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

  • * Corresponding author: Nadia Loy

    * Corresponding author: Nadia Loy 
This work was partially supported by Istituto Nazionale di Alta Matematica, Ministry of Education, Universities and Research, through the MIUR grant Dipartimenti di Eccellenza 2018-2022, Project no. E11G18000350001, and the Scientific Reseach Programmes of Relevant National Interest project n. 2017KL4EF3. NL also acknowledges Compagnia di San Paolo that funds her Ph.D. scholarship
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  • The aim of the article is to study the stability of a non-local kinetic model proposed in [17], that is a kinetic model for cell migration taking into account the non-local sensing performed by a cell in order to decide its direction and speed of movement. We show that pattern formation results from modulation of one non-dimensional parameter that depends on the tumbling frequency, the sensing radius, the mean speed in a given direction, the uniform configuration density and the tactic response to the cell density. Numerical simulations show that our linear stability analysis predicts quite precisely the ranges of parameters determining instability and pattern formation. We also extend the stability analysis to the case of different mean speeds in different directions.

    Mathematics Subject Classification: Primary: 35Q20, 35B36, 92B05; Secondary: 45K05, 92C15.


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  • 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 $

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