Survival probability for a diffusive process on a growing domain

We consider the motion of a diffusive population on a growing domain, $0<x<L\left(t\right)$, which is motivated by various applications in developmental biology. Individuals in the diffusing population, which could represent molecules or cells in a developmental scenario, undergo two different kinds of motion: (i) undirected movement, characterized by a diffusion coefficient, $D$, and (ii) directed movement, associated with the underlying domain growth. For a general class of problems with a reflecting boundary at $x=0$, and an absorbing boundary at $x=L\left(t\right)$, we provide an exact solution to the partial differential equation describing the evolution of the population density function, $C(x,t)$. Using this solution, we derive an exact expression for the survival probability, $S\left(t\right)$, and an accurate approximation for the long-time limit, $\mathcal{S}={lim}_{t\to \infty }S\left(t\right)$. Unlike traditional analyses on a nongrowing domain, where $\mathcal{S}\equiv 0$, we show that domain growth leads to a very different situation where $\mathcal{S}$ can be positive. The theoretical tools developed and validated in this study allow us to distinguish between situations where the diffusive population reaches the moving boundary at $x=L\left(t\right)$ from other situations where the diffusive population never reaches the moving boundary at $x=L\left(t\right)$. Making this distinction is relevant to certain applications in developmental biology, such as the development of the enteric nervous system (ENS). All theoretical predictions are verified by implementing a discrete stochastic model.

Figure 1

Comparison of continuum predictions (solid blue) and discrete estimates (red dashed) of $S\left(t\right)$ for (a) exponential domain growth $L\left(t\right)=L\left(0\right){\text{e}}^{\alpha t}$, and (b) linear domain growth $L\left(t\right)=L\left(0\right)+\beta t$. In all cases we have $L\left(0\right)=100$, and the initial condition is given by Eq. (5) with $\gamma =25$. Results in (a) correspond to $\alpha ={10}^{-5},5\times {10}^{-5},{10}^{-4}$, and $5\times {10}^{-4}$, with the arrow showing the direction of increasing $\alpha$. Results in (b) correspond to $\beta =5\times {10}^{-3},{10}^{-2},5\times {10}^{-2}$, and ${10}^{-1}$, with the arrow showing the direction of increasing $\beta$. To evaluate the exact continuum expressions for $S\left(t\right)$, given by Eqs. (11) and (14), we truncated the infinite series after 100 terms and checked that these results were insensitive to this choice of truncation. To generate the discrete results we set $N=80$ and $P=1$, giving $D=1/2$.