Given a population (vehicles in traffic lanes, predator-prey, store customers), we can multiply the transition matrix times the current population distribution to find the distribution after the passing of one or more time periods. if we have m as the transition matrix and xo is the initial population distribution, then mx,-x1 is the distribution after one time period, and )xo = m(mko-mx-x2 will be the distribution after two periods. notice that we can find successive x's recursively by performing matrix multiplication on a single column, rather than raising the transition matrix to high powers. over time, the populations can reach stability, which is the limit of the sequence of x's. for now we will do some calculations, observe trends and simply guess x, the equilibrium vector. then we can check our guess by one more multiplication: mx. o xo, if it is truly the equilibrium.