Figure 1

Morphological diagrams. Data points represent samples of different pairs of initial nutrient level $n_0$ and of “substrate softness” parameter $\tau$. Distinct pattern regimes (“phase”) deduced by inspection are labeled by numbers, wherein lines serve as guides that delineate phase boundaries. Representative patterns are shown for each phase. (A) active motility, low per capita phenotypic switching rate $\alpha =0.0098\Delta t^{-1}$; (B) active motility, high $\alpha =0.12\Delta t^{-1}$; and (C) passive motility, $\alpha =0.0098\Delta t^{-1}$. Phase boundaries A3–A4 and C3–C4 denote the onset of migrator dominance at low $\alpha$, whereas the phase boundary C1–C2 for $\tau \le 0.6$ is an outcome of a Mullins-Sekerka front instability. Recognized patterns: A3, B3 (CT); A4 (SD); B2 (DBM); C1 (RE); C2 (DLF). A composite diagram A3+A4+B2 (bottom)+C1+C2 approximates experimental morphological diagrams reported previously [3].Reuse & Permissions

Figure 2

Sketch of the plane-wave sharp interface in the comoving frame ($\zeta$,$x$), where $\zeta =y-vt$ with $v$ being the front velocity. The regions are designated as e (external uncolonized area), a (active region), and i (inactive region). (a) Unperturbed plane wave: $\zeta =0$ is designated as the front, whereas the tail is behind it at $\zeta =-d$; (b) perturbed plane wave: ${\zeta }_0\left(x\right)=\delta \cos qx$ and $d\left(x\right)=d-\delta \cos qx$; the velocity of the front is now designated as a function of $x$.Reuse & Permissions

Figure 3

Dispersion equation $\omega \left(q\right)$ for two extreme cases of nutrient level $n_0$. The plots are derived by setting $v=\lambda =0.01$. For the high-$n_0$ case, the plots are generated using Eq. (30), and $d$ is calculated through Eq. (28) with $n_0=20$. For the low-$n_0$ case, the plots are generated from Eq. (34), and $d$ is calculated through Eq. (33).Reuse & Permissions

Figure 4

(a) CT by active motility ($n_0=16$, $\tau =0.6$; $\gamma =0.2$; $\alpha =0.0098$; $\xi =0.2$; time $=600\Delta t\approx 5.8$ days). Circles indicate different times: solid ($t_1=150\Delta t$) corresponds to an onset of terrace formation; dashed ($t_2=252\Delta t$) to onset of rapid mass migration. (b) Phase scatterplot (migrator density $y$ vs proliferator density $x$) snapshots at $t_1$ (dark dots) and $t_2$ (bright dots). Nullclines are shown as black curves. The hollow circle is the unstable equilibrium $p_u^{*}$, and the solid circle is the stable equilibrium $p_s^{*}$. Wide gray arrow pointing toward $(0,0)$ indicates the effect of migration that induces a flow to the right and closer to $p_u^{*}$, which repels the trajectory and pushes it toward $p_s^{*}$. A migration-induced cycle thus generates the CT structure. The nonlinear mechanism is animated in video format [41].Reuse & Permissions