Figure 1

Dependence of the parameters in the reduced model of Sec. 2 in terms of the output from the stochastic simulations of the many-body problem. Panel (a) is a log-linear plot of the dimensionless $K/D$ versus the dimensionless quantity $Z=T_{\mathit{AS}}^{\prime }\rho {}^{\prime 2}$ where $\rho {}^{\prime }$ is the population density multiplied by the lattice spacing and $T_{\mathit{AS}}^{\prime }$ is the dimensionless quantity obtained by multiplying $T_{\mathit{AS}}$ with the random walk transfer rate $F$ between nearest neighbor sites. Panel (b) is a log-linear plot of the dimensionless quantity $s^{*}/L^2$ versus $Z^{1/4}$. The solid lines are least square fits of all the data points for, respectively, $\text{log}(K/D)=1.11-0.95Z$ in panel (a) and $\text{log}(\mathrm{s̄}/L^2)=0.64-2.32Z^{1/4}$ in panel (b). Each point in (a) is found by averaging over ${10}^5$ stochastic realisations of the simulation for as many time steps as needed to reach the boundary MSD’s asymptotic regime. The time to reach this asymptotic regime increases exponentially with both $\rho {}^{\prime }$ and $T_{\mathit{AS}}^{\prime }$. For lower values of $\rho {}^{\prime }$, a high $T_{\mathit{AS}}^{\prime }$ is required for the boundary MSD to reach the asymptotic regime before the limits of the box size are reached. Therefore using either very high or very low $\rho {}^{\prime }$ is not practically possible, allowing us only the limited set of values in this plot. For (b), a single simulation of ${10}^8$ time steps gave sufficient data for finding each data point.Reuse & Permissions

Figure 2

Comparison of the walker probability distribution from the simulated model (dashed lines) with the reduced model (solid lines) at time $\mathit{tF}=10\hspace{0.5em}500$ for four values of $Z$: (a) $Z=1.8$, (b) $Z=1.4$, (c) $Z=1.2$ and (d) $Z=0.6$. To plot the probability distribution of the reduced model, $M(x,t)$ in Eq. (B3) is multiplied by the lattice constant. In all panels the density has been kept fixed at $\rho {}^{\prime }=0.02$, whereas $T_{\mathit{AS}}^{\prime }=9000,7000,6000,3000$ for panels (a)–(d), respectively. Running the simulations requires a choice of initial scent spatial profile, i.e., the time value associated with each lattice site that defines how long before the scent stops being active. We have selected a biologically relevant scent profile with minimum values at the scent boundaries and a maximum in the middle of the territory. The initial location of the two walkers is consequently the middle of their respective territories (see Ref. [9] for further details).Reuse & Permissions

Figure 3

Nonmonotonicity of the walker’s MSD [Eq. (7) with $x_0=0$] for different parameter values. Panel (a) shows how the nonmonotonic behavior of the MSD, for two choices of $\alpha <{\alpha }_0$, tends to be washed away as $K^{\prime }$ decreases, while the $D^{\prime }$ and $\beta$ are kept fixed. In panel (b) the MSD dependence over $D^{\prime }$ is shown while keeping the other parameters fixed: $K^{\prime }=0.01$, $\alpha =0.05$, $\beta =0.1$, and in panel (c) the MSD dependence over $\beta$ is shown with $K^{\prime }=0.01$, $\alpha =0.05$, $D^{\prime }=1$.Reuse & Permissions

Figure 4

Lines in parameter space $D^{\prime }-\alpha$, for different values of $K^{\prime }\beta {}^{1-\alpha }$, delineating the regions below which the walker’s MSD is nonmonotonic.Reuse & Permissions