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Finding a good compromise between the exploitation of known resources and theexploration of unknown, but potentially more profitable choices, is a generalproblem, which arises in many different scientific disciplines. We propose astylized model for these exploration-exploitation situations, includingpopulation or economic growth, portfolio optimisation, evolutionary dynamics,or the problem of optimal pinning of vortices or dislocations in disorderedmaterials. We find the exact growth rate of this model for tree-like geometriesand prove the existence of an optimal migration rate in this case. Numericalsimulations in the one-dimensional case confirm the generic existence of anoptimum.

Explore or exploit? A generic model and an exactly solvable case