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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