Dementors do not breed (No, there'll be no sweet sweet love in Azkaban tonight!)
but rather grow like fungi under certain conditions.
They multiply by feeding off human despair, unhappiness, hopelessness, and are akin to depression.
They are pretty insidious creatures.
Dementors have no soul themselves, thus their constant drive to suck the souls from others.
According to J.K. Rowling: You cannot destroy Dementors, though you can limit their numbers if you eradicate the conditions in which they multiply, ie, despair and degradation.
Dementors are JKR's representation of depression: JKR (Rowling is asked about dementors being “a description of depression”):
“Yes.
That is exactly what they are.
It was entirely conscious.
And entirely from my own experience.
Depression is the most unpleasant thing I have ever experienced.
It is that absence of being able to envisage that you will ever be cheerful again.
The absence of hope.
That very deadened feeling, which is so very different from feeling sad.
Sad hurts
but it’s a healthy feeling.
It’s a necessary thing to feel.
Depression is very different.
I think [dementors] are the scariest things I’ve written.”
Leaky Lounge Web Chat Transcript with J.K. Rowling Depression, Bipolar Disorder, and Dementors The Dementor at the Harry Potter Companion
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The claim that dementors [don't] breed but grow like a fungus where there is decay has been canonically disproven , at least if you can trust Fudge's knowledge.
It seems to be an early idea by JKR, but has since evolved or been forgotten.
That quote by JKR was from an interview that happened in October, 2000, soon after the release of GoF.
Later, however, we learn from Fudge in the Half Blood Prince that dementors do breed : "
I thought dementors guard the prisoners in Azkaban," he said cautiously.
"They did," said Fudge wearily.
"But not anymore.
They've deserted the prison and joined He-Who-Must-Not-Be-Named.
I won't pretend that wasn't a blow.
"
"
But," said the Prime Minister, with a sense of dawning horror, "didn't you tell me they're the creatures that drain hope and happiness out of people?
"
"That's right.
And they're breeding.
That's what's causing all this mist."
So while we thankfully do not know the specific details about how dementors breed, we know that: Dementors do, in fact, breed
It happens when dementors are "all over the place" [Fudge]
It seems to generate a lot of mist.
We do not know if the mist is a byproduct or instrument of their breeding processes, but the mist is at least associated with it.
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I don't think it's answered in canon, but Rowling has been quoted as explaining that they "don't breed but grow like a fungus where there is decay."
Source
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Spontaneous Generation Dementors form spontaneously from an excess of despair, fear, and depression.
The Pottermore article on Boggarts states: Like a poltergeist, a Boggart is not and never has been truly alive.
It is one of the strange non-beings that populate the magical world, for which there is no equivalent in the Muggle realm.
Boggarts can be made to disappear, but more Boggarts will inevitably arise to take their place.
Like poltergeists and the more sinister Dementors , they seem to be generated and sustained by human emotions .
I stole the quote from ThruGog's answer to Did Ekrizdis
create the Dementors?
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Given the information, I'd say this mist acts as a spore-like cloud similar to what mushrooms and other Fungi release.
Thus I think what Fudge means by 'breeding' is that they are releasing their clouds of spores near places where hundreds and thousands of people are feeling depressed all at once.
Honestly, I don't think Fudge knows enough about science and biology to comprehend that a living creature could multiply in any way other than 'the human way' and some fellow nerds from the Dep. of Mysteries decided 'just tell him they are breeding, it's sort of true anyway'.
And it is.
breeding implies they are making more of their kind...
it's just not breeding like most creatures do.
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This is inspired by some discussions in  chat .  In the  Harry Potter  universe, how are Dementors born?  Or are they artificially created by some magical means?
How are Dementors created/born?
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<harry-potter><dementors>
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Rowling has been quoted as explaining that Dementors "don't breed but grow like a fungus where there is decay."  They multiply by feeding off human despair, unhappiness, hopelessness, and are akin to depression. It is claimed that dementors reproduce like an evil fungus, however this has been canonically disproven by Fudge.
Dementors spontaneously generate  from excess of despair, fear and depression. They act a bit like fungi Dementors breading like fungus has been canonically disproven by Fudge