Editing Quasispecies model (section)

===Alternative formulations===

The quasispecies formulae may be expressed as a set of linear differential equations. If we consider the difference between the new state <math>n'_i</math> and the old state <math>n_i</math> to be the state change over one moment of time, then we can state that the time derivative of <math>n_i</math> is given by this difference,   <math>\dot{n}_i = n'_i-n_i</math> we can write:

:<math>\dot{n}_i=\sum_j w_{ij}n_j-n_i</math>

The quasispecies equations are usually expressed in terms of concentrations <math>x_i</math> where

:<math>x_i\ \stackrel{\mathrm{def}}{=}\  \frac{n_i}{\sum_j n_j}</math>.
:<math>x'_i\ \stackrel{\mathrm{def}}{=}\  \frac{n'_i}{\sum_j n'_j}</math>.

The above equations for the quasispecies then become for the discrete version:

:<math>x'_i = \frac{\sum_j w_{ij}x_j}{\sum_{ij} w_{ij}x_j}</math>

or, for the continuum version:

:<math>\dot{x}_i =\sum_j w_{ij}x_j-x_i\sum_{ij}w_{ij}x_j.</math>