negative logic and positive logic gates
Why is AND gate equivalent of positive logic OR gate in negative logic?


There is no real 'why', it is simply the case. Take the truth table of an AND gate:


   0 0 => 0
   0 1 => 0
   1 0 => 0
   1 1 => 1


Now invert all values (as if you place inverters on the two inputs and the one output):


   1 1 => 1
   1 0 => 1
   0 1 => 1
   0 0 => 0


As you can see, this is the truth table of a OR gate.

If you want to think deeper, consider that an AND gate 'requires' all its inputs to be high to make its output high, a single low is sufficient to get a low on the output. You can almost hear the 'OR' logic in the last part of that sentence, but applied to 0's and resulting in a 0 at the output.


  I do not get why we express AND operation as OR gate in negative logic.


That's not a question.


  And what is the advantage of negative logic systems?


Logically speaking: none. But it can sometimes be enlighten the mind to think of a particular circuit in negative terms. And for electrical realizations of logic gates: the electronics do not care whether we call 0V a logic 0 or a logic 1.