In the first decades after the crucifixion of Jesus, the number of those who worshiped him as a resurrected savior was at most a few thousand, and probably many fewer. In 313, when Constantine issued the Edict of Milan mandating tolerance for Christianity, Christians numbered many millions. Thus, in two and a half centuries, Christian numbers had doubled at least ten times. On average, someone who lived to the age of seventy-five during this period could expect to live through an eight-fold increase in the numbers of Christians between her birth and her death.
Christianity began in confusion, controversy and schism and so it continued. A dominant orthodox church, with a recognizable ecclesiastical structure, emerged only very gradually and represented a process of natural selection — a spiritual survival of the fittest … The Darwinian image is appropriate: the central and eastern Mediterranean in the first and second centuries AD swarmed with an infinite multitude of religious ideas, struggling to propagate themselves. Every religious movement was unstable and fissiparous; and these cults were not only splitting up and multiplying but reassembling in new forms.
Paul Johnson A History of Christianity
If we’re going to apply Darwinian analogies to culture, we might want to distinguish between the selection of memes within people’s heads, and between them (intra- and inter-cephalic selection, if you will). Inside people’s heads, different ideas survive or fail depending on the benefits and costs they produce for their carriers, how well they fit with other ideas, and so on. Outside people’s heads, some ideas may spread because people work extra hard to propagate them. Christianity spread as rapidly as it did partly because it radically exalted the Spirit over the Flesh. As Clement (a relative moderate among early Christians) wrote “Our ideal is not to experience desire at all.” A consequence is that the new religion effectively sterilized a fraction of its hosts, who gave up on the normal business of marrying and raising children, and turned into cultural super-propagators.
Two ways of life were given by the Lord to his church. The one is above nature, and beyond common human living; it admits not marriage, childbearing, property nor the possession of wealth. . . . Like some celestial beings, these gaze down upon human life, performing the duty of a priesthood to Almighty God for the whole race. . . . And the more humble, more human way prompts men to join in pure nuptials, and to produce children, to undertake government, to give orders to soldiers fighting for right; it allows them to have minds for farming, for trade, and for the other more secular interests as well as for religion.
Eusebius, quoted in Peter Brown The Body and Society: Men, Women and Sexual Renunciation in Early Christianity
Women were another important vector for the spread of Christianity. Both pagans and Christians agreed they were over-represented among converts. Celsus, a pagan critic of Christianity, took a dim view of this, sounding like a modern critic of advertising for children: “Christians must admit that they can only persuade people destitute of sense, position, or intelligence, only slaves, women, and children, to accept their faith.” Tatian, a Christian apologist, was cheerier: “They say of us, that we gabble nonsense among females, half grown people, girls, and old women. Not so. Our maidens philosophize, and at their distaffs speak of things divine.”
Of course some of the same observations (e.g. regarding the overcoming of desire and religious celibacy) could be made about the spread of Buddhism during the same period, suggesting that there are general principles at work in the spread of religions, as in the spread of microorganisms.
For a sophisticated scholarly treatment along these lines of the interaction of selfish genes and selfish memes check out Mixed Messages: Cultural and Genetic Inheritance in the Constitution of Human Societies by Robert Paul. A lot of work on the coevolution of genes and culture has been done by mathematical modelers; Paul is something else, a long-time cultural anthropologist who brings wide knowledge and thick description of culture and symbolism to the topic.