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This paper is concerned with generalizations of the notion of principaleigenvalue in the context of space-time periodic cooperative systems. When thespatial domain is the whole space, the Krein-Rutman theorem cannot be appliedand this leads to more sophisticated constructions and to the notion ofgeneralized principal eigenvalues. These are not unique in general and we focuson a one-parameter family corresponding to principal eigenfunctions that arespace-time periodic multiplicative perturbations of exponentials of the spacevariable. Besides existence and uniqueness properties of such principaleigenpairs, we also prove various dependence and optimization resultsillustrating how known results in the scalar setting can, or cannot, beextended to the vector setting. We especially prove an optimization property onminimizers and maximizers among mutation operators valued in the set ofbistochastic matrices that is, to the best of our knowledge, new.

Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems