1. INTRODUCTION 5
0 1 2
· · ·
j 1 j j + 1
· · ·
1
−j j
Ehrenfest walk Yn
Birth and death chain
of Nowak and Schuster
Zn
1
2
1
2
1
2
.
.
.
.
.
.
i 1
i
i + 1
m 1
m
σi(m
i)e−a
m(σi + m i)
σi2
(
1
e−a
)
+ i(m i)
m(σi + m i)
Figure 2. Approximating process
Thus the Moran process is approximated by the process on
{ 0,..., } × { 0 } { 0 } × { 0,...,m }
described loosely as follows. On { 0,..., }×{ 0 }, the process follows the dynamics
of the Ehrenfest urn. On { 0 }×{ 0,...,m }, the process follows the dynamics of the
birth and death process of Nowak and Schuster [32]. When in (0, 0), the process can
jump to either axis. With this simple heuristic picture, we recover all the features
of our main result. We suppose that
+∞ , m +∞ , q 0 ,
in such a way that
q a ∈]0, +∞[ ,
m
α [0, +∞] .
The critical curve is then defined by the equation
discovery time persistence time
which can be rewritten as
α φ(a) = ln 4 .
This way we obtain an equation in the parameter space (a, α) separating the regime
where the equilibrium population is totally random from the regime where a qua-
sispecies is formed. We observe the existence of a critical population size necessary
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