Stochastic drift in discrete waves of nonlocally interacting particles

In this paper, we investigate a generalized model of $N$ particles undergoing second-order nonlocal interactions on a lattice. Our results have applications across many research areas, including the modeling of migration, information dynamics, and Muller's ratchet—the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as ${(lnN)}^{-2}$ and ${(lnN)}^{-3}$, respectively—much slower than the usual $N^{-1/2}$ factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.

Figure 1

The stochastic waves (colors) propagate faster than the deterministic solution (dashed). The position of the wave at a given time fluctuates from simulation to simulation, flattening the average curve (solid black line). The parameters used to generate the figure are: $N={10}^3$, $t=400$, $\alpha =0.5$, $\beta =0$, ${\lambda }_1=1$ and ${\lambda }_2=0$ (see Sec. 2 for the model description). Initial condition: $n_{50}\left(0\right)=N$.

Figure 2

Comparison between the stochastic and the deterministic systems for ${\lambda }_1={\lambda }_2$ at different instants of time, $t$. The solution of the ODE system, Eq. (9), (dashed) is faster than the stochastic waves (colours). The flattening of the simulations' average (solid black line) is caused by fluctuations in the position of the stochastic wave from realization to realization. Parameters: $N={10}^4$, $\alpha =1$, $\beta =0$, ${\lambda }_1={\lambda }_2=1$. Initial condition: $n_{250}\left(0\right)=N$.

Figure 3

The stochastic wave (turquoise, yellow, and orange) and the solution of Eq. (8) (brown) plotted at different times, $t$. The stochastic wave slowly drifts, whilst the deterministic solution remains static. Parameters: $N={10}^4$, $\alpha =1$, $\beta =0$, ${\lambda }_1=4$, ${\lambda }_2=1$. Initial conditions as explained in text.

Figure 4

A schematic representation of the definitions adopted in the cutoff approach to estimate the speed of propagation of the stochastic wavefront. The deterministic solution is set to zero if it assumes values below the cutoff threshold, ${\varepsilon }_N$ (hatched region). This occurs at a distance $L$ from the reference position, ${\mu }_t$. Note the log-scale in the $y$ axis.

Figure 5

The speed of propagation of the stochastic wavefront (squares) for different population sizes. The error bars give the standard deviation of the simulations for each population size. Upper figure: $\alpha =0.5$, $\beta =0$, ${\lambda }_1=1$, ${\lambda }_2=0$. Bottom figure: $\alpha =1,{\lambda }_1={\lambda }_2=\frac{1}{2}$, $\beta =0$.

Figure 6

The diffusion constant of the stochastic wavefront (squares) for different population sizes. Parameters: $\alpha =0.5$, $\beta =0$, ${\lambda }_1=1{\lambda }_2=0$.

Figure 7

Comparison between the deterministic solution of Eq. (4) (dashed) and stochastic simulations (colors) for $f\left(z\right)=\frac{1}{2\pi }arctan(-z)+\frac{1}{4}$ (inset). For this $f\left(z\right)$, ${lim}_{z\to -\infty }f\left(z\right)=\frac{1}{2}$ and ${lim}_{z\to \infty }f\left(z\right)=0$. Note the similarity with Fig. 1. The parameter values used to generate the figure are: $N={10}^3$, $t=400$, ${\lambda }_1=1$, ${\lambda }_2=0$. Initial condition: $n_{50}\left(0\right)=N$.

Figure 8

Comparison between the deterministic solution of Eq. (4) (dashed) and stochastic simulations (colors) for $f\left(z\right)=\theta (-z-5)+\frac{1}{2}\theta (z-5)$ (inset). For this $f\left(z\right)$, ${lim}_{z\to -\infty }f\left(z\right)=1$ and ${lim}_{z\to \infty }f\left(z\right)=\frac{1}{2}$, however, $f^{-}-f^{+}=\frac{1}{2}$ as before. The parameter values are: $N={10}^3$, $t=400$, ${\lambda }_1=1$, ${\lambda }_2=0$. Initial condition: $n_{50}\left(0\right)=N$ (same as Fig. 1).

Figure 9

Comparison between the deterministic solution of Eq. (4) (dashed) and stochastic simulations (colors) for the function $f\left(z\right)$ given by Eq. (D1) (inset). For this $f\left(z\right)$, ${lim}_{z\to -\infty }f\left(z\right)=1$ and ${lim}_{z\to \infty }f\left(z\right)=\frac{1}{2}$; however, $f^{-}-f^{+}=\frac{1}{2}$ as before. The parameter values are $N={10}^3$, $t=400$, ${\lambda }_1=1$, ${\lambda }_2=0$. Initial condition: $n_{50}\left(0\right)=N$.