Does "independently" refer to "(each species duplicating itself) independently"?

Context:

hopes on the origin of life. The root of evolution in non-biological
chemistry somehow seems to present a bigger gap than any par-
ticular transition during subsequent evolution. And in one sense it
is a bigger gap. That one sense is quite specific, and it offers no
comfort to the religious apologist. The origin of life only had to
happen once. We therefore can allow it to have been an extremely
improbable event, many orders of magnitude more improbable
than most people realize, as I shall show. Subsequent evolutionary
steps are duplicated, in more or less similar ways, throughout
millions and millions of species independently, and continually and repeatedly throughout geological time. Therefore, to explain the evolution of complex life, we cannot resort to the same kind of statistical reasoning as we are able to apply to the origin of life. The
events that constitute run-of-the-mill evolution, as distinct from its
singular origin (and perhaps a few special cases), cannot have been
very improbable.

-The God Delusion by Richard Dawkins