Figure 1

(a) Three-dimensional trajectory of a single bacterium showing the stereotypical run and tumble motion characterized by many persistent runs at constant velocity and occasional tumbles corresponding to changes in velocity. (Reprinted from Ref. [6], with permission.) (b) Image of a crowded colony of Escherichia coli. (Reprinted from Ref. [26], with permission.)Reuse & Permissions

Figure 2

One-dimensional schematic used to illustrate the key differences between the traditional noninteracting velocity-jump process [case (0)] and the three interacting velocity-jump processes [cases (1)–(3)] introduced in this work. Light gray (green) agents are shown with a left arrow and dark gray (red) right-moving agents are shown with a right arrow.Reuse & Permissions

Figure 3

Snapshots of a one-dimensional noninteracting velocity-jump process. The distribution of agents in 20 identically prepared realizations are shown at (a) $t=0$, (b) $t=15$, and (c) $t=30$. Each independent realization is presented as a separate row in (a)–(c). Each realization is performed on a one-dimensional lattice with $1\le x\le 200$ and the parameters in the velocity-jump process are $\Delta =\tau =P=1$ with $v=2$ and $\lambda =0.01$. Each snapshot shows light gray (green) left-moving agents and dark gray (red) right-moving agents. The initial distribution of agents is given by Eqs. (4) and (5) with $q=4$, $w=0.5$, and $x_c=100$.Reuse & Permissions

Figure 4

Comparison of discrete and continuum density profiles for various noninteracting velocity-jump processes with small turning rates. Simulations are performed on a one-dimensional lattice with $1\le x\le 2000$ and the parameters in the velocity-jump process are $\Delta =\tau =P=1$ with $v=2$ and various values of $\lambda$. The initial distribution of agents is given by Eqs. (4) and (5) with $q=150$, $w=0.5$, and $x_c=1000$. Density profiles from the discrete simulations are computed using Eqs. (6) and (7) with $M=1000$ identically prepared realizations. The profile shown in (a) corresponds to the initial condition and the three profiles shown in (b)–(d) correspond to $\lambda ={10}^{-4},{10}^{-3}$, and ${10}^{-1}$, respectively. The stochastic density profiles are shown by the thin solid, dark gray (red) line and the thin solid, light gray (green) line for the left-moving and right-moving subpopulations, respectively. The solution of the continuum PDE model Eqs. (12) and (13) is shown by the superimposed, thick dashed, dark gray (red) line and the thick dashed, light gray (green) line for $L(x,t)$ and $R(x,t)$, respectively. The numerical solution of the PDE models was obtained using the technique outlined in the manuscript with $\delta x=0.5$, $\delta t=0.05$, and $\epsilon =1\times {10}^{-6}$.Reuse & Permissions

Figure 5

Comparison of discrete and continuum density profiles for various noninteracting velocity-jump processes with larger turning rates. Simulations are performed on a one-dimensional lattice with $1\le x\le 2000$ and the parameters in the velocity-jump process are $\Delta =\tau =P=1$ with $v=2$ and various values of $\lambda$. The initial distribution of agents is given by Eqs. (4) and (5) with $q=150$, $w=0.5$, and $x_c=1000$. Density profiles from the discrete simulations are computed using Eqs. (6) and (7) with $M=2000$ identically prepared realizations and the profiles shown in (a) and (b) correspond to $\lambda =0.8$ and $0.9$, respectively. The stochastic density profiles are shown by the thin solid, dark gray (red) line and the thin solid, light gray (green) line for the left-moving and right-moving subpopulations, respectively. The solution of the continuum PDE model Eqs. (12) and (13) is shown by the superimposed, thick dashed, dark gray (red) line and the thick dashed, light gray (green) line for $L(x,t)$ and $R(x,t)$, respectively. Results in (c) and (d) are for the total population $S(x,t)=L(x,t)+R(x,t)$ and these profiles were obtained by summing the corresponding profiles in (a) and (b) with the discrete results shown by the thin solid line and the continuum results shown by the thick dotted line. The numerical solution of the PDE models was obtained using the technique outlined in the paper with $\delta x=0.5$, $\delta t=0.05$, and $\epsilon =1\times {10}^{-6}$.Reuse & Permissions

Figure 6

Snapshots of several one-dimensional noninteracting and interacting velocity-jump processes. The distribution of agents in 20 identically prepared realizations are shown at (a) $t=0$ and in (b)–(e) at $t=20$ for a range of interacting and noninteracting cases. Noninteracting results are shown in (b) while interacting cases (1)–(3) are shown in (c)–(e), respectively. Each independent realization is presented as a separate row in (a)–(e). All simulations are performed on a one-dimensional lattice with $1\le x\le 200$ and the parameters $\Delta =\tau =P=1$ with $v=2$ and $\lambda =0.01$. Each snapshot shows light gray (green) left-moving agents and dark gray (red) right-moving agents. The initial distribution of agents is given by Eqs. (4) and (5) with $q=4$, $w=0.5$, and $x_c=100$.Reuse & Permissions

Figure 7

Comparison of discrete and continuum density profiles for various interacting velocity-jump processes with no turning $\lambda =0$. Simulations are performed on a one-dimensional lattice with $1\le x\le 2000$ and the parameters $\Delta =\tau =P=1$ with $v=2$. In all cases the initial distribution of agents is given by Eqs. (4) and (5) with $q=150$, $w=0.25$, and $x_c=1000$. Density profiles from the discrete simulations are computed using Eqs. (6) and (7) with $M=1000$ identically prepared realizations. The profiles in (a) and (d) are for case (1) at $t=50$ and 100, respectively. The profiles in (b) and (e) are for case (2) at $t=50$ and 100, respectively. The profiles in (c) and (f) are for case (3) at $t=50$ and 100, respectively. The discrete profiles for the left-moving subpopulations are shown by the thin light gray (green) line. The solution of the continuum PDE models (Eqs. (17), (18), (22), (23), (27), and (28) are by the superimposed, thick dashed, light gray (green) line for $L(x,t)$. (g) and (h) Comparison of the continuum initial condition (dotted black line) with the solution of the noninteracting (solid black line) model and the three interacting models [light gray (green) solid line] for this particular problem at $t=50$ and 100, respectively. The profiles for cases (0)–(3) are shown with the number indicating the case. The numerical solution of the PDE models was obtained using the technique outlined in the paper with $\delta x=0.5$, $\delta t=0.05$, and $\epsilon =1\times {10}^{-6}$.Reuse & Permissions

Figure 8

Comparison of discrete and continuum density profiles for various interacting velocity-jump processes with a small turning probability $\lambda =0.01$. Simulations are performed on a one-dimensional lattice with $1\le x\le 2000$ and the parameters $\Delta =\tau =P=1$ with $v=2$. In all cases the initial distribution of agents is given by Eqs. (4) and (5) with $q=150$, $w=0.25$, and $x_c=1000$. Density profiles from the discrete simulations are computed using Eqs. (6) and (7) with $M=1000$ identically prepared realizations. The profiles in (a) and (d) are for case (1) at $t=15$ and 30, respectively. The profiles in (b) and (e) are for case (2) at $t=15$ and 30, respectively. The profiles in (c) and (f) are for case (3) at $t=15$ and 30, respectively. The stochastic density profiles for the left-moving subpopulations and the right-moving subpopulations are shown by the thin light gray (green) and dark gray (red) lines, respectively. The solutions of the continuum PDE models Eqs. (17), (18), (22), (23), (27), and (28) are shown by the superimposed, thick dashed, light gray (green) line for $L(x,t)$ and the thick dashed, dark gray (red) line for $R(x,t)$. (g) and (h) Comparison of the continuum initial condition (dotted black) with the solution of the noninteracting model (black line) and the three interacting models [light gray (green) line] for this particular problem at $t=15$ and 30, respectively. The profiles of the left-moving subpopulation are shown for cases (0)–(3) with the number indicating the case. The numerical solution of the PDE models was obtained using the technique outlined in the paper with $\delta x=0.5$, $\delta t=0.05$, and $\epsilon =1\times {10}^{-6}$.Reuse & Permissions