content_image/323822.jpg
The approximate solutions for system (6): 

<img src='content_image/323822.jpg'>

If we take q 11 (0) = 0 . 1 , q 22 (0) = 0 . 2 , q 21 (0) = 0 . 3 , r 1 = 0 . 01 , r 2 = 0 . 1 , r 3 = 1 . 1 , r 4 = − 0 . 3 , and r 5 = 0 . 4 . 

We find that the interior equilibrium point is unstable where the condi- tions (2.7) is not satisfied. 

If we take q 11 (0) = 0 . 1 , q 22 (0) = 0 . 2 , q 21 (0) = 0 . 3 , r 1 = 0 . 2 , r 2 = 0 . 5 , r 3 = 1 . 5 , r 4 = − 0 . 3 , and r 5 = 0 . 4 . 

We find that the interior equilibrium point (1 . 13636 , 0 . 454545 , 0 . 772727) is locally asymptotically stable where the conditions (2.7) is satisfied. 

## 3 Cooperation on graphs 

In natural gas, electricity and water transport a new situation exists namely transit countries. Those where transmission lines pass e.g. Belarus in the line Russia-Belarus- EU gas line. In this case a Prisoner’s Dilemma (PD) game [6] exists between both producers and end users from one side and 

