# Resource allocation in self-replicative systems

**Classical Mechanics**Level 5

Think of \(\mathbf{S}_R\) like factories and \(\mathbf{S}_O\) like street sweepers. One part is creating new things and the other is doing maintenance on what already exists. The catch is, the material in \(\mathbf{S}_R\) must not only make the rest of the system, but also itself!

Suppose that the materials in \(\mathbf{S}_R\) and the materials in \(\mathbf{S}_O\) cost the same amount of energy for \(\mathbf{S}_R\) to make per unit amount. Suppose the material in \(\mathbf{S}_R\) can convert raw material from the environment into system mass at the rate \(\gamma_R = 3 \mbox{ kg } \mathbf{S}/\mbox{hr}/\mbox{kg }\mathbf{S}_R\). If the system doubles in size once every 2 \(\mbox{hrs}\), what fraction of the material in \(\mathbf{S}\) is devoted to \(\mathbf{S}_O\)?

**Assumptions**

- The fact that the system continuously doubles in size in a fixed time means that this system is in exponential growth, i.e. \(\dot{\mathbf{S}} = \lambda \mathbf{S}\)

**Note**: This problem was originally prepared for the IPhOO.

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