  Finding a good compromise between the exploitation of known resources and the
exploration of unknown, but potentially more profitable choices, is a general
problem, which arises in many different scientific disciplines. We propose a
stylized model for these exploration-exploitation situations, including
population or economic growth, portfolio optimisation, evolutionary dynamics,
or the problem of optimal pinning of vortices or dislocations in disordered
materials. We find the exact growth rate of this model for tree-like geometries
and prove the existence of an optimal migration rate in this case. Numerical
simulations in the one-dimensional case confirm the generic existence of an
optimum.
