Einstein’s Brownian motion model for chemotactic system and traveling band

The candidate reproduces the paper’s traveling-wave reduction and integration for the limited-substrate model with crowd effect: starting from ∂t v + k u v = 0, the traveling-band ansatz yields (ln v)' = (k/c)u; substituting into the u-equation collapses it to a first-order logistic equation for u, which integrates to u(ζ) = [2τc^2]/[k(β+γτ)]·(1 + C5 e^{(2τc/μ)ζ})^{-1} and, upon integrating (ln v)' and imposing v(+∞)=v∞, gives v(ζ) = v∞(1 + C7 e^{−(2τc/μ)ζ})^{−μ/(β+γτ)} with the expected asymptotics u(+∞)=0, u(−∞)=2τc^2/[k(β+γτ)], v(+∞)=v∞, v(−∞)=0. These are exactly equations (5.5)–(5.7) obtained in the paper from equations (5.2)–(5.4) under the traveling-wave reduction , and the boundary behavior listed there . The governing PDE system and traveling-band setup agree with the paper’s derivation (4.8), (5.1), and (4.9) . For the special case γ0 = 1/τ (no crowd effect), the limiting plateau K reduces to τc^2/(kβ), matching the paper’s Theorem 4 (5.9)–(5.15) .