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In this paper we propose some Harris-like criteria in order to study the longtime behavior of general positive and periodic semiflows. These criteria allowus to obtain new existence results of principal eigenelements, and theirexponential attractiveness. We present applications to two biological models ina space-time varying environment: a non local selection-mutation equation and agrowth-fragmentation equation. The particularity of this article is to studysome inhomogeneous problems that are periodic in time, as it appears forinstance when the environment changes, due for instance to the seasonal cycleor circadian rhythms.

Harris's method for non-conservative periodic semiflows and application to some non-local PDEs