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