Resource allocation in self-replicative systems

Consider a very simple model for an open self-replicative system such as a cell, or an economy. A system $$\mathbf{S}$$ is comprised of two kinds of mass: one kind is $$\mathbf{S}_R$$ that is capable of of taking raw material that comes from outside the system, and converting it into components of the system, and the other is $$\mathbf{S}_O$$ that takes care of other things.

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. $$\dot{\mathbf{S}} = \lambda \mathbf{S}$$.
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