= Solution
Take $u>0$. Despite the source's terminology, $P(x)$ is a dimensionless conditional bound-state <probability> at phase $x$, not a spatially normalized <probability density>; the phase-average bound fraction is $\Delta^{-1}\int_0^\Delta P\,dx$. The given equation has <advection> and binding reactions but no diffusion term. Large $N$ permits the prescribed steady $u$ approximation.
For <spatially gated myosin attachment>, define $A=e^{-k_1x_1/u}$, $B=e^{-k_2(\Delta-x_2)/u}$, $p_0=P(0)$ and $p_b=P(x_1)$. Continuity and periodicity give $p_b=1-(1-p_0)A$, $p_0=Bp_b$, hence $p_b=(1-A)/(1-AB)$. The full periodic steady solution is
$$
\boxed{P_{\rm ss}(x)=\begin{cases}
1-(1-p_0)e^{-k_1x/u},&0\le x<x_1,\\
p_b,&x_1\le x<x_2,\\
p_b e^{-k_2(x-x_2)/u},&x_2\le x<\Delta,
\end{cases}\qquad
p_0=\frac{B(1-A)}{1-AB}.}
$$
The requested limit makes $B\ll1$, so $p_0\simeq0$ and $p_b\simeq1-e^{-k_1x_1/u}$. The curve rises exponentially over the binding window, remains flat to $x_2$, then decays over length $u/k_2$ to an exponentially small value before the next period. The piecewise expression is continuous at both switching points and at the period boundary.
\Image[/past-exam-of-the-flows-fluctuations-and-complexity-and-symmetry-and-relativity-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011-motor-occupancy.png]
{title=Periodic bound-head probability with a binding window, plateau and detachment tail}
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