Table 1 Summary of the different coordinate systems used to describe the population in Model 1.

From: First-passage times and normal tissue complication probabilities in the limit of large populations

 

Coordinate

Interpretation

Relations

(A)

Nt

number of individuals in population at time t

—

(B)

nt

population density

nt = Nt/M

(C)

ϕ(t)

deterministic (mean-field) trajectory

nt = ϕ(t) + M−1/2ξt

ξt

deviation from mean-field path due to linear noise

(D)

ζτ

re-scaled population near boundary \(\ell =L/M\)

\({n}_{\tau }=\ell +{M}^{-\mathrm{1/2}}{\zeta }_{\tau }\)

τ

re-scaled time near deterministic crossing time t*

\(t={t}^{\ast }+\frac{{M}^{-\mathrm{1/2}}}{-\mu (\ell ,{t}^{\ast })}\tau \)

  1. Original coordinates (A) appear in the master Eq. (5), while coordinates (B) and (C) are used in the Kramers–Moyal expansion and linear-noise approximation, respectively [see Eqs. (9) and (11)]. Coordinates (D) are used for our analysis of the dynamics in the narrow, boundary-crossing region. The subscript t (or τ) is used to denote random processes.
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