Semelparous organisms breed only once during their lifetime. Examples of this type of reproduction can be found in Pacific salmon and bamboo. The per capita rate of increase, r, can be thought of as a measure of reproductive fitness. The greater the value of r, the more offspring an individual produces. The intrinsic rate of increase is typically a function of age x. Models for age-structured populations of semelparous organisms predict that the intrinsic rate of increase as a function of x is given by

r(x)=(ln[(x)m(x)]/x)

where (x) is the probability of surviving to age x and m(x) is the number of female offspring at age x. The optimal age of reproduction is the age xx that maximizes r(x). Find the optimal age of reproduction for

(x)=e^ax

and

m(x)=bx^c

where a,b, and c are positive constants.

Optimal age of reproduction is