I confirm tomleslie's observation that pdsolve gets stuck here; I cancelled it after lunch break (about 1h).
However, you can still find special types (or forms) of solutions. Just to give you a start:
infolevel[pdsolve]:=1: # optionally
pde := diff(u(x, t), t)-DD*diff(u(x, t), x, x) = alpha*u(x, t)*(1-v(x, t));
pde := diff(v(x, t), t)-EE*diff(v(x, t), x, x) = beta*v(x, t)*(1-u(x, t));
pdesys := [pde,pde]:
TWSols := TWSolutions(pdesys,parameters=[DD,EE,alpha,beta]);
SimSols := SimilaritySolutions(pdesys);
I've stopped here, but there are several other specialized commands in PDEtools, as you can see from its export list. You may have to play with command options to expand or to restrict their output.
Note: I replaced constant factor D by DD, as the former is the differential operator in Maple. For cosmetic reasons, I then replaced E by EE as well.