I have recently started making mead as a bit of a hobby to use up our honey supply. Making the mead has been quite a fun experience but I was wondering. Since your using yeast and feeding it sugars to keep it alive. Could you just keep part of a batch each time as your yeast source? It works for bread starters and it's relatively the same concept just with a drink.
Yes, you can reuse the yeast, under some conditions.
If you use it quickly enough, there's nothing to worry about. I brew a batch of beer, let it ferment for 2 weeks, bottle it, and keep the leftover trub and yeast sediment for the next batch. It sits on the counter or in the chest freezer I use for fermenting for a bout a day or two. This is basically a big starter: you have lots of healthy yeast cells ready to go. Keep them at the same temperature as your fermenter, if possible. I've brewed four batches (of usually ~5% abv beers) from the same pack of Saflager / Safale dry yeast without any problems. Fermentation is really quick. I didn't try a fifth batch as this maxed out my storage capacity.
If you want to keep yeast for a longer period, it gets more complicated. You can either get lab equipment and harvest only the healthy cells, discarding the dead ones and the trub, and put them on a growth medium in the fridge. This goes beyond my chemistry skills and the effort I want to put in.
You could also try feeding it on a regular basis and see how that works. I don't know how much you need to feed and if there are any negative consequences in the long run from dead yeast cells or lack of nutrients (assuming you just feed sugar).
AFAIK you should not freeze yeast, as ice crystals can puncture the cells, destroying them.
Short answer: Yes
Just rack the new batch over the old "fresh" cake. Best to rack on it asap. No need to oxygenate, it's a full pitch yeast will go directly into feeding phase. The resulting batch will be very clean almost no yeast profile. if you want the yeast profile then only use some of the old cake and oxygenate so you get a new growth phase to produce esters. You can reuse the same cake as a full pitch about 5 times before viable cell count goes down and it needs oxygen again for a new growth cycle. You can do the partial indefinitely since there's always new yeast. Lack of O2 triggers yeast to stop reproducing and enter feeding phase. If you have the cell count needed to complete the batch in x time then no need to add O2.
As the others have already responded, yes, you can reuse the yeast from a previous batch. There's a caveat though.
Depending on the gravity of your mead, your yeast may or may not be usable. High gravity, high ABV meads will either kill off your yeast or leave it in poor condition, deteriorating the quality of your future meads. So as a general rule, if you want to reuse yeast, make a low gravity mead, rack it (collecting the yeast) and then add more honey.
As for keeping a portion of mead, as opposed to collecting yeast cake, you need to ensure there's enough yeast there to kickstart a new fermentation. I'd say it's more difficult to pull off after racking to secondary.