In animals it is called parthenogenesis and happens in lizards, snakes, birds and sharks. All of them have been around quite some time on earth, also.
What helped these species to be so resilient to a devastating infection when other animals that reproduce sexually (Central American frogs, North American bats) are being wiped away so badly by fungal infections?
We started mass production (cloning) of cavendish bananas around the the 1900s, and REALLY started mass production (cloning) in the 1950s, we only started seeing strains of banana blight effect it in the last few decades. Obviously there are all sorts of differences between these cases, but it's probably worth thinking about a bit.
Oh, also it looks like most animals capable of parthenogenesis can also under go sexual reproduction, helping them hedge against certain types of threats.
Also remember that we're looking at animal population on human generation timescales. One might say that a species with a population of millions being wiped out to a population of thousands (that are fungal resistant) over a time scale of 60 years as a 'disaster'. It in our life time it probably is, cause our activities would probably completely fuck over whatever is left. But in the absense of our activity, those thousands or so could still form a viable population and if they still have a good ecological niche to occupy, could easily within a few hundred/thousand years regain their original population. And we'd be all completely blind to that, cause that's not how we tend to think.
It was likely an oversight to not mention why we started to clone the Cavendish.
Panama disease devastated the also cloned Gros Michel cultivar over 60 years ago. That's why, today, we eat a different banana than those who came before us.
Well for starters asexual reproduction is an option for them, not the sole means of reproduction.
Secondly, diseases and catastrophes strike all sorts of critters and sometimes no amount of genetic variation will save them. For example you mention snakes and their resilience: http://www.bbc.com/news/science-environment-42428528
Thirdly, snakes, lizards, sharks, make up a huge number of species and genetic variability with global distribution. That provides some protection. And for most of history global pandemics were probably less common or were slow enough that it didn't happen at once everywhere. And if enough individuals survive for the species to recover, then that's that.
Fourth, sexual reproduction isn't a guarantee of immunity to disease, it's a survival strategy to help increase the odds of surviving as a species or having individuals with immunity because of all the possible combinations of genes being tried. For example, AIDS (without treatment) is lethal for most of the human population. But there are a few Scandinavian groups that have partial or full immunity, something like less than 1% of the total human population. That's what sexual reproduction gets us as a species.
At least some plants have genes that randomise genetic code if I recall my high school biology well. I also read that polyploid plants have something to randomly shuffle chromosome sets.
Quite alike a stack randomisation it is
Asexual species have been around for "quite some time" by comparison to, say, all of recorded history.
They tend to be very recent on a geological timescale.
Asexual reproduction was there first and predates sexual reproduction by several billion years. Some species then evolved back asexual reproduction, like this marbled crayfish.
Which would make sense, if an individual has great fitness they can "boom" extremely rapidly, but then any change in condition or emergence of specialised predator or disease will take down the entire population just as quickly because it has very low variability and flexibility.
When you see a statistic like that it points to a filtering event in a species. Much like the previous type of banana was clones, once a disease rises up that attacks that type of clone there just isn't the genetic diversity and rate of change to save the species.
> Normal sex cells contain a single copy of each chromosome. But the mutant crayfish sex cell had two.
> Somehow the two sex cells fused and produced a female crayfish embryo with three copies of each chromosome instead of the normal two. Somehow, too, the new crayfish didn’t suffer any deformities as a result of all that extra DNA.
> It grew and thrived. But instead of reproducing sexually, the first marbled crayfish was able to induce her own eggs to start dividing into embryos. The offspring, all females, inherited identical copies of her three sets of chromosomes. They were clones.
> Now that their chromosomes were mismatched with those of slough crayfish, they could no longer produce viable offspring. Male slough crayfish will readily mate with the marbled crayfish, but they never father any of the offspring.
Also, since humans are now very numerous and so good at surviving, makes you wonder if humans could mutate this way too. Makes it seem like you'd just get outright pregnant instead of ovulating, which may be a bummer.
It looks like there is some evidence, from mice experiments, that epigenetic information could be synthesized to induce asexual reproduction in mammals. I also found a recent gene editing technique to directly edit mice epigenomes, in theory that might be a path to design an asexual mouse, or another asexual mammal.
I guess technology and modern medicine could counteract that but it's still not ideal in the long-term. Nor would genetic clones create a very interesting society.
Mass-produced clothing would fit well. Public schools would not need to worry much about the extremes of ability. Shared mental traits would encourage a high level of trust, but identification difficulty would undermine it. With everybody having the same innate ability and interests, the pay differences between different jobs would be much higher.
One of the more interesting attempts at creating a human society made up of clones is the James Tiptree, Jr. story, "Houston, Houston, Do You Read?" 
"Krebs" means either cancer (the diseases) or crabs (the animals). The tumors that early physicians saw reminded them of crabs, so that's what they called them. English forked cancer (Latin word for crabs) to mean the diseases and uses the evolved colloquial word for the animals, but German uses its evolved colloquial word for both. "Krebstiere" is often used to refer to crabs as a group, probably to avoid misunderstandings.
Prof. Dr. Lyko seems to primarily study epigenetics, and it's just coincidence that the animals he's studied here are crabs: http://www.dkfz.de/en/epigenetik/
Right now spicy crawfish boils are fashionable in China, and I'm pretty sure the other East/Southeast Asian countries like crawfish also.
Just like with jellyfish. Chinese people already eat jellyfish, so if there's an infestation of jellyfish in the ocean, figure out a way to eat that species. A billion people can really put a dent in any wild population.
We have been eating crayfish for a long time; indigenous French (or European?) crayfish were delicious but they have been wiped out by this species.
There are various streams near me where you can go out in summer with a piece of bacon, some string and a box (maybe a net too) and catch signal crayfish. You don't need a permit as they're an invasive species - in fact you're not allowed to put them back if you catch any.
(But do put them back if you catch the native species! The signal crayfish are larger and have big red claws with white flashes which give them their name.)
edit: was writing my now largely redundant post as masklinn posted. I hadn't realised the Marmorkrebs were smaller - does that make them less of a threat? It does say they're already a problem for native species in Madagascar.
I had thought you only needed the licence to set up traps, which I'd rather not myself as there's a risk of catching/injuring/killing other species, particularly native crayfish or otters. Sorry for the inaccurate post. Sadly it's too late to edit it or I'd put strikethrough all over it...
More details about catching (mostly with traps) and licence details below.
To answer your questions, I'd wait until at least spring; I've only been in summer. Not sure about speed or depth (I've seen them in fast shallow streams and in a slower metre-deep river) but they like lots of hiding places like rocks or overhangs.
AFAIK indigenous european species have mostly been displaced by various american species (mostly spinycheek, signal, and louisiana crawfishes) — and severely hit by diseases those brought — over the course of the 20th century. The marbled crayfish is a recent development.
I don't think the marbled crayfish is great cooking fare, it's pretty small.
 especially the crayfish plague: https://en.wikipedia.org/wiki/Crayfish_plague
US-origin crayfish only look like shrimp-sized lobsters, they don't taste as good as either. They're more like if you boiled a lobster for too long and smeared it with river-bottom mud instead of butter. And at that size, the shells are a real pain.
This species was found here in 2012, but only sporadically it seems after that, so we might not have a problem yet.
Naw, every person is likely to carry at least one de novo mutation in their genome. Mutations happen constantly, they're just usually unremarkable.