Foraging animals tend to optimize the amount of time they spend at a food patch, so as to maximize the amount of energy gained. For example, a bird might forage at a berry bush; stay too long and the initially abundant fruit is increasingly difficult to find, requiring more energy. Suppose the patch energy function is given by

f(t)=Emaxt/k+t where Emax, k>0 are constant

Further, the amount of time spent going out to forage is described by

g(t)=(T+t)

where

is a fixed travel time between patches andt0

t0is the time spent foraging at a patch. We can combine these to form a function that describes the amount of energy gained per unit time spent foraging:

R(t)=Energy gained/Time spent forging=f(t)/g(t)

R(t)=Energy gained

Time spent foraging

=f(t)

g(t)

.

R(t)=Energy gained/Time spent foraging=f(t)g(t).

Plot this function and find the optimal foraging time.