Figure 4.

Oscillatory dynamics generated from the Turing instability for the go-or-grow model. Spatio-temporal evolution of (a) motile and (b) static populations. The horizontal axis is used for space and the vertical one for time. Both curves correspond to the long-time evolution of the simulations presented in Figures 3(b) and (c). (a) The dashed lines represent the contour level ${\rho }_1={\stackrel{‒}{\rho }}_1$, the value of the initial uniform steady-state (${\stackrel{‒}{\rho }}_1=0.31$ for the simulation). These contour levels suggest an almost spatially uniform distribution of motile cells that increases after the initial instability depicted in Figure 3(b) and (c) until a threshold (reached at t ≃ 140), where motile cells become static (i.e. ρ₁ decreases by feeding the static population). This corresponds to the development of a new instability of the static population, which later stabilizes and feeds again the motile population, thus increasing again until similar dynamics are repeated at t ≃ 500. (b) The dotted and dashed lines represent the contour levels ρ₂ = 0.01 and ρ₂ = 0.1, respectively. They suggest that low values of the static population correspond to almost uniform distributions (i.e. ρ₂(x, t) ≃ ρ₂(t)). When the immotile population increases enough due to an influx from the motile population (at times t ≃ 140 and t ≃ 500), an instability occurs and briefly leads to a non-uniform distribution of immotile aggregates. The latter disappear shortly by releasing cells into the motile population due to the lack of density until the process repeats.