For a given nonnegative n × n matrix A consider the following quantity as long as the denominator is positive. It is simply the ratio between the smallest and the largest entries of Am. We call s(Am) the inverse spread of Am which is interpreted as a measure of the maximum variation among the entries of Am in the multiplicative and reciprocal sense. Smaller s(Am) means a larger variation for Am. Clearly 0 = s(Am) = 1 for all m = 1, 2, . . . We study the asymptotic behavior of s(Am), that is, the behavior of s(Am) as m ? 8. The study arises from evolutionary biology.