Derivation of equation (2-2) on page 23
First note the errata sheet that this equation was botched by the printer.
We consider template replication to involve basically two processes, although this is a
considerable oversimplification of many intermediate steps which, if modeled, do not
change the basic form. First, template plus energy-rich mononucleotides (the resources in
this simple system) form a template-resource complex. Second, this complex dissociates
into two templates. Let 
denote
replicator 
and 
its limiting resource (or substrate). The
derivation follows the Michaelis-Menten kinetics common in the chemistry of enzyme
kinetics (see, for example, a popular text in biochemistry, that of Leninger, pp. 154-155
in an old version that is on my shelves). The template functions as the
"enzyme," the resources function as the substrate, and the product is then a new
complementary strand plus the old template (which is unused in the reaction, although it
catalyzes the formation of the complementary strand). The notation 
means concentration of replicator ; etc. for 
.
We consider here only template replication with no enzyme, although the approach can be
extended to enzyme mediated replication. The overall birth process is described by the
following diagram:
intermediary complex: 
The rate of formation of complex is 
.
The rate of breakdown of the complex is 
.
The reaction is in steady-state when the concentration of replicator-resource complex is constant or setting the rate of formation of complex to its rate of breakdown gives
,
or 
.
Following the Michaelis-Menten kinetics it can be seen that 
is the concentration of resources (mononucleotide building
blocks) at which the rate of growth is half maximum. Solving this for the steady-state
value of template resource complex 
gives
.
Since the initial rate of replication of 
is proportional to the concentration of complex:
.
When resources are common, the rate of replication no longer depends on resource
concentration and is directly proportional to the concentration of replicators 
substitute into 
above
gives:
Upon dividing both sides of the last equation by the maximum rate of change in
concentration of replicator i, 
, gives
or solving for
recall that on 
, so we have
However there is also the death process 
which is
independent of resources. The total is then the sum of
the birth process and the death process to give
.
Note that in the absence of resource competition, the growth rate of the replicator is
exponential, 
with rate 
. I
refer to such replicators as "Malthusian" because they obey Malthus' law of
exponential increase when uninhibited by resource limitation or competition. Another
derivation (Szathmary 1989) of template replication gives a square root growth law: 
. This can be referred to as
subexponential growth, since the basic growth process is less than exponential. The reason
for this is that the template will readily form a duplex with its complement and is then
temporarily unavailable for replication, in other words each template interferes with its
own replication more than it interferes with the replication of a competitor. The form of
the replication dynamic dramatically affects the condition for selection. Malthusian
growth leads to "survival of the fittest" while subexponential growth leads to a
cost of commonness or "survival of anybody." More than exponential growth (as
exists with protein catalyzed replication or with sexual reproduction), say 
, leads to a cost of rarity and "survival of the
first."