  We present a simple mathematical model that captures the evolutionary
capabilities of a prebiotic compartment or protocell. In the model the
protocell contains an autocatalytic set whose chemical dynamics is coupled to
the growth-division dynamics of the compartment. Bistability in the dynamics of
the autocatalytic set results in a protocell that can exist with two distinct
growth rates. Stochasticity in chemical reactions plays the role of mutations
and causes transitions from one growth regime to another. We show that the
system exhibits `natural selection', where a `mutant' protocell in which the
autocatalytic set is active arises by chance in a population of inactive
protocells, and then takes over the population because of its higher growth
rate or `fitness'. The work integrates three levels of dynamics: intracellular
chemical, single protocell, and population (or ecosystem) of protocells..
