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A species.
Yes
They produce offspring so that organisms species doesn't become extinct!
A group of organisms that can mate and produce offspring which can themselves mate is called a species.
No, organisms of different species can breed and produce offspring. Donkeys and horses can breed and produce mules but mules cannot produce offspring.
Species
It is possible because they are the exact same species, Canis lupus familiaris.
once they can no longer breed and produce a fertile offspring with the other group they become a separate species
It is where two ducks of the same species breed and produce hybrid offspring. Commonly in ducks, hybrids are most likely fertile.
Yes.
they can inter breed and produce fertile offspring.
This might be thought of as a species if the group indicated was large enough to include all of the potential members that can breed and produce viable fertile offspring. This would mean that animals which can breed and produce infertile offspring such as horses and donkeys which can mate and produce offspring are not of the same species. This situation would be within the bounds of the question. When a group which is of one species but is of limited such a limited population that the only can breed with a small number of individuals and produce a fertile offspring it would be described as a bottlenecked population. This can lead to severe genetic drift in that population.