The concept of Maximum Sustainable Yield (MSY) have been lying at the core of the theory of sustainable harvesting a fishery for decades and have become a key reference point for many fishing administrations, including the European Union. However, the existence of a MSY relies on the stability of a population equilibrium. This hypothesis, though always true in the original Schaeffer model, is still challenging in more realistic and recent population models. However, recent advances shows that fish population can exhibit complex dynamics that are ill described by the classical theory. In particular, processes occurring at intra-annual time scales can affect the stability of a population equilibrium even in a strictly single species case. Associated to stability, the resilience of the equilibrium (defined as an inverse return-time following a perturbation) also matters in a management purpose. Here, we introduce an analytical single population model in discrete time with a monthly time-step allowing temporal distinction between maturation and recruitment with density-dependent mortality and fishing exploitation. We show that, thanks to an appropriate population structure, we can easily derive inter-annual population equilibrium, and study their resilience and stability properties. Then, we show that under classical hypothesis concerning density-dependence, equilibrium stability is not guaranteed and that MSY can, in theory, be associated to unstable or low resilient states. However such destabilisation seems unlikely with realistic sets of parameters. Finally, a numerical illustration for sole (Solea solea) of the Bay of Biscay suggests that the value of MSY was sensitive to maturation period whereas viability, stability and resilience was more sensitive to timing of recruitment. The value of appeared robust to uncertainty concerning maturation and recruitment. We conclude by saying that even if the risk of destabilisation is low for real populations, the risk of decreased resilience near the border of extinction should be cared of.