For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v^3 . It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u ( u < v ) , then the time required to swim a distance L is L / ( v − u ) and the total energy E required to swim the distance is given by:

E (v) = av^3 ⋅ L/v − u

where a is the proportionality constant.