Sand and oil ,but thin the oil and it may nourish it enough to swell and close gap.
Difficult to repair since the construction design is at fault, any decent table will have timber at right angle to main timber to lock the timbers together to prevent such fault occurring.
You could add a metal strap to underside to help prevent further spread movement but it wont help shrinkage.
Have a look on YouTube at adding a 'bow-tie' or two, for a traditional style repair.
Izzy Swan, Frank Howarth or Cosmas Bauer have some good videos on that, it doesn't have to be in a different colour wood to become an 'accent' like they all do though. Failing that, kitchen worktop bolts that pull together from underneath would work.
Sand and oil as suggested, but in the second pic there appears to be a timber at right angles to the surface so the movement to date has already overcome the restraint it offered.
Wood twists and cups, as well as splitting, so dog bolts (which are designed to join engineered worktops) may not solve the problem of a natural hardwood, and possibly create stresses elsewhere in the table.
Maybe experiment with packing the split with sawdust dipped in the oil to be used, and keep under review.
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