there be organisms with sequence i. Let's say that each of these organisms asexually gives rise to offspring. Some are duplicates of their parent, having sequence i, but some are mutant and have some other sequence. Let the mutation rate correspond to the probability that a j type parent will produce an i type organism. Then the expected fraction of offspring generated by j type organisms that would be i type organisms is , where . Then the total number of i-type organisms after the first round of reproduction, given as , is Sometimes a death rate term is included so that: where is equal to 1 when i=j and is zero otherwise. Note that the n-th generation can be found by just taking the n-th power of W substituting it in place of W in the above formula. This is just a system of linear equations. The usual way to solve such a system is to first diagonalize the W matrix. Its diagonal entries will be eigenvalues corresponding to certain linear combinations of certain subsets of sequences which will be eigenvectors of the W matrix. These subsets of sequences are the quasispecies. Assuming that the matrix W is a primitive matrix (irreducible and aperiodic), then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate. The components of this eigenvector give the relative abundance of each sequence at equilibrium. Note about primitive matrices W being primitive means that for some integer , that the power of W is > 0, i.e. all the entries are positive. If W is primitive then each type can, through a sequence of mutations (i.e. powers of W) mutate into all the other types after some number of generations. W is not primitive if it is periodic, where the population can perpetually cycle through different disjoint sets of compositions, or if it is reducible, where the dominant species (or quasispecies) that develops can depend on the initial population, as is the case in the simple example given below. Alternative formulations The quasispecies formulae may be expressed as a set of linear differential equations. If we consider the difference between the new state and the old state to be the state change over one moment of time, then we can state that the time derivative of is given by this difference, we can