecumbe was linked with Tauranga and Auckland by rail, with the opening of the East Coast Main Trunk Railway and Taneatua Express in 1928.

In 1987, a large earthquake centred on Edgecumbe shook the Bay of Plenty, causing widespread damage and causing much of the population to leave Edgecumbe.

In July 2004, May 2005 and April 2017, the town experienced heavy flooding that ruined many homes. In the 2005 flood, the river swelled to within 5 cm of breaching the flood banks that had been put in place seven years prior.

1987 Edgecumbe earthquake
The earthquake was a major turning point in the history of Edgecumbe. It measured 6.3 on the Richter magnitude scale and struck the Bay of Plenty region of New Zealand on 2 March 1987, just after 1.42 pm. The earthquake was very shallow, being centred approximately 8 km from the Earth's surface. The earthquake was one of the most damaging New Zealand's North Island has experienced in recent decades, with approximately 50% of the houses in Edgecumbe being damaged by the quake. There was extensive damage to a local milk factory, with large storage tanks toppled. Kawerau was another nearby town that suffered damage and Whakatāne was also badly shaken. An 80-tonne New Zealand Railways DC class locomotive was thrown onto its side.

The earthquake caused no fatalities.

However, one person died at the time as a result of a heart attack, possibly due to the quake. A foreshock just minutes before had cut the power supply and many people had moved away from heavy machinery and out of their houses. The largest aftershock was measured at 5.2 and struck at 1.52 pm.

A crack 7 km long opened in the Rangitaiki Plains near Edgecumbe, as a result of the earthquake. It is now known as the 'Edgecumbe Fault'. At one point, the land close to the fault dropped 2 m.

The epicentre of the quake was approximately 2.24 km south-south-east of the town of Matata, or 15 km north-north-west of Edgecumbe. The intense ground-shaking caused by the earthquake led to a large number of ground surface failures, including sand boils, ridge-top shatters and debris avalanches on steeper slopes. Because of the earthquake, many people left and therefore Edgecumbe's population dropped considerably. Edgecumbe is slowly recovering as it has increased by 21 in the years 2006 to 2013, possibly due to many kiwifruit orchards being affected by PSA.

2017 Edgecumbe flood
In April 2017, remnants of Cyclone Debbie caused heavy rainfall in the region. At 8:30 am on April 6, water from the Rangitaiki River breached a stopbank next to Edgecumbe, causing widespread flooding across the township. A state of emergency was declared for the Whakatāne District, and around 2000 people were evacuated to Kawerau and Whakatāne. More than 300 homes were damaged, with some becoming uninhabitable.

There were no deaths due to the flood, yet, over 50 percent of people there were highly effected by it.

The largest animal rescue in New Zealand history was recorded during the aftermath of the flood. Approximately 1000 animals were rescued by an SPCA led operation, with support from local civil defence rescue teams and veterinarians from Massey University. The event highlighted a significant gap in animal welfare emergency management capability in New Zealand.

Demographics
Edgecumbe covers 2.01 km2 and had an estimated population of as of  with a population density of  people per km2.

Edgecumbe had a population of 1,644 at the 2018 New Zealand census, an increase of 6 people (0.4%) since the 2013 census, and an increase of 18 people (1.1%) since the 2006 census. There were 546 households, comprising 855 males and 786 females, giving a sex ratio of 1.09 males per female. The median age was 34.5 years (compared with 37.4 years nationally), with 402 people (24.5%) aged under 15 years, 330 (20.1%) aged 15 to 29, 654 (39.8%) aged 30 to 64, and 255 (15.5%) aged 65 or older.

Ethnicities were 66.6% European/Pākehā, 46.4% Māori, 3.3% Pacific peoples, 4.7% Asian, and 2.0% other ethnicities. People may identify with more than one ethnicity.

The percentage of people born overseas was 9.3, compared with 27.1% nationally.

Although some people chose not to answer the census's question about religious affiliation, 56.2% had no religion, 28.5% were Christian, 4.6% had Māori religious beliefs, 0.5% were Hindu, 0.2% were Muslim, 0.2% were Buddhist and 2.9% had other religions.

Of those at least 15 years old, 126 (10.1%) people had a bachelor's or higher degree, and 318 (25.6%) people had no formal qualifications. The median income was $25,200, compared with $31,800 nationally. 144 people (11.6%) earned over $70,000 compared to 17.2% nationally. The employment status of those at least 15 was that 618 (49.8%) people were employed full-time, 165 (13.3%) were part-time, and 66 (5.3%) were unemployed.

Geography
Edgecumbe is located inland from the coast on the end of the fertile Rangitaiki Plains. The nearby beach is part of Maketu. The volcanic cone of Mount Edgecumbe, 15 kilometres to the south and close to the town of Kawerau, is visible from Edgecumbe.

Clubs and organisations
The Association Football Club Plains Rangers AFC is based at the Edgecumbe Domain, along with rugby and hockey.

The Edgecumbe Volunteer Fire Brigade is located on SH2 and services the township and surrounding area.

The town has a local rugby club, and a Kart Sport track.

Education
Edgecumbe School is a co-educational state primary school for Year 1 to 8 students, with a roll of  as of.

Edgecumbe College is a co-educational state high school for Year 9 to 13 students, with a roll of.

Notable people

 * Eve Rimmer, paraplegic athlete

title: Edgecumbe


text: User talk:128.178.71.132

title: User talk:128.178.71.132


text:
Volker Beck (politician)

Volker Beck (born 12 December 1960) is a German politician. From 1994 to 2017, he was a member of the Bundestag, the German federal parliament, for the Green Party. Beck served as the Green Party Speaker for Legal Affairs from 1994 to 2002, and as the Green Party Chief Whip in the Bundestag till 2013. He was also spokesperson for the Green Parliamentary Group for interior affairs and religion.

Political career
Beck served as spokesman of the Association of Lesbians and Gays in Germany (Lesben- und Schwulenverband in Deutschland) LSVD for over ten years. He is a supporter of same-sex marriage and has been referred to as the "Father of the German Registered Partnership Act".

Beck served as spokesperson of the Green Party's parliamentary group on legal affairs from 1994 to 2002, and as the Green Party Chief Whip in the Bundestag until 2013. He was also spokesperson for the Green Parliamentary Group for interior affairs and religion.

In a German Green leadership crisis he created the expression candystorm in support of Claudia Roth.

On his last day in session the Bundestag decided to vote on the gay marriage bill which Beck struggled for since 1989. On the last day of his parliamentary career, he received a standing ovation following the approval of same-sex marriage bill in the German Bundestag. Beck was credited as one of the most influential and important legislators campaigning for this in the German federal Parliament.

Human and gay rights activism
On 27 May 2006, Beck was attacked and injured during a gay rights demonstration in Moscow, called Moscow Pride. His attack as well as his participation at the Moscow Pride Festival is featured in the documentary Moscow Pride '06.

In May 2007, he was arrested and put in a bus in front of Moscow City Hall by the police. He had wanted to hand over a petition signed by several Members of Parliament at Moscow City Hall. He was attacked and had eggs thrown at his head. As in 2007, his partner Jacques Teyssier tried to protect Beck from attacks by anti gay rights protesters.

Alongside fellow MP Marieluise Beck, Volker Beck has been a vocal critic of Russia's human rights situation. During his time as his parliamentary group's spokesperson on human rights, he described the repeated prison sentences against Mikhail Khodorkovsky and Platon Lebedev as "disproportionate." He has often urged the German government and the European Union to make unequivocally clear to the Russian government that "the only Russia that can be a strategic partner is one that observes the rule of law." He also called on Western investors to "put an end to their opportunistic silence."

Beck spoke in favor of continuing European Union sanctions on Uzbekistan, originally instated after the Uzbek government did not allow an international investigation of the Andijan massacre, on 17 October 2006.

In June 2016, Beck was among 19 people detained by Turkish police central in Istanbul; the police had tried to disperse dozens of activists attempting to gather to mark the annual Istanbul Pride week after authorities banned their march.

Compensation for victims of Nazism
Beck is also considered an important figure in securing compensation for victims of Nazism. Alongside Otto Graf Lambsdorff, he negotiated the $4.6 billion fund to compensate people enslaved by the Nazis and later served as one of the trustees of the Foundation "Remembrance, Responsibility and Future". In 2008, he claimed that the German Parliament should raise more money to fight against right wing extremists. In 2015, he was awarded the Leo-Baeck-Award by the Central Council of Jews in Germany.

In 2013, Beck complained to R&S, the maker of condoms that were used in a far-right party's campaign against immigrant births. The company apologized and pledged to donate 10,000 condoms as well as the proceeds from the NPD order to a German foundation against right-wing extremism and anti-Semitism.

Foreign affairs
In 2015, following the agreement on a Joint Comprehensive Plan of Action with Iran, Beck criticized Vice Chancellor Sigmar Gabriel for calling the country a "friend" and argued that "with its position on Israel and its human rights situation this Iranian regime cannot be Germany's friend or strategic partner."

In October 2023, he wanted to stop the financial support for the Palestinian authorities, as well for the water supply, hospitalized and traumatized Palestinian children in hospitals.

Sex with minors
In 2013, Beck received major media attention, when his 1988 essay: "Amending criminal law? An appeal for a realistic, new orientation of sexuality politics" which promoted the decriminalization of pedosexuality, was released in full by Spiegel Online, showing that editorial changes had not altered the message of the essay as he had claimed. In response to the essay's initial discovery amidst revelations within the Green Party regarding members' prior involvement in the pedophilia movement, Beck distanced himself from his previous writing and mentioned that reports from German child care facilities combating child sexual abuse dramatically changed his position.

Subsequent career
Since November 2017 Beck is lecturer at the Center for Studies in Religious Sciences (CERES) at the Ruhr University Bochum.

Other activities

 * Aktion Deutschland Hilft (Germany's Relief Coalition), Member of the Board of Trustees (since 2005)
 * American Jewish Committee's Ramer Institute Berlin, Member of the Advisory Board (since 2002)
 * Association of German Foundations, Member of the Parliamentary Advisory Board (since 2005)
 * Foundation Flight, Expulsion, Reconciliation, Member of the Board of Trustees (since 1998)
 * Center for Information and Documentation of Work Against Racism (IDA), Member of the Advisory Board
 * German Coordinating-Council for Christian-Jewish Cooperation Organizations, Member of the Board of Trustees
 * Foundation "Remembrance, Responsibility and Future", Member of the Board of Trustees
 * Heinrich Heine University (HHU), Institut für Deutsches und Internationales Parteienrecht und Parteienforschung, Member of the Board of Trustees (since 2009)
 * Hirschfeld Eddy Foundation, Member of the Board of Trustees (since 2005)
 * Leo Baeck Foundation, Member of the Board of Trustees (since 2005)
 * Magnus Hirschfeld Foundation, Member of the Board of Trustees (since 2009)
 * Memorial to the Murdered Jews of Europe Foundation, Member of the Board of Trustees (since 1998)
 * Société des amis du Louvre, Member
 * Max Planck Institute for the Study of Societies (MPIfG), Member of the Board of Trustees (2002-2008)
 * President of the German-Israeli Society.

Personal life
Beck is openly gay. He lived in a long-term partnership with Jacques Teyssier until his death from cancer in Berlin on 25 July 2009. The couple had officially registered their partnership in 2008, after 16 years. Beck later entered into a relationship with architect Adrian Petkov, and after registering their partnership in the summer of 2017, the two married on the first day of legal same-sex marriage in Kreuzberg.

Possession of drugs
In March 2016, a police search based on reasonable suspicion that Beck had purchased and was in possession of a small amount (lowest classification under German narcotics law) of a controlled substance, lead to the seizure of 0.6 grams of an alleged drug. Media reports have claimed that the found substance was crystal meth. Legal proceedings ended with a fine of €7,000 under the classification of "minimal guilt" in April 2016. In the aftermath the Green Party decided for him to remain their spokesperson for religious affairs and elected him to become spokesperson for immigration policy. Irene Mihalic replaced him as spokesperson for domestic policy.

title: Volker Beck (politician)


text: Talk:Sharpie

title: Talk:Sharpie


text: User talk:69.134.85.87

title: User talk:69.134.85.87


text:
AirPort Extreme

AirPort Extreme is a line of residential gateways made by Apple Inc. that combine the functions of a router, network switch, wireless access point and NAS as well as varied other functions. It is one of Apple's former AirPort products. The latest model, the 6th generation, supports 802.11ac networking in addition to older standards. Versions of the same system with a built-in network-accessible hard drive are known as the AirPort Time Capsule.

Apple discontinued developing its lineup of wireless routers in 2016, but continues limited hardware and software support.

History
The first AirPort Extreme was announced at the MacWorld expo in San Francisco on January 7, 2003. It featured 802.11g wireless technology for the first time in an AirPort base station.

The name "AirPort Extreme" originally referred to any one of Apple's AirPort products that implemented the (then) newly introduced 802.11g Wi-Fi standard, differentiating it from earlier devices that ran the slower 802.11a and b standards. At that time (circa 2003) the gateway part of this lineup was known as the AirPort Extreme Base Station. With the addition of the even faster Draft-N standards in early 2009 the naming of "Base Station" was dropped, and was renamed to AirPort Extreme. Several minor upgrades followed with the wireless-N models, mostly to change antenna and wireless output power. In 2013, a major upgrade changed the physical structure of the device, added 802.11ac support, and added more antennas.

The AirPort Extreme has gone through three distinct physical forms. The earliest models had a similar plastic housing to the original AirPort Base Station, in a round "flying saucer" shape. From 2007 to 2013, the devices took on a flat rounded rectangle shape, similar in layout and size to the Mac mini or early models of the Apple TV. The 2013 model had a more vertical form, taller than it was wide.

Discontinuation
In approximately 2016, Apple disbanded the wireless router team that developed the AirPort Time Capsule and AirPort Extreme router. In 2018, Apple formally discontinued both products, exiting the router market. Bloomberg News noted that "Apple rarely discontinues product categories" and that its decision to leave the business was "a boon for other wireless router makers."

Overview

 * Fully featured 802.11ac Wi-Fi base station
 * Sleep Proxy Service
 * 4 Ethernet ports (3 LAN ports, 1 WAN port) — all ports are gigabit Ethernet on newer versions
 * USB 2.0 interface for disk and printer sharing
 * Built-in file server (AFP and SMB)
 * Runs VxWorks Operating System by WindRiver or a customized version of NetBSD.

AirPort Disk
The AirPort Disk feature allows users to plug a USB hard drive into the AirPort Extreme for use as a network-attached storage (NAS) device for Mac OS X and Microsoft Windows clients. Users may also connect a USB hub and printer. The performance of USB hard drives attached to an AirPort Extreme is slower than if the drive were connected directly to a computer. This is due to the processor speed on the AirPort Extreme. Depending on the setup and types of reads and writes, performance ranges from 0.5 to 17.5 MB/s for writing and 1.9 to 25.6 MB/s for reading. Performance for the same disk connected directly to a computer would be 6.6 to 31.6 MB/s for writing and 7.1 to 37.2 MB/s for reading. NTFS-formatted drives are not supported.

Original generation (2003)
The original AirPort Extreme Base Station in 2003 was so named because of its support for the 802.11g standard of the day, as well as for its ability to serve up to 50 Macs or PCs simultaneously. One feature found in most models of this generation was an internal 56K dial-up modem, allowing homes that lacked a broadband connection to enjoy wireless connectivity, albeit at dial-up speeds. It was the last generation to retain the "flying saucer" form factor. Later generations would adopt the short, rounded-square form factor that would be seen until 2013.

1st generation
On January 9, 2007 the AirPort Extreme began shipping, with support for 802.11n draft specification, and built-in wireless print and storage server.

2nd generation
On March 19, 2008, Apple released a firmware update for both models of the AirPort Extreme that, according to third-party reports, allowed AirPort Disks to be used in conjunction with Time Machine, similar to the functionality provided by AirPort Time Capsule.

3rd generation
On March 3, 2009, Apple unveiled a new AirPort Extreme with simultaneous dual-band 802.11 Draft-N radios. This allowed full 802.11 Draft-N 2x2 communication in both 802.11 Draft-N bands at the same time.

4th generation
On October 20, 2009, Apple unveiled an updated AirPort Extreme with antenna improvements.

5th generation
On June 21, 2011, Apple unveiled an updated AirPort Extreme, referred to as AirPort Extreme 802.11n (5th Generation).

The detailed table of output power comparison between the 4th generation model MC340LL/A and the 5th generation model MD031LL/A can be seen below:

 * {| class="wikitable" border="1" style="text-align:center"

! Frequency range (MHz) ! Mode ! AirPort Extreme model ! Output power (dBm) ! Output power (mW) ! Comparison (percent) ! Difference (percent) Note: A 3dB increase is equivalent to a doubling of power output.
 * rowspan="6"| 2412–2462
 * rowspan="2"| 802.11b
 * 4th generation
 * 24.57
 * 286.42
 * 100
 * rowspan="2" style="background:none; color:red"| -10.3
 * 5th generation
 * 24.10
 * 257.04
 * 89.7
 * rowspan="2"| 802.11g
 * 4th generation
 * 21.56
 * 143.22
 * 100
 * rowspan="2" style="background:none; color:green"| +114.8
 * 5th generation
 * 24.88
 * 307.61
 * 214.8
 * rowspan="2"| 802.11n HT20
 * 4th generation
 * 21.17
 * 130.92
 * 100
 * rowspan="2" style="background:none; color:green"| +96.8
 * 5th generation
 * 24.11
 * 257.63
 * 196.8
 * rowspan="2"| 5745–5825
 * rowspan="2"| 802.11a
 * 4th generation
 * 23.07
 * 202.77
 * 100
 * rowspan="2" style="background:none; color:green"| +61.1
 * 5th generation
 * 25.14
 * 326.59
 * 161.1
 * rowspan="2"| 5745–5805
 * rowspan="2"| 802.11n HT20
 * 4th generation
 * 22.17
 * 164.82
 * 100
 * rowspan="2" style="background:none; color:green"| +104.6
 * 5th generation
 * 25.28
 * 337.29
 * 204.6
 * rowspan="2"| 5755–5795
 * rowspan="2"| 802.11n HT40
 * 4th generation
 * 21.44
 * 139.32
 * 100
 * rowspan="2" style="background:none; color:green"| +181.8
 * 5th generation
 * 25.94
 * 392.64
 * 281.8
 * }
 * rowspan="2"| 802.11n HT40
 * 4th generation
 * 21.44
 * 139.32
 * 100
 * rowspan="2" style="background:none; color:green"| +181.8
 * 5th generation
 * 25.94
 * 392.64
 * 281.8
 * }
 * }

6th generation
On June 10, 2013, Apple unveiled an updated AirPort Extreme, referred to as AirPort Extreme 802.11ac (6th Generation). The 6th generation AirPort Extreme (and 5th generation AirPort Time Capsule) featured three-stream 802.11ac Wi-Fi technology with a maximum data rate of 1.3 Gbit/s, which is nearly three times faster than 802.11n. Time Machine was now supported using an external USB hard drive connected to AirPort Extreme (802.11ac model only).

Comparison chart
* 802.11n draft-specification support in 1st- to 3rd-generation models.

** 802.11ac draft-specification support in 6th-generation model.

*** All models support IPv6 tunnel mode.

**** Supported by Apple.

Discontinuation and support
According to a Bloomberg report on November 21, 2016, "Apple Inc. has disbanded its division that develops wireless routers, another move to try to sharpen the company’s focus on consumer products that generate the bulk of its revenue, according to people familiar with the matter."

In an April 2018 statement to 9to5Mac, Apple announced the discontinuation of its AirPort line, effectively leaving the consumer router market. Apple continued supporting the AirPort Extreme.

title: AirPort Extreme


text:
Sukhmani Sahib

Sukhmani Sahib (ਸੁਖਮਨੀ ਸਾਹਿਬ), known under the title of Gauri Sukhmani in the scripture (named after the Gauri raga musical measure it belongs to), is usually translated to mean Prayer of Peace is a set of 192 padas (stanzas of 10 hymns) present in the holy Guru Granth Sahib, the main scripture and living Guru of Sikhism from Ang 262 to Ang 296 (about 35 count). This Gurbani text (writing of the Gurus) was written by the 5th Guru, Guru Arjan (1563–1606) at Amritsar in around 1602. Guru Arjan first recited the bani at Gurdwara Barth Sahib in the Gurdaspur district of Punjab, India.

Content
The composition deals with such topics such as Simran (general meditation that leads to merging with God) and Nam Japna (meditation of Naam), the greatness of Saints and Sadh Sangat (holy congregation), true devotion, doing good deeds, the nature of the mind, the badness of slandering, concepts relating to Brahmvidya, Advaita, Sargun and Nirgun, materialism and death, Hukam, and other similar topics.

Recitation
Sukhmani Sahib is frequently recited by Sikhs is one of the popular Banis (compositions of the Guru). It can be done individually or as a group usually in either a place of worship (Gurdwara) or at home. Reciting the entire Sukhmani Sahib takes about 60 to 90 minutes, and is sometimes undertaken by everyone in a smaller congregation with turn by turn reading. Reciting the Gurbani of Sukhmani Sahib is believed to bring peace to one's mind and aid help remembering God constantly.

Form
Sukhmani Sahib belongs to the Raag Gauri with Gauri meaning pure. The term Sukhmani comprises two words: Sukh (peace) and Mani (the treasure or jewel of the mind). It typically is found in Gutka form (small prayer book).

Structure
Sukhmani Sahib is divided into 24 Ashtpadi (Section). The Ashtpada is the Sanskrit word for a verse that has eight (Asht) metrical feet (pādi). Before the Ashtpadi begins there is a Salok of two lines and then each Ashtpadi contains eight padas of 10 hymns per pada.

History
Sukhmani Sahib was composed by Guru Arjan around 1602 before he compiled the Adi Granth. The Guru compiled it at Ramsar Sarovar (Sacred pool), Amritsar which at the time was in thick woods.

It is believed Sri Chand came to Amritsar to meet Guru Arjan and he engaged in the creation of Sukhmani. The Guru had written 16 canto and requested Sri Chand to finish the composition. Sri Chand out of humility only recited the Salok from the Mul Mantar by Guru Nanak, his father. Therefore this Salok was put by the Guru in the start of the 17 canto.

Prominent Sikh saint Baba Nand Singh would tell Sikhs to recite Sukhmani Sahib twice daily.

title: Sukhmani Sahib