Why don't the critters ever age?
Edit: Please see the comments for why this does not address the question, however, I think I should not delete it in order to preserve the discussion...

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Not an expert, but my impression is that reversible computation does not by itself result in a change in entropy: the amount of information in the system has not changed, only the format in which it is stored. In other words such processes obey the second law with only a weak inequality, not a strict one. To get a strict increase in entropy, one has to erase or destroy information, i.e. irreversible computation. I don't see why the rest of the details in your question -- two dimensions, cellular automata, chaos, recurrence -- would affect this.

Happy to be corrected by any experts, however.