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The population growth can be modeled by the differential equation $\frac{dP}{dt} = kP$, where $P$ is the population size and $k$ is a constant. Solving this equation with the given initial condition gives $P(t) = 100e^{kt}$. Since the population doubles in 10 years, we have $200 = 100e^{10k}$, which implies $k = \frac{\ln 2}{10}$. Therefore, the population after 20 years is $P(20) = 100e^{20k} = 100e^{2\ln 2} = 1600$ rabbits.