Yes, in the presence of oxygen (anerobic conditions), the fermentation process, however, DOES NOT MAKE USE OF THAT OXYGEN, thus it is still an ANEROBIC process!
This is a little old, but what I have available and part of what I have researched that has built my understanding yeast.
“It was shown by Swanson & Clifton (1948) for the first time that most of the aerobic growth of a yeast (Saccharomyces cerevisiae) on glucose proceeds via fermentation. A quantitative study of the aerobic growth of yeast made by Lemoigne, Aubert & Millet (1954) showed a diauxic growth on high concentrations of glucose: a fast growth involving intensive aerobic fermentation, followed by a slow growth involving oxidation of the accumulated ethanol. At lower glucose concentrations (5-10 mg./100 ml.) the absence of any noticeable diauxie suggested that the pathway used for the degradation of glucose might depend on the glucose concentration.
Slonimski (1956) studied the rate of respiratory adaptation in Saccharomyces
cerevisiae as a function of the glucose concentration. At low concentrations of glucose (below ~ x ~ O - ~ Mthe) rate of respiratory adaptation increased as the concentration of glucose was increased. This relationship was to be expected since the aerobic fermentation of glucose proceeds very slowly a t these glucose concentrations and severely limits the supply of energy to the organisms. But a t concentrations of glucose higher than 6x10-3~, the rate of adaptation to aerobic conditions decreased as the concentration of glucose was increased. Simultaneously, as the concentration of glucose increased, the rate of aerobic fermentation also increased. It is this inhibition of the synthesis of respiratory enzymes by the high fermentation rates which occur a t high glucose concentrations which is known as the contre-effet Pasteur. Slonimski’s work, which was done with suspensions of organisms, was
followed by a more extensive study of the contre-effet Pasteur in a growing
yeast (Ephrussi, Slonimski, Yotsuyanagi & Tavlitzki, 1956). Starting a culture in the presence of 3 yo (w/v) glucose with organisms fully adapted to aerobic conditions, the following was observed : the rate of aerobic fermentation ( Qco, ferm.) increased sharply during the phase of exponential growth; a t the same time the rate of respiration (Qo,) decreased to a low value. A few generations before the end of the exponential phase the glucose no longer saturated the fermentation system, and as a result the Qco, ferm. decreased to a low value; simultaneously the Qo, increased (respiratory adaptation). This experiment showed clearly that the contre-effet Pasteur is an important part of the physiology of a yeast like S. cerevisiae growing in a high concentration of glucose.”
“If one compares a yeast strain which degrades glucose by aerobic fermentation, like S. cerevisiae normal strain, with a strain which degrades glucose by respiration, like Candida tropicalis (an organism which has no Crabtree effect), one can eliminate the first possibility. Growing S. cerevisiae
organisms have an aerobic fermentation rate of 78 pl. CO,/lO min./107 rganisms, which corresponds to a degradation rate of 1.74 pmoles glucose/lO min./107 organisms. Candida tropicalis grew at about the same rate with a respiration rate of 27.7 p l . O,/lO min./107 organisms, i.e. a degradation rate of only 0.21 pmole glucose/lO min./107 organisms. It is thus clear that the growth-limiting factor for organisms when degrading a sugar by aerobic fermentation is not the rate of intermediary metabolites synthesis but rather the rate of energy production. Therefore the Crabtree effect must be considered as the repression of an energy-producing system, respiration, by another energy-producing system, fermentation. Thus, when respiration occurs simultaneously with aerobic fermentation, as is the case with the normal strain of S . cerevisiae growing on galactose, the compensation that he respiration brings to the deficient rate of fermentation is actually an energy compensation.”
De Deken, R. H. (1966). The crabtree effect: A regulatory system in yeast. Journal of General Microbiology, 44, 149-156. Retrieved from
http://mic.sgmjournals.org/content/44/2/149.full.pdf
Also noted in this article that of the strains they tested, 50% had the crabtree effect.
If you have a newer study that redefines this I'm all ears, I am always happy to learn something new...I just don't throw out what I know that is based off research unless its newer research.