Orang Utans

Orangutans are the only exclusively Asian genus of extant great ape. The largest living arboreal animals, they have proportionally longer arms than the other, more terrestrial, great apes. They are among the most intelligent primates and use a variety of sophisticated tools, also making sleeping nests each night from branches and foliage. Their hair is typically reddish-brown, instead of the brown or black hair typical of other great apes.
Native to Indonesia and Malaysia, orangutans are currently found only in rainforests on the islands of Borneo and Sumatra, though fossils have been found in Java, the Thai-Malay Peninsula, Vietnam and Mainland China. There are only two surviving species, both of which are endangered: the Bornean Orangutan (Pongo pygmaeus) and the critically endangered Sumatran Orangutan (Pongo abelii). The subfamily Ponginae also includes the extinct genera Gigantopithecus and Sivapithecus. The word "orangutan" comes from the Malay words "orang" (man) and "(h)utan" (forest); hence, "man of the forest".
Taxonomic classification

Genus Pongo[1]
Bornean Orangutan (Pongo pygmaeus)
Pongo pygmaeus pygmaeus - northwest populations
Pongo pygmaeus morio - east populations
Pongo pygmaeus wurmbii - southwest populations
Sumatran Orangutan (Pongo abelii)
The populations on the two islands were classified as subspecies until recently, when they were elevated to full specific level, and the three distinct populations on Borneo were elevated to subspecies. The population currently listed as P. p. wurmbii may be closer to the Sumatran Orangutan than the Bornean Orangutan. If confirmed, abelii would be a subspecies of P. wurmbii (Tiedeman, 1808).[3] Regardless, the type locality of pygmaeus has not been established beyond doubts, and may be from the population currently listed as wurmbii (in which case wurmbii would be a junior synonym of pygmaeus, while one of the names currently considered a junior synonym of pygmaeus would take precedence for the northwest Bornean taxon).[3] To further confuse, the name morio, as well as various junior synonyms that have been suggested,[1] have been considered likely to all be junior synonyms of the population listed as pygmaeus in the above, thus leaving the east Bornean populations unnamed.[3]


Pongo pygmaeus
In addition, a fossil species, P. hooijeri, is known from Vietnam, and multiple fossil subspecies have been described from several parts of southeastern Asia. It is unclear if these belong to P. pygmaeus or P. abeli or, in fact, represent distinct species.
Anatomy and physiology



Size relative to a 6 foot (1.8 m) man
An orangutan's standing height averages from 4 to 5 ft (1.2 to 1.5 m). On average, an orangutan weighs between 73 to 180 pounds (33 to 82 kg).[4] Males can weigh up to 250 lb (110 kg) or more.[5] Orangutan hands are similar to humans' hands; they have four long fingers and an opposable thumb. Their feet have four long toes and an opposable big toe. Orangutans can grasp things with both their hands and their feet. The largest males have an arm span of about 7.5 ft (2 m).
Orangutans have a large, bulky body, a thick neck, very long, strong arms, short, bowed legs, and no tail. They are mostly covered with long reddish-brown hair, although this differs between the species: Sumatran Orangutans have a more sparse and lighter coloured coat.[6]
The orangutan has a large head with a prominent mouth area. Adult males have large cheek flaps (which get larger as the ape ages)[7] that show their dominance to other males and their readiness to mate. The age of maturity for females is approximately 12 years. On average, orangutans may live about 35 years in the wild, and up to 60 years in captivity (though it is unknown what the typical lifespan of the orangutan in the wild is and many would certainly live much longer).[8] Both sexes have throat pouches located near their vocal chords that make their calls resonate through the forest, although the males' pouches are more developed.[8] There is significant sexual dimorphism: females can grow to around 4 ft 2 in or 127 cm and weigh around 100 lb (45 kg) while flanged adult males can reach 5 ft 9 in or 175 cm in height and weigh over 260 lb (118 kg).[9]
The arms of orangutans are twice as long as their legs, and an adult orangutan's arms can be well over seven feet from fingertip to fingertip.[8] Much of the arm's length has to do with the length of the radius and the ulna rather than the humerus.[10] Their fingers and toes are curved, allowing them to better grip onto branches. Orangutans have less restriction in the movements of their legs than humans and other primates, due to the lack of a hip joint ligament which keeps the femur held into the pelvis. Unlike gorillas and chimpanzees, orangutans are not true knuckle-walkers, and are instead fist-walkers.[11]
Ecology and behavior

Orangutans live in primary and old secondary forests, particularly dipterocarp forests and peat swamp forests.[12][13] Both species can be found in both mountainous and lowland swampy areas.[12] Sumatran orangutans live in elevations as high as 1500 m (4921 ft), while Bornean orangutans live no higher than 1000 m (3281 ft). The latter will sometimes enter grasslands, cultivated fields, gardens, young secondary forest, and shallow lakes.[14] Orangutans are the most arboreal of the great apes, spending nearly all of their time in the trees. Most of the day is spent feeding, resting, and moving between feeding and resting sites. They start the day feeding for 2–3 hours in the morning. They rest during midday followed by traveling in the late afternoon. When evening arrives, they begin to prepare their nest for the night.[15] Tigers are the major predatory threat to orangutans in Sumatra. Orangutans may also be preyed on by clouded leopards and crocodiles. The former can kill adolescents and small adult females but have not been recorded killing adult males.[15] In Borneo, orangutans are not threatened by tigers and seem to descend to the ground more often than their Sumatran relatives.[14][15] Orangutans do not swim, although least one population at a conservation refuge on Kaja island in Borneo have been photographed wading in deep water.[16]
Diet


Flanged adult male
Fruit makes up 65–90 percent of the orangutan diet. Fruits with sugary or fatty pulp are favored. Ficus fruits are commonly eaten, because they are easy to harvest and digest. Lowland dipterocarp forests are preferred by orangutans because of their plentiful fruit. Bornean orangutans consume at least 317 different food items that include young leaves, shoots, bark, insects, honey and bird eggs.[17][18]
Orangutans are opportunistic foragers, and their diets vary markedly from month to month.[18] Bark is eaten as a last resort in times of food scarcity; fruits are always more popular.
A decade-long study of urine and faecal samples at the Gunung Palung Orangutan Conservation Project in West Kalimantan has shown that orangutans give birth during and after the high fruit season (though not every year), during which they consume various abundant fruits, totalling up to 11,000 calories per day. In the low fruit season they eat whatever fruit is available in addition to tree bark and leaves, with daily intake at only 2,000 calories. Together with a long lactation period, orangutans also have a long birth interval.[19]
Orangutans are thought to be the sole fruit disperser for some plant species including the climber species Strychnos ignatii which contains the toxic alkaloid strychnine.[20] It does not appear to have any effect on orangutans except for excessive saliva production.
Geophagy, the practice of eating soil or rock, has been observed in orangutans. There are three main reasons for this dietary behavior; for the addition of minerals nutrients to their diet; for the ingestion of clay minerals that can absorb toxic substances; or to treat a disorder such as diarrhea.[21]
Orangutans use plants of the genus Commelina as an anti-inflammatory balm.[22]
Social life


Orangutans, Gunung Leuser NP, Sumatra
Orangutans live a more solitary lifestyle than the other great apes. Most social bonds occur between adult females and their dependent and weaned offspring. Adult males and independent adolescents of both sexes tend to live alone.[23] The society of the orangutan is made up of resident and transient individuals of both sexes. Resident females live with their offspring in defined home ranges that overlap with those of other adult females, who may be their relatives like mothers and sisters. One to several resident female home ranges are encompassed within the home range of a resident male, who is their primary breeder.[24] Transient males and females range broadly.[15][23][24] They usually travel alone, but as sub-adults they may travel in small groups. However this behavior does not extend to adulthood. The social structure of the orangutan can be best described as solitary but social. As the ranges of males and females overlap, they commonly encounter each other while traveling and feeding and may have brief social interactions.[23] Interactions between adult females range from friendly, to avoidance to antagonistic.[15][25] Resident males may have overlapping ranges and interactions between them tend to be hostile.
During dispersal, females tend to settle in home ranges that overlap with their mothers. However, they do not interact with them any more than the other females and they do not seem to form bonds though affiliation, grooming, or agonistic support.[25][26] Males disperse much farther from their mothers and enter into a transient phase. This phase lasts until a male can challenge and displace a dominant, resident male from his home range.[27] There are dominance hierarchies between adult males that regularly encounter each other with the most dominant males being the largest and having the best body conditions.[28] Adult males dominate sub-adult males.[29] Both resident and transient orangutans aggregate on large fruiting trees to feed. The fruits tend to be abundant, so competition is low and individuals may benefit from social contacts.[28] Orangutans will also form travelling groups in which members coordinate travel between food sources for a few days at a time.[27] These groups tend to be made of only a few individuals. They also tend to be mating consortships, each made of an adult male and female traveling and mating.[28]
Reproduction and parenting
Male orangutans exhibit arrested development. They mature at around 15 years of age by which they have fully descended testicles and can reproduce. However they do not develop the cheek pads, pronounced throat pouches, long fur or long-calls of more mature males until they gain a home range,[15][30] which occurs when they are between 15 and 20 years old.[15] These sub-adult males are known as unflanged males in contrast to the more developed flanged males. The transformation from unflanged to flanged can occur very quickly. Unflanged and flanged males have two different mating strategies. Flanged males use long calls to advertise their location which attract estrous females.[31] Unflanged males wander widely in search of estrous females and upon finding one, will force copulation on her. Both strategies are successful,[31] however females prefer to mate with flanged males and will seek them out for protection from unflanged males.[29] Resident males may form consortships with females that can last days, weeks or months after copulation.[31]


A two-week old orangutan
Female orangutans experience their first ovulatory cycle between 5.8 and 11.1 years. These occur earlier in larger females with more body fat than in thinner females.[32][33] Like other great apes, female orangutans have a period of adolescent infertility which may last for 1–4 years.[33][34] Female orangutans also have a 22-30 day menstrual cycle. Gestation lasts for nine months with females giving birth to their first offspring between 14 and 15 years old. Female orangutans have the longest interbirth intervals of the great apes, having eight years between births.[34]
Male orangutans play almost no role in raising the young. Females are the primary caregivers for the young and are also instruments of socialization for them. A female often has more than one offspring with her, usually an adolescent and an infant, and the older of them can also help in socializing their younger sibling.[35] Infant orangutans are completely dependent on their mothers for the first two years of their lives. The mother will carry the infant during traveling, as well as feed it and sleep with it in the same night nest.[15] The infant doesn’t even break physical contact with its mother for the first four months and is carried on her belly. The amount of physical contact soon wanes in the following months.[35] When an orangutan reaches the age of two, its climbing skills are more developed and will hold the hand of another orangutan while moving through the canopy, a behavior known as "buddy travel".[35] Orangutans are juveniles from about two to five years of age and start to exploratory trips from their mothers.[15] Juveniles are usually weaned at about four years of age. Adolescent orangutans seek peers and play and travel with peer groups while still having contact with their mothers.[15]
Tool use and culture


A young Orangutan at the Toledo Zoo in Ohio. The Orangutan's opposable toes and fingers give them the ability to use tools.
Like the other great apes, orangutans are among the most intelligent primates.[36] Wild chimpanzees have been known since the 1960s to use tools.[37] Tool use in orangutans was observed by Birutė Galdikas in ex-captive populations.[38]
Evidence of sophisticated tool manufacture and use in the wild was reported from a population of orangutans in Suaq Balimbing (Pongo pygmaeus abelii) in 1996.[39] These orangutans developed a tool kit for use in foraging that consisted of insect-extraction tools for use in the hollows of trees, and seed-extraction tools which were used in harvesting seeds from hard-husked fruit. The orangutans adjusted their tools according to the nature of the task at hand and preference was given to oral tool use.[40] This preference was also found in an experimental study of captive orangutans (P. pygmaeus).[41]
Carel P. van Schaik from the University of Zurich and Cheryl D. Knott from Harvard University further investigated tool use in different wild orangutan populations. They compared geographic variations in tool use related to the processing of Neesia fruit. The orangutans of Suaq Balimbing (P. abelii) were found to be avid users of insect and seed-extraction tools when compared to other wild orangutans.[42][43] The scientists suggested that these differences are cultural. The orangutans at Suaq Balimbing live in dense groups and are socially tolerant; this creates good conditions for social transmission.[42] Further evidence that highly social orangutans are more likely to exhibit cultural behaviors came from a study of leaf-carrying behaviors of ex-captive orangutans that were being rehabilitated on the island of Kaja in Borneo.[44] The above evidence is consistent with the existence of orangutan culture as geographically distinct behavioral variants which are maintained and transmitted in a population through social learning.[45]


Orangutan at Columbus Zoo and Aquarium
In 2003, researchers from six different orangutan field sites who used the same behavioral coding scheme compared the behaviors of the animals from the different sites.[46] They found that the different orangutan populations behaved differently. The evidence suggested that the differences in behavior were cultural: first, because the extent of the differences increased with distance, suggesting that cultural diffusion was occurring, and second, because the size of the orangutans’ cultural repertoire increased according to the amount of social contact present within the group. Social contact facilitates cultural transmission.[46] Carel P. van Schaik suggests that young orangutans (P. abelii) acquire tool use skills and cultural behaviors by observing and copying older orangutans.[45]
Orangutans do not limit their tool use to foraging, displaying or nest-building activities. Wild orangutans (P. pygmaeus wurmbii) in Tuanan, Borneo, were reported to use tools in acoustic communication.[47] They use leaves to amplify the kiss squeak sounds that they produce. Some have suggested that the apes employ this method of amplification in order to deceive the listener into believing that they are larger animals.[47]
Communication
A two year study of orangutan symbolic capability was conducted from 1973-1975 by Gary L. Shapiro with Aazk, a juvenile female orangutan at the Fresno City Zoo (now Chaffee Zoo) in Fresno, California. The study employed the techniques of David Premack who used plastic tokens to teach the chimpanzee, Sarah, linguistic skills. Shapiro continued to examine the linguistic and learning abilities of ex-captive orangutans in Tanjung Puting National Park, in Indonesian Borneo, between 1978 and 1980. During that time, Shapiro instructed ex-captive orangutans in the acquisition and use of signs following the techniques of R. Allen and Beatrix Gardner who taught the chimpanzee, Washoe, in the late-1960s. In the only signing study ever conducted in a great ape's natural environment, Shapiro home-reared Princess, a juvenile female who learned nearly 40 signs (according to the criteria of sign acquisition used by Francine Patterson with Koko, the gorilla) and trained Rinnie, a free-ranging adult female orangutan who learned nearly 30 signs over a two year period. For his dissertation study, Shapiro examined the factors influencing sign learning by four juvenile orangutans over a 15-month period.[48]


Orangutan "laughing"
The first orangutan language study program, directed by Dr. Francine Neago, was listed by Encyclopædia Britannica in 1988. The Orangutan language project at the Smithsonian National Zoo in Washington, D.C., uses a computer system originally developed at UCLA by Neago in conjunction with IBM.[49]
Zoo Atlanta has a touch screen computer where their two Sumatran Orangutans play games. Scientists hope that the data they collect from this will help researchers learn about socializing patterns, such as whether they mimic others or learn behavior from trial and error, and hope the data can point to new conservation strategies.[50]
A 2008 study of two orangutans at the Leipzig Zoo showed that orangutans are the first non-human species documented to use 'calculated reciprocity' which involves weighing the costs and benefits of gift exchanges and keeping track of these over time.[51]
Orangutans, along with chimpanzees, gorillas, and other apes, have even shown laughter-like vocalizations in response to physical contact, such as wrestling, play chasing, or tickling.[52]
Sexual interest in humans
Male orangutans have been known to display sexual attaction to human women to the point of rape. The cook of noted primatologist Birutė Galdikas was raped by an orangutan.[53] An orangutan tried to have sex with actress Julia Roberts but was prevented by a film crew.[54]
Conservation status



Male, child, and female Sumatran orangutans
The Sumatran species is critically endangered[55] and the Bornean species of orangutans is endangered[2] according to the IUCN Red List of mammals, and both are listed on Appendix I of CITES. The total number of Bornean orangutans is estimated to be less than 14% of what it was in the recent past (from around 10,000 years ago until the middle of the twentieth century) and this sharp decline has occurred mostly over the past few decades due to human activities and development.[2] Species distribution is now highly patchy throughout Borneo: it is apparently absent or uncommon in the south-east of the island, as well as in the forests between the Rejang River in central Sarawak and the Padas River in western Sabah (including the Sultanate of Brunei).[2] The largest remaining population is found in the forest around the Sabangau River, but this environment is at risk.[56] A similar development have been observed for the Sumatran orangutans.[55]

Orangutans are the only exclusively Asian genus of extant great ape. The largest living arboreal animals, they have proportionally longer arms than the other, more terrestrial, great apes. They are among the most intelligent primates and use a variety of sophisticated tools, also making sleeping nests each night from branches and foliage. Their hair is typically reddish-brown, instead of the brown or black hair typical of other great apes.
Native to Indonesia and Malaysia, orangutans are currently found only in rainforests on the islands of Borneo and Sumatra, though fossils have been found in Java, the Thai-Malay Peninsula, Vietnam and Mainland China. There are only two surviving species, both of which are endangered: the Bornean Orangutan (Pongo pygmaeus) and the critically endangered Sumatran Orangutan (Pongo abelii). The subfamily Ponginae also includes the extinct genera Gigantopithecus and Sivapithecus. The word "orangutan" comes from the Malay words "orang" (man) and "(h)utan" (forest); hence, "man of the forest".
Taxonomic classification

Genus Pongo[1]
Bornean Orangutan (Pongo pygmaeus)
Pongo pygmaeus pygmaeus - northwest populations
Pongo pygmaeus morio - east populations
Pongo pygmaeus wurmbii - southwest populations
Sumatran Orangutan (Pongo abelii)
The populations on the two islands were classified as subspecies until recently, when they were elevated to full specific level, and the three distinct populations on Borneo were elevated to subspecies. The population currently listed as P. p. wurmbii may be closer to the Sumatran Orangutan than the Bornean Orangutan. If confirmed, abelii would be a subspecies of P. wurmbii (Tiedeman, 1808).[3] Regardless, the type locality of pygmaeus has not been established beyond doubts, and may be from the population currently listed as wurmbii (in which case wurmbii would be a junior synonym of pygmaeus, while one of the names currently considered a junior synonym of pygmaeus would take precedence for the northwest Bornean taxon).[3] To further confuse, the name morio, as well as various junior synonyms that have been suggested,[1] have been considered likely to all be junior synonyms of the population listed as pygmaeus in the above, thus leaving the east Bornean populations unnamed.[3]


Pongo pygmaeus
In addition, a fossil species, P. hooijeri, is known from Vietnam, and multiple fossil subspecies have been described from several parts of southeastern Asia. It is unclear if these belong to P. pygmaeus or P. abeli or, in fact, represent distinct species.
Anatomy and physiology



Size relative to a 6 foot (1.8 m) man
An orangutan's standing height averages from 4 to 5 ft (1.2 to 1.5 m). On average, an orangutan weighs between 73 to 180 pounds (33 to 82 kg).[4] Males can weigh up to 250 lb (110 kg) or more.[5] Orangutan hands are similar to humans' hands; they have four long fingers and an opposable thumb. Their feet have four long toes and an opposable big toe. Orangutans can grasp things with both their hands and their feet. The largest males have an arm span of about 7.5 ft (2 m).
Orangutans have a large, bulky body, a thick neck, very long, strong arms, short, bowed legs, and no tail. They are mostly covered with long reddish-brown hair, although this differs between the species: Sumatran Orangutans have a more sparse and lighter coloured coat.[6]
The orangutan has a large head with a prominent mouth area. Adult males have large cheek flaps (which get larger as the ape ages)[7] that show their dominance to other males and their readiness to mate. The age of maturity for females is approximately 12 years. On average, orangutans may live about 35 years in the wild, and up to 60 years in captivity (though it is unknown what the typical lifespan of the orangutan in the wild is and many would certainly live much longer).[8] Both sexes have throat pouches located near their vocal chords that make their calls resonate through the forest, although the males' pouches are more developed.[8] There is significant sexual dimorphism: females can grow to around 4 ft 2 in or 127 cm and weigh around 100 lb (45 kg) while flanged adult males can reach 5 ft 9 in or 175 cm in height and weigh over 260 lb (118 kg).[9]
The arms of orangutans are twice as long as their legs, and an adult orangutan's arms can be well over seven feet from fingertip to fingertip.[8] Much of the arm's length has to do with the length of the radius and the ulna rather than the humerus.[10] Their fingers and toes are curved, allowing them to better grip onto branches. Orangutans have less restriction in the movements of their legs than humans and other primates, due to the lack of a hip joint ligament which keeps the femur held into the pelvis. Unlike gorillas and chimpanzees, orangutans are not true knuckle-walkers, and are instead fist-walkers.[11]
Ecology and behavior

Orangutans live in primary and old secondary forests, particularly dipterocarp forests and peat swamp forests.[12][13] Both species can be found in both mountainous and lowland swampy areas.[12] Sumatran orangutans live in elevations as high as 1500 m (4921 ft), while Bornean orangutans live no higher than 1000 m (3281 ft). The latter will sometimes enter grasslands, cultivated fields, gardens, young secondary forest, and shallow lakes.[14] Orangutans are the most arboreal of the great apes, spending nearly all of their time in the trees. Most of the day is spent feeding, resting, and moving between feeding and resting sites. They start the day feeding for 2–3 hours in the morning. They rest during midday followed by traveling in the late afternoon. When evening arrives, they begin to prepare their nest for the night.[15] Tigers are the major predatory threat to orangutans in Sumatra. Orangutans may also be preyed on by clouded leopards and crocodiles. The former can kill adolescents and small adult females but have not been recorded killing adult males.[15] In Borneo, orangutans are not threatened by tigers and seem to descend to the ground more often than their Sumatran relatives.[14][15] Orangutans do not swim, although least one population at a conservation refuge on Kaja island in Borneo have been photographed wading in deep water.[16]
Diet


Flanged adult male
Fruit makes up 65–90 percent of the orangutan diet. Fruits with sugary or fatty pulp are favored. Ficus fruits are commonly eaten, because they are easy to harvest and digest. Lowland dipterocarp forests are preferred by orangutans because of their plentiful fruit. Bornean orangutans consume at least 317 different food items that include young leaves, shoots, bark, insects, honey and bird eggs.[17][18]
Orangutans are opportunistic foragers, and their diets vary markedly from month to month.[18] Bark is eaten as a last resort in times of food scarcity; fruits are always more popular.
A decade-long study of urine and faecal samples at the Gunung Palung Orangutan Conservation Project in West Kalimantan has shown that orangutans give birth during and after the high fruit season (though not every year), during which they consume various abundant fruits, totalling up to 11,000 calories per day. In the low fruit season they eat whatever fruit is available in addition to tree bark and leaves, with daily intake at only 2,000 calories. Together with a long lactation period, orangutans also have a long birth interval.[19]
Orangutans are thought to be the sole fruit disperser for some plant species including the climber species Strychnos ignatii which contains the toxic alkaloid strychnine.[20] It does not appear to have any effect on orangutans except for excessive saliva production.
Geophagy, the practice of eating soil or rock, has been observed in orangutans. There are three main reasons for this dietary behavior; for the addition of minerals nutrients to their diet; for the ingestion of clay minerals that can absorb toxic substances; or to treat a disorder such as diarrhea.[21]
Orangutans use plants of the genus Commelina as an anti-inflammatory balm.[22]
Social life


Orangutans, Gunung Leuser NP, Sumatra
Orangutans live a more solitary lifestyle than the other great apes. Most social bonds occur between adult females and their dependent and weaned offspring. Adult males and independent adolescents of both sexes tend to live alone.[23] The society of the orangutan is made up of resident and transient individuals of both sexes. Resident females live with their offspring in defined home ranges that overlap with those of other adult females, who may be their relatives like mothers and sisters. One to several resident female home ranges are encompassed within the home range of a resident male, who is their primary breeder.[24] Transient males and females range broadly.[15][23][24] They usually travel alone, but as sub-adults they may travel in small groups. However this behavior does not extend to adulthood. The social structure of the orangutan can be best described as solitary but social. As the ranges of males and females overlap, they commonly encounter each other while traveling and feeding and may have brief social interactions.[23] Interactions between adult females range from friendly, to avoidance to antagonistic.[15][25] Resident males may have overlapping ranges and interactions between them tend to be hostile.
During dispersal, females tend to settle in home ranges that overlap with their mothers. However, they do not interact with them any more than the other females and they do not seem to form bonds though affiliation, grooming, or agonistic support.[25][26] Males disperse much farther from their mothers and enter into a transient phase. This phase lasts until a male can challenge and displace a dominant, resident male from his home range.[27] There are dominance hierarchies between adult males that regularly encounter each other with the most dominant males being the largest and having the best body conditions.[28] Adult males dominate sub-adult males.[29] Both resident and transient orangutans aggregate on large fruiting trees to feed. The fruits tend to be abundant, so competition is low and individuals may benefit from social contacts.[28] Orangutans will also form travelling groups in which members coordinate travel between food sources for a few days at a time.[27] These groups tend to be made of only a few individuals. They also tend to be mating consortships, each made of an adult male and female traveling and mating.[28]
Reproduction and parenting
Male orangutans exhibit arrested development. They mature at around 15 years of age by which they have fully descended testicles and can reproduce. However they do not develop the cheek pads, pronounced throat pouches, long fur or long-calls of more mature males until they gain a home range,[15][30] which occurs when they are between 15 and 20 years old.[15] These sub-adult males are known as unflanged males in contrast to the more developed flanged males. The transformation from unflanged to flanged can occur very quickly. Unflanged and flanged males have two different mating strategies. Flanged males use long calls to advertise their location which attract estrous females.[31] Unflanged males wander widely in search of estrous females and upon finding one, will force copulation on her. Both strategies are successful,[31] however females prefer to mate with flanged males and will seek them out for protection from unflanged males.[29] Resident males may form consortships with females that can last days, weeks or months after copulation.[31]


A two-week old orangutan
Female orangutans experience their first ovulatory cycle between 5.8 and 11.1 years. These occur earlier in larger females with more body fat than in thinner females.[32][33] Like other great apes, female orangutans have a period of adolescent infertility which may last for 1–4 years.[33][34] Female orangutans also have a 22-30 day menstrual cycle. Gestation lasts for nine months with females giving birth to their first offspring between 14 and 15 years old. Female orangutans have the longest interbirth intervals of the great apes, having eight years between births.[34]
Male orangutans play almost no role in raising the young. Females are the primary caregivers for the young and are also instruments of socialization for them. A female often has more than one offspring with her, usually an adolescent and an infant, and the older of them can also help in socializing their younger sibling.[35] Infant orangutans are completely dependent on their mothers for the first two years of their lives. The mother will carry the infant during traveling, as well as feed it and sleep with it in the same night nest.[15] The infant doesn’t even break physical contact with its mother for the first four months and is carried on her belly. The amount of physical contact soon wanes in the following months.[35] When an orangutan reaches the age of two, its climbing skills are more developed and will hold the hand of another orangutan while moving through the canopy, a behavior known as "buddy travel".[35] Orangutans are juveniles from about two to five years of age and start to exploratory trips from their mothers.[15] Juveniles are usually weaned at about four years of age. Adolescent orangutans seek peers and play and travel with peer groups while still having contact with their mothers.[15]
Tool use and culture


A young Orangutan at the Toledo Zoo in Ohio. The Orangutan's opposable toes and fingers give them the ability to use tools.
Like the other great apes, orangutans are among the most intelligent primates.[36] Wild chimpanzees have been known since the 1960s to use tools.[37] Tool use in orangutans was observed by Birutė Galdikas in ex-captive populations.[38]
Evidence of sophisticated tool manufacture and use in the wild was reported from a population of orangutans in Suaq Balimbing (Pongo pygmaeus abelii) in 1996.[39] These orangutans developed a tool kit for use in foraging that consisted of insect-extraction tools for use in the hollows of trees, and seed-extraction tools which were used in harvesting seeds from hard-husked fruit. The orangutans adjusted their tools according to the nature of the task at hand and preference was given to oral tool use.[40] This preference was also found in an experimental study of captive orangutans (P. pygmaeus).[41]
Carel P. van Schaik from the University of Zurich and Cheryl D. Knott from Harvard University further investigated tool use in different wild orangutan populations. They compared geographic variations in tool use related to the processing of Neesia fruit. The orangutans of Suaq Balimbing (P. abelii) were found to be avid users of insect and seed-extraction tools when compared to other wild orangutans.[42][43] The scientists suggested that these differences are cultural. The orangutans at Suaq Balimbing live in dense groups and are socially tolerant; this creates good conditions for social transmission.[42] Further evidence that highly social orangutans are more likely to exhibit cultural behaviors came from a study of leaf-carrying behaviors of ex-captive orangutans that were being rehabilitated on the island of Kaja in Borneo.[44] The above evidence is consistent with the existence of orangutan culture as geographically distinct behavioral variants which are maintained and transmitted in a population through social learning.[45]


Orangutan at Columbus Zoo and Aquarium
In 2003, researchers from six different orangutan field sites who used the same behavioral coding scheme compared the behaviors of the animals from the different sites.[46] They found that the different orangutan populations behaved differently. The evidence suggested that the differences in behavior were cultural: first, because the extent of the differences increased with distance, suggesting that cultural diffusion was occurring, and second, because the size of the orangutans’ cultural repertoire increased according to the amount of social contact present within the group. Social contact facilitates cultural transmission.[46] Carel P. van Schaik suggests that young orangutans (P. abelii) acquire tool use skills and cultural behaviors by observing and copying older orangutans.[45]
Orangutans do not limit their tool use to foraging, displaying or nest-building activities. Wild orangutans (P. pygmaeus wurmbii) in Tuanan, Borneo, were reported to use tools in acoustic communication.[47] They use leaves to amplify the kiss squeak sounds that they produce. Some have suggested that the apes employ this method of amplification in order to deceive the listener into believing that they are larger animals.[47]
Communication
A two year study of orangutan symbolic capability was conducted from 1973-1975 by Gary L. Shapiro with Aazk, a juvenile female orangutan at the Fresno City Zoo (now Chaffee Zoo) in Fresno, California. The study employed the techniques of David Premack who used plastic tokens to teach the chimpanzee, Sarah, linguistic skills. Shapiro continued to examine the linguistic and learning abilities of ex-captive orangutans in Tanjung Puting National Park, in Indonesian Borneo, between 1978 and 1980. During that time, Shapiro instructed ex-captive orangutans in the acquisition and use of signs following the techniques of R. Allen and Beatrix Gardner who taught the chimpanzee, Washoe, in the late-1960s. In the only signing study ever conducted in a great ape's natural environment, Shapiro home-reared Princess, a juvenile female who learned nearly 40 signs (according to the criteria of sign acquisition used by Francine Patterson with Koko, the gorilla) and trained Rinnie, a free-ranging adult female orangutan who learned nearly 30 signs over a two year period. For his dissertation study, Shapiro examined the factors influencing sign learning by four juvenile orangutans over a 15-month period.[48]


Orangutan "laughing"
The first orangutan language study program, directed by Dr. Francine Neago, was listed by Encyclopædia Britannica in 1988. The Orangutan language project at the Smithsonian National Zoo in Washington, D.C., uses a computer system originally developed at UCLA by Neago in conjunction with IBM.[49]
Zoo Atlanta has a touch screen computer where their two Sumatran Orangutans play games. Scientists hope that the data they collect from this will help researchers learn about socializing patterns, such as whether they mimic others or learn behavior from trial and error, and hope the data can point to new conservation strategies.[50]
A 2008 study of two orangutans at the Leipzig Zoo showed that orangutans are the first non-human species documented to use 'calculated reciprocity' which involves weighing the costs and benefits of gift exchanges and keeping track of these over time.[51]
Orangutans, along with chimpanzees, gorillas, and other apes, have even shown laughter-like vocalizations in response to physical contact, such as wrestling, play chasing, or tickling.[52]
Sexual interest in humans
Male orangutans have been known to display sexual attaction to human women to the point of rape. The cook of noted primatologist Birutė Galdikas was raped by an orangutan.[53] An orangutan tried to have sex with actress Julia Roberts but was prevented by a film crew.[54]
Conservation status



Male, child, and female Sumatran orangutans
The Sumatran species is critically endangered[55] and the Bornean species of orangutans is endangered[2] according to the IUCN Red List of mammals, and both are listed on Appendix I of CITES. The total number of Bornean orangutans is estimated to be less than 14% of what it was in the recent past (from around 10,000 years ago until the middle of the twentieth century) and this sharp decline has occurred mostly over the past few decades due to human activities and development.[2] Species distribution is now highly patchy throughout Borneo: it is apparently absent or uncommon in the south-east of the island, as well as in the forests between the Rejang River in central Sarawak and the Padas River in western Sabah (including the Sultanate of Brunei).[2] The largest remaining population is found in the forest around the Sabangau River, but this environment is at risk.[56] A similar development have been observed for the Sumatran orangutans.[55]

Function Matematics

"f(x)" redirects here. For the band, see f(x) (band).

Graph of example function,

The input to a function need not be a number, it can be any well defined object. For example, a function might associate the letter A with the number 1, the letter B with the number 2, and so on. There are many ways to describe or represent a function, such as a formula or algorithm that computes the output for a given input, a graph that gives a picture of the function, or a table of values that gives the output for certain specified inputs. Tables of values are especially common in statistics, science, and engineering.

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. f(x) is said, "F of X." The argument and the value may be real numbers, but they can also be elements from any given set. An example of a function is f(x) = 2x, a function which associates with every number the number twice as large. Thus, with the argument 5 the value 10 is associated, and this is written f(5) = 10.

The set of all inputs to a particular function is called its domain. The set of all outputs of a particular function is called its image. In modern mathematics functions also have a codomain. For every function, its codomain includes its image. For instance, a codomain of the function squaring real numbers f(x) = x2 usually is defined to be the set of all real numbers. The function f does not have the real numbers − 1, − 2, − 3 as outputs, thus they are in its codomain but not in its image. Codomains are useful for function composition. Composition  (f followed by g) is defined if the codomain of g is the same as the domain of f. Thus the codomain of g defines what functions may follow g. The word range is used in some texts to refer to the image and in others to the codomain, in particular in computing it often refers to the codomain. The domain and codomain are often "understood." Thus for the example given above, f(x) = 2x, the domain and codomain were not stated explicitly. They might both be the set of all real numbers, but they might also be the set of integers. If the domain is the set of integers, then image consists of just the even integers.

The set of all the ordered pairs or inputs and outputs (x, f(x)) of a function is called its graph. A common way to define a function is as the triple (domain, codomain, graph), that is as the input set, the possible outputs and the mapping for each input to its output.

Overview

A function may sometimes be described through its relationship to other functions, for example, as the inverse function of a given function, or as a solution of a differential equation. Functions can be added, multiplied, or combined in other ways to produce new functions. An important operation on functions, which distinguishes them from numbers, is the composition of functions. There are uncountably many different functions, most of which cannot be expressed with a formula or an algorithm.

Collections of functions with certain properties, such as continuous functions and differentiable functions, are called function spaces and are studied as objects in their own right, in such mathematical disciplines as real analysis and complex analysis.

Because functions are so widely used, many traditions have grown up around their use. The symbol for the input to a function is often called the independent variable or argumentand is often represented by the letter x or, if the input is a particular time, by the letter t. The symbol for the output is called the dependent variable or value and is often represented by the letter y. The function itself is most often called f, and thus the notation y = f(x) indicates that a function named f has an input named x and an output named y.

A function ƒ takes an input, x, and returns an output ƒ(x). One metaphor describes the function as a "machine" or "black box" that converts the input into the output.

The set of all permitted inputs to a given function is called the domain of the function. The set of all resulting outputs is called the image orrange of the function. The image is often a subset of some larger set, called the codomain of a function. Thus, for example, the functionf(x) = x² could take as its domain the set of all real numbers, as its image the set of all non-negative real numbers, and as its codomain the set of all real numbers. In that case, we would describe f as a real-valued function of a real variable. Sometimes, especially in computer science, the term "range" refers to the codomain rather than the image, so care needs to be taken when using the word.

It is usual practice in mathematics to introduce functions with temporary names like ƒ. For example, ƒ(x) = 2x+1, implies ƒ(3) = 7; when a name for the function is not needed, the form y = 2x+1 may be used. If a function is often used, it may be given a more permanent name as, for example,

Functions need not act on numbers: the domain and codomain of a function may be arbitrary sets. One example of a function that acts on non-numeric inputs takes English words as inputs and returns the first letter of the input word as output. Furthermore, functions need not be described by any expression, rule or algorithm: indeed, in some cases it may be impossible to define such a rule. For example, the association between inputs and outputs in a choice function often lacks any fixed rule, although each input element is still associated to one and only one output.

A function of two or more variables is considered in formal mathematics as having a domain consisting of ordered pairs or tuples of the argument values. For example Sum(x,y) = x+y operating on integers is the function Sum with a domain consisting of pairs of integers. Sum then has a domain consisting of elements like (3,4), a codomain of integers, and an association between the two that can be described by a set of ordered pairs like ((3,4), 7). Evaluating Sum(3,4) then gives the value 7 associated with the pair (3,4).

A family of objects indexed by a set is equivalent to a function. For example, the sequence 1, 1/2, 1/3, ..., 1/n, ... can be written as the ordered sequence <1/n> where n is a natural number, or as a function f(n) = 1/n from the set of natural numbers into the set of rational numbers.

Definition

One precise definition of a function is an ordered triple of sets, written (X, Y, F), where X is the domain, Y is the codomain, and F is a set of ordered pairs (a, b). In each of the ordered pairs, the first element a is from the domain, the second element b is from the codomain, and a necessary condition is that every element in the domain is the first element in exactly one ordered pair. The set of all b is known as the image of the function, and need not be the whole of the codomain. Most authors use the term "range" to mean the image, while some use "range" to mean the codomain.

The notation ƒ:X→Y indicates that ƒ is a function with domain X and codomain Y, and the function f is said to map or associate elements of X to elements of Y.

If the domain and codomain are both the set of real numbers, using the ordered triple scheme we can, for example, write the function y = x² as

,

In most situations, the domain and codomain are understood from context, and only the relationship between the input and output is given.

In set theory especially, a function f is often defined as a set of ordered pairs, with the property that if (x,a) and (x,b) are in f, then a = b. In this case statements such as (2,3) є f are appropriate when, say, f is defined by f(x) = x + 1, for all x є R.

The graph of a function is its set of ordered pairs. Part of such a set can be plotted on a pair of coordinate axes; for example, (3, 9), the point above 3 on the horizontal axis and to the right of 9 on the vertical axis, lies on the graph of y = x².

A specific input in a function is called an argument of the function. For each argument value x, the corresponding unique y in the codomain is called the function value at x, output of ƒ for an argument x, or the image of x under ƒ. The image of x may be written as ƒ(x) or as y.

A function can also be called a map or a mapping. Some authors, however, use the terms "function" and "map" to refer to different types of functions. Other specific types of functions include functionals and operators.

A function is a special case of a more general mathematical concept, the relation, for which the restriction that each element of the domain appear as the first element in one and only one ordered pair is removed. In other words, an element of the domain may not be the first element of any ordered pair, or may be the first element of two or more ordered pairs. A relation is "single-valued" when if an element of the domain is the first element of one ordered pair, it is not the first element of any other ordered pair. A relation is "left-total" or simply "total" if every element of the domain is the first element of some ordered pair. Thus a function is a total, single-valued relation.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for x = 0, and so is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph. In other parts of mathematics, non-single-valued relations are similarly conflated with functions: these are called multivalued functions, with the corresponding term single-valued function for ordinary functions.

Many operations in set theory, such as the power set, have the class of all sets as their domain, and therefore, although they are informally described as functions, they do not fit the set-theoretical definition outlined above, because a class is not necessarily a set.

Notation

Formal description of a function typically involves the function's name, its domain, its codomain, and a rule of correspondence. Thus we frequently see a two-part notation, an example being

where the first part is read:

• "ƒ is a function from N to R" (one often writes informally "Let ƒ: X → Y" to mean "Let ƒ be a function from X to Y"), or
• "ƒ is a function on N into R", or
• "ƒ is an R-valued function of an N-valued variable",

and the second part is read:

•  maps to 

Here the function named "ƒ" has the natural numbers as domain, the real numbers as codomain, and maps n to itself divided by π. Less formally, this long form might be abbreviated

where f(n) is read as "f as function of n" or "f of n". There is some loss of information: we no longer are explicitly given the domain N and codomain R.

It is common to omit the parentheses around the argument when there is little chance of confusion, thus: sin x; this is known as prefix notation. Writing the function after its argument, as in x ƒ, is known as postfix notation; for example, the factorial function is customarily written n!, even though its generalization, the gamma function, is written Γ(n). Parentheses are still used to resolve ambiguities and denote precedence, though in some formal settings the consistent use of either prefix or postfix notation eliminates the need for any parentheses.

Functions with multiple inputs and outputs

The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets.

For example, consider the function that associates two integers to their product: ƒ(x, y) = x·y. This function can be defined formally as having domain Z×Z , the set of all integer pairs; codomain Z; and, for graph, the set of all pairs ((x,y), x·y). Note that the first component of any such pair is itself a pair (of integers), while the second component is a single integer.

The function value of the pair (x,y) is ƒ((x,y)). However, it is customary to drop one set of parentheses and consider ƒ(x,y) a function of two variables, x and y. Functions of two variables may be plotted on the three-dimensional Cartesian as ordered triples of the form (x,y,f(x,y)).

The concept can still further be extended by considering a function that also produces output that is expressed as several variables. For example, consider the integer divide function, with domain Z×N and codomain Z×N. The resultant (quotient, remainder) pair is a single value in the codomain seen as a Cartesian product.

Currying

Main article: Currying

An alternative approach to handling functions with multiple arguments is to transform them into a chain of functions that each takes a single argument. For instance, one can interpret Add(3,5) to mean "first produce a function that adds 3 to its argument, and then apply the 'Add 3' function to 5". This transformation is called currying: Add 3 is curry(Add) applied to 3. There is a bijection between the function spaces C^A×B and (C^B)^A.

When working with curried functions it is customary to use prefix notation with function application considered left-associative, since juxtaposition of multiple arguments—as in (ƒ x y)—naturally maps to evaluation of a curried function. Conversely, the → and ⟼ symbols are considered to be right-associative, so that curried functions may be defined by a notation such as ƒ: Z → Z → Z = x ⟼ y ⟼ x·y

Binary operations

The familiar binary operations of arithmetic, addition and multiplication, can be viewed as functions from R×R to R. This view is generalized in abstract algebra, where n-ary functions are used to model the operations of arbitrary algebraic structures. For example, an abstract group is defined as a set X and a function ƒ from X×X to X that satisfies certain properties.

Traditionally, addition and multiplication are written in the infix notation: x+y and x×y instead of +(x, y) and ×(x, y).

Injective and surjective functions

Three important kinds of function are the injections (or one-to-one functions), which have the property that if ƒ(a) = ƒ(b) then a must equal b; the surjections (or onto functions), which have the property that for every y in the codomain there is an x in the domain such that ƒ(x) = y; and the bijections, which are both one-to-one and onto. This nomenclature was introduced by the Bourbaki group.

When the definition of a function by its graph only is used, since the codomain is not defined, the "surjection" must be accompanied with a statement about the set the function maps onto. For example, we might say ƒ maps onto the set of all real numbers.

Function composition

Main article: Function composition

A composite function g(f(x)) can be visualized as the combination of two "machines". The first takes input x and outputs f(x). The second takes f(x) and outputs g(f(x)).

The function composition of two or more functions takes the output of one or more functions as the input of others. The functions ƒ: X → Y and g: Y → Z can be composed by first applying ƒ to an argument x to obtain y = ƒ(x) and then applying g to y to obtain z = g(y). The composite function formed in this way from general ƒ and g may be written

This notation follows the form such that

The function on the right acts first and the function on the left acts second, reversing English reading order. We remember the order by reading the notation as "g of ƒ". The order is important, because rarely do we get the same result both ways. For example, suppose ƒ(x) = x² and g(x) = x+1. Then g(ƒ(x)) = x²+1, while ƒ(g(x)) = (x+1)², which is x²+2x+1, a different function.

In a similar way, the function given above by the formula y = 5x−20x³+16x⁵ can be obtained by composing several functions, namely the addition, negation, and multiplication of real numbers.

An alternative to the colon notation, convenient when functions are being composed, writes the function name above the arrow. For example, if ƒ is followed by g, where g produces the complex number e^ix, we may write

A more elaborate form of this is the commutative diagram.

Identity function

Main article: Identity function

The unique function over a set X that maps each element to itself is called the identity function for X, and typically denoted by id_X. Each set has its own identity function, so the subscript cannot be omitted unless the set can be inferred from context. Under composition, an identity function is "neutral": if ƒ is any function from X to Y, then

Restrictions and extensions

Main article: Restriction (mathematics)

Informally, a restriction of a function ƒ is the result of trimming its domain.

More precisely, if ƒ is a function from a X to Y, and S is any subset of X, the restriction of ƒ to S is the function ƒ|_S from S to Y such that ƒ|_S(s) = ƒ(s) for all s in S.

If g is a restriction of ƒ, then it is said that ƒ is an extension of g.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of gand f|_X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f|_X ∩ W and g|_X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

Inverse function

Main article: Inverse function

If ƒ is a function from X to Y then an inverse function for ƒ, denoted by ƒ⁻¹, is a function in the opposite direction, from Y to X, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible. The inverse function exists if and only if ƒ is a bijection.

As a simple example, if ƒ converts a temperature in degrees Celsius C to degrees Fahrenheit F, the function converting degrees Fahrenheit to degrees Celsius would be a suitable ƒ⁻¹.

The notation for composition is similar to multiplication; in fact, sometimes it is denoted using juxtaposition, gƒ, without an intervening circle. With this analogy, identity functions are like the multiplicative identity, 1, and inverse functions are like reciprocals (hence the notation).

For functions that are injections or surjections, generalized inverse functions can be defined, called left and right inverses respectively. Left inverses map to the identity when composed to the left; right inverses when composed to the right.

Image of a set

Main article: Image (mathematics)

The concept of the image can be extended from the image of a point to the image of a set. If A is any subset of the domain, then ƒ(A) is the subset of im ƒ consisting of all images of elements of A. We say the ƒ(A) is the image of A under f.

Use of ƒ(A) to denote the image of a subset A⊆X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinalsare also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is ƒ[A] for the set { ƒ(x): x ∈ A }.

Notice that the image of ƒ is the image ƒ(X) of its domain, and that the image of ƒ is a subset of its codomain.

Inverse image

The inverse image (or preimage, or more precisely, complete inverse image) of a subset B of the codomain Y under a function ƒ is the subset of the domain X defined by

So, for example, the preimage of {4, 9} under the squaring function is the set {−3,−2,2,3}.

In general, the preimage of a singleton set (a set with exactly one element) may contain any number of elements. For example, if ƒ(x) = 7, then the preimage of {5} is the empty set but the preimage of {7} is the entire domain. Thus the preimage of an element in the codomain is a subset of the domain. The usual convention about the preimage of an element is that ƒ⁻¹(b) means ƒ⁻¹({b}), i.e

In the same way as for the image, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write ƒ⁻¹[B] and ƒ⁻¹[b] for the preimage of a set and a singleton.

The preimage of a singleton set is sometimes called a fiber. The term kernel can refer to a number of related concepts.