The phrase “Red Queen hypothesis” comes from Chapter 2 in Through the Looking Glass (Carroll 1872). In Alice’s dream about the looking glass house, she first finds that things appear left-to-right as if shown in a mirror. She then finds that chess pieces are alive. She will later encounter several of these pieces (most notably the Red Queen), after she leaves the looking glass house to see the garden.
Alice decides that it would be easier to see the garden if she first climbs the hill, to which there appears to be a very straight path. However, as she follows the path, she finds that it leads her back to the house. When she tries to speed up, she not only returns to the house, she crashes into it. Hence, the forward movement takes Alice back to her starting point (Red Queen dynamics), and rapid movement causes abrupt stops (extinction).
Eventually, Alice finds herself in a patch of very vocal and opinionated flowers; the rose is especially vocal. The flowers tell Alice that someone like her (the Red Queen) often passes through, and Alice decides to seek this person, mostly as a way to escape more verbal abuse. When Alice spots the Red Queen, she begins moving toward her. But, the Red Queen quickly disappears from sight. Alice decides to follow the advice of the rose, and go the other way (“I should advise you to walk the other way”). Immediately she comes face-to-face with the Red Queen (see Lythgoe and Read 1998).
The Red Queen then leads Alice directly to the top of the hill. Along the way, the Red Queen explains that hills can become valleys, which confuses Alice. Already, in this world, straight can become curvy, and progress can be made only by going the opposite direction; now, according to the Red Queen, hills can become valleys and valleys can become hills.
At the top of the hill, the Red Queen begins to run, faster and faster. Alice runs after the Red Queen but is further perplexed to find that neither one seems to be moving. When they stop running, they are in exactly the same place. Alice remarks on this, to which the Red Queen responds: “Now, here, you see, it takes all the running you can do to keep in the same place”. And so it may be with coevolution. Evolutionary change may be required to stay in the same place. Cessation of change may result in extinction.
Source : http://www.indiana.edu/~curtweb/Research/Red_Queen%20hyp.html