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 S\mathbf{S} is comprised of two kinds of mass: one kind is SR\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 SO\mathbf{S}_O that takes care of other things.

Think of SR\mathbf{S}_R like factories and SO\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 SR\mathbf{S}_R must not only make the rest of the system, but also itself!

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

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