Stability of a non-local kinetic model for cell migration with density dependent orientation bias

Figure 1.  Stability diagram for a localized sensing kernel $ \gamma_R = \delta(\lambda-R) $. The red dashed line delimits the unstable region, in (a) when $ b'(\rho_{\infty})<0 $ and in (b) when $ b'(\rho_{\infty})>0 $. The blue dotted lines evidentiate the dimensionless wave numbers $ k_{max}R $ with local maxima of the growth rate (given by (34)) in the unstable regime. The lowest curves correspond to the most dangerous wave numbers in both cases

Figure 2.  Stability diagram for a uniform (a) and a ramp (b) sensing kernel. The blue dotted line represents $ k_{max}R $ given respectively by (a) (41) and (b) (44). The unstable region is the one to the left of the red dashed line, $ {\it i.e.} $ the values of $ k_{max}R $ also satisfy (40) in (a) and (43) in (b)

Figure 3.  Temporal evolution of $ \rho(t,x) $ from $ \rho_0(x) = 0.2\left(1+0.1\sin(\pi x/5)\right) $ (a) in the unstable case $ \mathcal{V}_b\approx -0.1125 $ and (b) in the stable case $ \mathcal{V}_b\approx -0.3376 $. (c) Wavelength of the most unstable mode as obtained from the simulation (black line) and from Eq.(34) (green line)

Figure 4.  Evolution of the density distribution in the adhesion case starting from the initial condition $ \rho_0(x) = 0.2\left(1+0.1\sin(\pi x/5)\right) $, so that $ \rho_{\infty}\approx 0.2127 $, for a delta kernel (top row) and a Heaviside kernel (bottom row). In the left column the values of $ \mathcal{V}_b $ correspond to stable cases, while in the right column $ \mathcal{V} $ correspond to the unstable case. Specifically, in (a) $ \mathcal{V}_b\approx 1.1364 $, in (b) $ \mathcal{V}_b\approx 0.7812 $, in (c) $ \mathcal{V}_b\approx 0.5681 $, and in (d) $ \mathcal{V}_b\approx 0.4032 $

Figure 5.  Comparison of unstable evolutions for the same value of $ \mathcal{V} = 0.0625 $, (given by $ V = 0.25, R = 0.04, \mu = 100 $) starting form the initial condition $ \rho_0(x) = 0.2\left(1+0.1\sin(\pi x/5)\right) $, so that $ \rho_{\infty}\approx 0.2127 $. In (a) $ \gamma_R(\lambda) = \delta(\lambda-R) $, in (b) $ \gamma_R(\lambda) = H(R-\lambda) $ and in (c) $ \gamma_R = \left(1-\frac{\lambda}{R}\right)_+ $

Figure 6.  Evolution in the asymmetric case for a localised sensing kernel. The initial condition is always $ \rho_0(x) = 0.2\left(1+0.01\sin(\pi x/5)\right) $, so that $ \rho_{\infty}\approx 0.2013 $. (a)-(c): Adhesion, $ \mu = 3, R = 0.44, V^+ = 0.5, V^- = 1, \mathcal{V}_b\approx 0.5682 $. (d)-(f): Volume filling, $ \mu = 200, R = 0.4, V^+ = 0.25,V^- = 0.5,\rho_{th} = 1, \mathcal{V}_b\approx -0.018 $. (g): Volume filling, $ \mu = 200, R = 0.4, V^+ = V^- = 0.375,\rho_{th} = 1, \mathcal{V}_b\approx -0.018 $. (h)-(i): Adhesion, $ \mu = 1, R = 0.44, V^+ = 0.5, V^- = 1, \mathcal{V}_b\approx 1.7045 $