Diffusive wave dynamics beyond the continuum limit

Scientists have observed and studied diffusive waves in contexts as disparate as population genetics and cell signaling. Often, these waves are propagated by discrete entities or agents, such as individual cells in the case of cell signaling. For a broad class of diffusive waves, we characterize the transition between the collective propagation of diffusive waves, in which the wave speed is well described by continuum theory, and the propagation of diffusive waves by individual agents. We show that this transition depends heavily on the dimensionality of the system in which the wave propagates and that disordered systems yield dynamics largely consistent with lattice systems. In some system dimensionalities, the intuition that closely packed sources more accurately mimic a continuum can be grossly violated.

Figure 1

Discrete lattice wave dynamics in 1D. (a) Schematic demonstrating our solution method for Fisher-type waves with $n=1$. Here, we examine the region well beyond the wave front and make an ansatz that the concentration at source $N+j$ is given by $c(x=(N+j)d,t=0)\sim e^{-(N+j)\gamma d}$. This ansatz allows a range of solutions $v\left(\gamma \right)$, the minimum of which is the realized wave speed. (b) Schematic demonstrating the solution method for Heaviside waves with $n\to \infty$. Here, the concentration at the wave front is given by $C_{\text{th}}$ while the source a distance $kd$ from the wave front has been emitting since time $t=-kd/v$. (c) Discrete wave speeds divided by continuum wave speed as a function of $ad/D{C}_{\text{th}}=\varphi$ for the $n=1$ theory (blue line), $n\to \infty$ theory (black line), and finite $n$ numerics (colored circles). For all $n>1$, large separation between sources leads to a subcontinuum wave speed while for $n=1$ the wave speed increases compared with its continuum value. At small values of $\varphi$, the wave can be understood as a collective effect that is well described by a continuum theory. (d) Theoretical wave speed for $n=1$ as stipulated by Eq. (23). At large values of $\varphi$, the wave speed attains a $d$- and $D$-independent value of $a/2C_{\text{th}}$. (e) Concentration at source $N$ generated by source $N+j$ for various values of $\varphi$ and for $n=1$. At small $\varphi$, many sources contribute to the concentration at source $N$ while as $\varphi$ increases, the wave propagation becomes a few-body phenomenon. For $\varphi =1000$, we see the extreme limit of a pulled wave, in which the concentration at the wave front is almost entirely due to self-amplification. (f) Concentration at the wave front generated by source $j$ as $n\to \infty$. As with the case of $n=1$, many sources contribute to the concentration at the wave front when $\varphi$ is small. As $\varphi$ increases, the wave propagation becomes a few-body problem. In contrast to the case of $n=1$, for $\varphi =1000$ the wave front is pushed by the nearest neighbor behind the front.

Figure 2

Comparison of lattice and disordered wave speeds in 1D. (a) Solution methodology for finding the wave speed propagated by a discrete, disordered system in 1D. The wave speed is determined by the average distance between sources $d$, divided by the mean hopping time $\overline{\tau }$. In the limit $d\ll D{C}_{\text{th}}/a$, $\overline{\tau }$ can be solved for analytically. (b) Discrete wave speed for $n=1$ divided by the continuum theory wave speed as a function of $ad/D{C}_{\text{th}}$. When $v/v_{1,1,n}\approx 1$, the discrete theory is in the continuum limit. Here, we see that the lattice theory (solid line) agrees well with the disordered numerics (circle markers), both of which agree with the nearest-neighbor disordered theory (dashed gray line) in the limit of $ad/D{C}_{\text{th}}\gg 1$. (c) Same as $\mathbf{B}$, but for $n\to \infty$. Unlike for $n=1$, here we see significant deviation of the disordered numerics (circular markers) and the lattice theory (solid line). In the limit $ad/D{C}_{\text{th}}\gg 1$, this discrepancy is well characterized by our nearest-neighbor disordered theory (dashed gray line).

Figure 3

Comparison of lattice and Poisson-disordered wave speeds in higher dimensions. (a) Wave speed divided by the continuum theory value with $(N,M)=(1,2)$ for disordered systems (circles) and lattice systems (black line); we plot this quantity for varying $a/D{C}_{\text{th}}$ (top) and varying $d$ (bottom). When $v/v_{N,M,n\to \infty }\approx 1$, a discrete system is in the continuum limit. As predicted by dimensional analysis, $v/v_{1,2,n\to \infty }$ does not change at all with $d$ (bottom panel), but decreases as $a/D{C}_{\text{th}}$ increases (top panel). Results in the bottom panel are shown for $a/D{C}_{\text{th}}=10$. The disordered and continuum theories show substantial deviation, which can be understood through a disordered nearest-neighbor theory (dashed gray line) at large $a/D{C}_{\text{th}}$. (b) Same as (a), but for $(N,M)=(2,2)$. Here, we see close correspondence between the lattice and disordered system wave speeds. As with $(N,M)=(1,2)$, the correspondence with the continuum theory is determined solely by $a/D{C}_{\text{th}}$ (top panel) and does not vary with $d$ (bottom panel). Results in the bottom panel are shown for $a/D{C}_{\text{th}}=1000$. (c) Wave speed divided by the continuum theory value as a function of $a/dD{C}_{\text{th}}$ with $(N,M)=(2,3)$. Here, we see a close correspondence between the continuum theory (solid line) and the discrete numerical simulations (circles). Unlike the cases of $M=1,2$, decreasing $d$ results in larger deviations from continuum theory. (d) Same as (c) but for $(N,M)=(3,3)$. Here, we see deviations between the continuum theory and numerical simulation for intermediate values of $a/dD{C}_{\text{th}}$. Again, we observe that decreasing $d$ results in larger deviations from the continuum theory, as predicted by dimensional analysis.