f:={k1 u2 (a1-u1)-k2 u1 u3,k3 u1 (a2-u2),k5 (a3-u3)-k6 u1 u3}
u:={u1,u2,u3}

JacobianMatrix[f_List,x_List]:=Outer[D,f,x]

JacobianMatrix[f,u]//MatrixForm