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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: 

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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 

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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: 

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