With x(r)=population size

the nonautonomous equation x(r+1)=x(r)expf(r

x(r)) provides a very general description of population growth in which any of the many factors that influence the growth rate may vary through time. If there is some fixed length of time such that during any interval of this length the population experiences environmental variability representative of the variation that occurs in all time

then definite conclusions about the population's long term behavior apply. Specifically

conditions that produce population presistence can be distinguished from conditions that cause extinction. These attributes resemble corresponding features of the related autonomous population growth model x(τ+1)=x(r)expf(x(τ)).