From Osher Doctorow

COPYRIGHT NOTICE
Biological Evolution Rank Statistics Approximate Riccati Differential
Equation
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005

E. Ben-Naim (Los Alamos National Laboratory, New Mexico) and P. L.
Krapivsky (Boston U. Physics Dept. and Center for Molecular
Cybernetics) in "Rank statistics in biological evolution,"
q-bio.PE/0508023 v1 18 Aug 2005, obtain as one of their two main
results:

1) P_k ~ M/k (where ~ is "approximately equals for large k")

where P_k is the average size of the kth descendent species population
where they are ranking species in the order of their creation (lst,
2nd, 3rd, ..., kth, ...) from a common ancestor k = 1 by mutation and M
is the average total population size at time 0.

The equation (1) was not related to the Riccati Differential equation
by Ben-Naum and Krapivsky, but if we consider the continuous model with
k proportional to time t (since rank increases with time) or vice
versa, we get:

2) y(t) = (definition) P(t) = M/t
3) dy/dt = -M/t^2 = -y^2/M

which is the Riccati Differential equation dy/dt = A(t) + B(t)y +
C(t)y^2 with A(t) = B(t) = 0 and C(t) = -1/M.

In fact, Ben-Naum and Krapivsky get:

4) dP_k/dt = (a + b - c)P_k + bMf_(k-1)

where a, b, c are respectively the rates of replication, mutation, and
death, and f_k is the probability that the total number of distinct
species originated up to time t equals k and itself satisfies:

5) df_k/dt = bM(f_(k-1) - f_k)

Osher Doctorow

From Osher Doctorow

Ben-Naim and Krapivsky study the standard Replication-Mutation-Death
process (RMD) which has been commonly used to model genome evolution,
population genetics, species differentiation. They list 11 references
for the model of genome evolution and 1 each for population genetics
and species differentiation or "speciation". The rank is the
chronological order by which the species are created, and they consider
their main result that both total population size and total descendant
population size o a species of a given rank decay algebraically with
the rank. Each mutation event creates a distinct new species.

Osher Doctorow