The main problem is that Log2 is a function, not a number. I substituted Log[2] for it and moved the diff. eq. outside Manipulate to avoid recalculating the solution many times.
step[k_, a_] := k HeavisidePi[t/a - 1/2];
psol = ParametricNDSolveValue[{
u'[t] == step[10, 20] + \[Lambda]1*u[t]^2/(1 + u[t]^2) - Log[2]/120*u[t] - v[t]*\[Alpha]*u[t],
w'[t] == \[Lambda]1*u[t]^2/(1 + u[t]^2) - Log[2]/120*w[t],
v'[t] == \[Lambda]1*1/(1 + w[t]^2)*step[10, 20] + \[Beta] - Log[2]/30*v[t],
u[0] == w[0] == v[0] == 0},
{u[t], w[t], v[t]},
{t, 0, 100},
{\[Alpha], \[Beta], \[Lambda]1}];
Manipulate[
Plot[Evaluate@psol[\[Alpha], \[Beta], \[Lambda]1], {t, 0, 100}],
{{\[Alpha], 0, TraditionalForm[auxin]}, 0, 10},
{{\[Beta], 0, TraditionalForm[b]}, 0, 10},
{{\[Lambda]1, 0, TraditionalForm[lamda1]}, 0, 10}]
