exactsol = Simplify[
k[t]/.DSolve[{k'[t]==s ((k[t])^α) -(δ+n) k[t],k[0]==a},k[t],t][[1]]]

VectorPlot[{1,s (k^α) -(δ+n)  k/.{δ->0.1,n->0.01,s->0.15,α->0.25}},{t,0,30},{k,0,2.5},VectorPoints->Fine]

exactsol/.{α->0.25,δ->0.1,n->0.01,s->0.1}//Simplify

Plot[Table[exactsol/.{α->0.25,δ->0.1,n->0.01,s->0.1},{a,0.1,2,0.1}],{t,0,40},PlotRange->Full]

Plot[Table[exactsol/.{δ->0.1,n->0.01,s->0.15,a->0.1},{α,0.1,0.9,0.1}],{t,0,80},PlotRange->Full]