Hi,

it appears that NSolve does a good job:

f[u, v] := u (1 - u + (-1 + E^(-u \[Alpha])) v);
g[u, v] := v (\[Beta] - E^(-u \[Alpha]) \[Beta] - \[CapitalGamma]);
NSolve[{f[u, v] == 0, g[u, v] == 0}, {u, v}, Reals]

gives

Cheers,

M.

I'm getting errors when I try to evaluate both your Manipulate expressions. For he first:

Part::take: Cannot take positions 1 through 3 in sols. >>
ReplaceAll::reps: {sols[[1;;3]]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

etc.

For the second:

Select::normal: Nonatomic expression expected at position 1 in Select[sols,!MemberQ[#1[[All,2]],Undefined]&]. >>
ReplaceAll::reps: {sols} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>
ReplaceAll::reps: {sols} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>
ReplaceAll::reps: {!MemberQ[#1[[All,2]],Undefined]&} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

etc.