1. The problem asks for the ratio of $R_0$ to $S_0$ for $X_0$ in a stable population. 2. In population dynamics, $R_0$ often represents the net reproductive rate, $S_0$ the initial survival rate, and $X_0$ the initial population size. 3. For a stable population, the growth rate is constant, meaning the population neither grows nor declines over time. 4. The key formula relating these quantities is $$\lambda = R_0 \times S_0$$ where $\lambda$ is the population growth rate. 5. For stability, $\lambda = 1$, so $$1 = R_0 \times S_0$$ 6. Rearranging for the ratio $\frac{R_0}{S_0}$ gives $$\frac{R_0}{S_0} = \frac{1}{S_0^2}$$ but since $S_0$ is survival rate, the ratio simplifies to $$\frac{R_0}{S_0} = \frac{1}{S_0}$$ 7. Therefore, the ratio $\frac{R_0}{S_0}$ for $X_0$ in a stable population is $$\boxed{\frac{1}{S_0}}$$. 8. This means the net reproductive rate must be the reciprocal of the survival rate to maintain population stability.