This is a huge and complex subject, in which I am a mere novice; fanciers have devoted decades to exploring the pros and cons of in-breeding. Linda Hogan's book 'Canary Tales'  has one of the best chapters on the genetics of canaries that I have seen to date.  In-breeding has been used by horse breeders to produce champions for centuries; every British thoroughbred racehorse is directly descended from one of only four Arab stallions imported to England sometime in the 1700s - one of them was called The Godolphin.

The general theory is that in-breeding concentrates and 'fixes' family traits - both good and bad ones, and therein lies both the opportunity and the risk. You might intend to 'fix' good inherited traits, like superior form or colour, but actually 'fix' bad characteristics like infertility, blindness, or just an inferior posture.

Beginners in the canary fancy often adopt a 'pik 'n mix' approach; buying a pair here and a pair there, from different breeders.
The positive side of this is that unrelated birds will have widely differing genes and this will generate strong 'hybrid vigour' - tending to produce healthy, vigorous, fertile offspring. BUT, the offspring of such matings will be extremely 'hetero-zygous' - meaning 'differently -gened'.  They will exhibit every variation of shape, colour and song you can imagine. A better strategy for would-be show exhibitors, is to buy matched-pairs from the long-established strain of a champion. Birds from such a stud will be closely related (honozygous) and the owner will have 'weeded-out' many faults and reinforced the good traits in his strain over years of careful pairing. You are buying the results of his/her years of hard work but it is vital to confirm that this stud is healthy, fertile and vigorous, before you buy; ask around.

In the old days, a breeder would gradually refine his strain over many years and then, when he was winning at shows consistently, year after year, he would inbreed strongly - father to daughter, mother to son, sister to brother etc. This would produce birds which were strongly 'homo-zygous' - sharing the same concentrated genes ( rather like boiling down a soup-stock to enrich it). If the inbreeding went well, and he managed to 'fix' all the good traits,  the chicks came out like peas in a pod, with near-identical family resemblances, producing consistent show-winners, year after year. Phil Warne, the paragon of Border breeders in this country produces birds which are so consistently alike that even I can spot a Warne bird from twenty feet away in a row of show cages. They are strongly  homo-zygous - bear the same genetic stamp, like identical twins, and I assume this has been achieved by an elaborate system of in-breeding.

Now the down-side. In-breeding concentrates BAD genes as well as the good: ALL birds carry some defective genes for traits such as: poor fertility, physical abnormalities, blindness, deafness  but usually in a 'recessive' form. Recessive genes are ones that do not 'express' themselves visibly, but are 'latent'/ dormant within the DNA of the bird or person. They don't become active - i.e. 'dominant', unless they meet an identical recessive gene in the sperm/ eggs of a partner. If your mother has blue eyes and your father has brown eyes - you will generally be born with brown eyes; since brown is dominant and blue is recessive, BUT you still carry your mother's genes for blue eyes in a recessive, unexpressed form. If you marry another brown eyed person, who carries a recessive blue-eye-gene, your children may well have blue eyes. Now, lets say a hen carries a recessive gene for blindness - in normal pairings to unrelated birds this would never surface. But if that hen is paired (inbred) to its father or brother, which also carries the defective gene - bingo - you get blind chicks.

Many experts use inbreeding and insist that it is the only way to go if you want to consistently breed champion birds. BUT they caution that you MUST cull ALL defective birds which appear. In other words as long as you are prepared to kill any weaklings, or poor mothers, or infertile fathers etc. inbreeding strongly fixes and accelerates family traits, producing a stud of birds which look-alike, sing-alike, are coloured-alike and which maintain these qualities generation after generation. But the bad qualities WILL appear and you MUST cull ruthlessly.

Nature's Rule is: - out-crosses and heterozygous genes good -Inbreeding and homozygous genes bad.
A classic example is the African cheetah. When biologists analysed the DNA of Africa's cheetahs they were shocked to discover that ALL cheetahs are descended from a SINGLE female. Sometime in the past an epidemic killed every cheetah in the world, except one female, and her cubs, who were either immune or very isolated. After the epidemic they found themselves alone on an empty continent and they populated it with near-clones of themselves. But cheetahs are massively in-bred, as a result, and have so little genetic variability that they could easily become extinct as a result of a single epidemic.

Many human diseases are caused by recessive genes, forced to the surface by kinship marriage to close cousins(in-breeding).  Some of Queen Victoria's children and grand children were haemophiliacs because European royalty became too-closely related; they married off their children to cousins or second cousins over several centuries. The German Kaiser was Victoria's cousin as was the Tsar Nicholas; Queen Elizabeth's REAL family name is Elizabeth Saxe-Coburg Von Gotha (they opted for 'Windsor' at the outbreak of World War One) . So haemophilia affected the children of the British, Russian and German royal families as a result of 'inbreeding'. The terrible disease called Porphyria, also passed on as a recessive gene, portrayed in the film 'The Madness of King George' caused other British Royals to be declared 'mad' in the 18th and 19th centuries.

Thousands of years ago, humans realised that breeding with your sisters or cousins produced weak children with mental and physical deficiencies. The taboo on incest is almost universal in human societies - and 'exogamy'  - marrying-outside the village became traditional. But in isolated rural communities like Appalachia in the 18th and 19th centuries - poor communications, isolation and close-kinship marriage devastated the health of many communities.

In a BBC documentary about the late Sir Stanley Whittle - (who invented the jet engine circa 1933),  his son was asked what Sir Stanley would have valued about the world which his jet-engine has changed beyond recognition. He answered that the jet had not just given millions of people better holidays - it may well improve human health more than any invention since proper sanitation.  The jet has brought people together, all over the world, to meet and marry, and their children are strongly hetero-zygous - 'differently-gened' - and often display the best qualities of both their parents' races. Which is why America, as a country of immigrants, produces more Olympic champions than any nation on earth - as well as bigger, taller, healthier people with better teeth and better eyesight - who live longer lives. Its not JUST wealth and nutrition - its the result of all those different genes from across the globe.

Arguably the reason why the small island of Britain coloured the world's map red in the 18th and 19th centuries and invented everything from the steam engine, railway, antibiotics, radar, the jet etc. etc. was the endless waves of conquerors and refugees who mixed their genes in the UK: Romans, Celts, Anglo-Saxons, Vikings, Normans, Huguenots, Poles, Jews, Afro-Caribbeans, Asians etc. All brought something new to the feast in terms of ideas, culture and abilities.

Evolution has produced behaviours that strongly militate against inbreeding. Lions, monkeys, swans (and CANARIES!) are territorial and drive their sons away at puberty - to spread their genes elsewhere. Darwin spent 20 years visiting and corresponding with breeders of fancy pigeons, canaries, rabbits, cattle etc. because what these breeders were doing, whether they realised it or not, was experimenting with accelerated evolution by inbreeding. The insights he derived are all contained in: The Origin of Species, which contains vast amounts of data gathered from bird fanciers.

In conclusion, I recommend Linda Hogan's book - not just for the chapter on genetics - those on management are first rate.  The great budgerigar pioneer of the 1940s and 50s Cyril Watmough outlined the principles of in-breeding in his classic book  'The Cult of the Budgerigar' - regularly available on Ebay or through abebooks.com (beautiful illustrations).  He also wrote  'Inbreeding Budgerigars' - very intense and detailed - worth a look if you can find it, but it is now rare. I think, from what I have read, that if you want to seriously compete in type or song canaries ( I know nothing about colour canaries) - you have to inbreed to get up there with the champions consistently. But the price of producing such 'thoroughbreds' by inbreeding is to have no compunction in culling any sign of weakness in your stock.

Below is an abstract from an excellent article on the subject.
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HETEROSIS THE MAGIC BREEDING BULLET By Helmut Herder (BSc.Honors Genetics).
Obtain full article from  http://sta-ines-loft.limaperu.net/heterosis.html

ABSTRACT:
So what is heterosis?
Heterosis is the genetic term for hybrid vigor. It is the result of cross breeding two different and distant related plants or animals of the same
species. The offspring of such crosses are more vigorous and often more healthy and disease resistant. Why is this? There are many theories but the exact mechanism of heterosis is not entirely understood. Most of the genetic loci are in a heterozygous state and this somehow results in hybrid vigor. There have been some population genetic studies done that indicate even in the wild, nature favors heterosis, probably because it leads to greater genetic variation.

OUT CROSSING
There is no doubt about it, cross breeding is exciting and often produces thrilling results. It is also very frustrating because it leads to greater
genetic variation in a breeding program. In my own loft it has been my experience over the years that the best racers that have been bred have been from outcrosses and I think that the majority of champion racers that are produced by other lofts are also due to cross breeding.

INBREEDING AND HETEROSIS
Inbreeding by definition is the breeding of closely related plants or animals. All of us who breed racing pigeons are practising inbreeding because our birds have come into being only within the last 150 years. Close inbreeding, which I call harsh inbreeding, involves brother-sister, father-daughter, and son-mother matings. Even half brother and half sister matings are still harsh inbreeding.

Cousin and second cousin matings are more distant related matings and are the outer limit of what I call gentle inbreeding. In my opinion, there is no need to practice harsh inbreeding, and it is a disadvantage to do so for two reasons: 1)the loss of heterosis
2) the offspring will be no better than the parents from which they have been produced.

Harsh inbreeding leads to homozygous genetic loci. This is how recessive lethal and deleterious genes are expressed. Indiscriminate harsh inbreeding produces offspring that are generally weak, less disease resistant, runts and behaviorly stupid individuals. However, gentle inbreeding is often desireable and necessary. It is needed to fix certain desireable characteristics.

FIXING DESIRABLE CHARACTERISTICS
The most productive breeding strategy to breed champion pigeons is to:

1) outcross matings
2) produce a champion racer
3) embark on a gentle inbreeding program.

When you have a champion racer from an outcross mating, you have a genetically unique bird that has never existed before. There are gene combinations in that bird that do not exist in any other bird in the world. The trick is to pass some of those gene combinations on to as many offspring as possible. Very few, if any of the favorable gene combinations will be passed on to the offspring intact. However, if some of the genes that are responsible for making the bird a champion are dominant genes, then some of the offspring from the champion, will
also show signs of being superior racers. This sometimes happens if you are lucky.

Breed the champion to as many different mates as possible, while keeping only the best youngsters. Next let these youngsters mate to distant related mates. Keep only the best youngsters. These second generation youngsters then should be mated to cousins that have the champion in their pedigree. Hopefully, some of the gene combinations that made the champion a champion will combine in these cousin to cousin matings. If this happens, then desirable characteristics have been fixed in the resulting birds and a family based on the champion genes can
be established through further gentle inbreeding.

ABSTRACT ENDS

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