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 \(\mathbf{S}_O\)?


  • 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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