r = H + F, (2) H = − ∑ ı,ℓ π$_{ı}$p$_{ıℓ}$ ln p$_{ıℓ}$ F = ∑ ı,ℓ π$_{ı}$p$_{ıℓ}$ ln A$_{ℓı}$ (3) p$_{ıℓ}$ = A$_{ℓı}$ V$_{ℓ}$ Λ$_{0}$ V$_{ı}$ (4) r = sup ν ∈ M ( H$_{ν}$ + ∫ γdν ) (5) P$_{F}$ ( y ) = 1 − ( 1 − Δ σ 2 σ 2 M y )$^{2}$ 〈 N 〉 s Δ σ 2 +1 1 − ( 1 − Δ σ 2 σ 2 M )$^{2}$ 〈 N 〉 s Δ σ 2 +1 (6) s = Δ r − Δ σ 2 〈 N 〉 (7) where r = log(Λ$_{0}$) is the growth rate (or Malthusian parameter), with Λ$_{0}$ is the dominant eigenvalue of A , whereas H and F are the entropy and the proliferative potential, defined as: where p$_{ıℓ}$ is defined as: with V A = Λ$_{0}$ V , and π is the stationary distribution associated to P = ( p$_{ij}$ ): π$_{i}$ = V$_{i}$U$_{i}$ , with U given by A U = Λ$_{0}$ U . Eqs. (2)-(4) are derived from a variational principle [16], namely where ν is a Markov measure H$_{ν}$ = − ∑ n j $_{=1}$ν ( A$_{j}$ ) log ν ( A$_{j}$ ) and γ = log a$_{x}$$_{0}$$_{x}$$_{1}$ , where A$_{j}$, j = 1 , . . . , n is a partition of the phase space of the system and a$_{x}$$_{0}$$_{x}$$_{1}$ is the transition probability between two states of the system, x$_{0}$ and x$_{1}$ . The solution to this variational problem produces Eqs. (2)-(4) [16, 21]. According to Demetrius et al. [18] the diffusion approximation yields the following equa- tion for the fixation or invasion probability as a function of the initial concentration of mutants, y is given by: where the total population N is assumed to be constant and s is defined by: 11