If there are 2 hares and 100 rabbits initially, that is, if and , then and . Now, and . There will be 8752 rabbits and 3184 hares after two years.
You could continue that until k= 20 or look for a general solution of the form , . Putting those into the two equations gives and . Dividing both sides of the first equation by gives and dividing both sides of the second equation by gives . We can solve that second equation for : . Now put that into the previous equation to get an equation for and .
By the way, are you sure you have copied the problem correctly? It seems strange that a large number of hares would reduce both rabbit and hare populations while a large number of rabbits would increase both. Typically, for competing species, x and y, the equations are , , reversing the signs.