Self-organized clusters in diffusive run-and-tumble processes

Figure 4.  Homoclinic trajectories in phase space (black), their projections onto the $ \rho $-$ \rho' $ plane (red) and the region $ G $ in the $ \rho $-$ \rho' $ plane (blue) for $ \gamma = 1/16 $ and $ \mu = 1 $. The vertical plane $ \rho' = 0 $ is the reversibility plane $ \mathrm{Fix}\,M $; trajectories are symmetric with respect to reflections at this plane. See Section 3.1 for details on how these solutions were found numerically

Figure 1.  The Hamiltonian potential, $ V(w) $, with region spanned by homoclinic (left) and heteroclinic (right) orbit

Figure 2.  Left column, top to bottom: Potentials $ W(\rho,c) $ for cases with homoclinics to $ \rho_- $, $ \rho_+ $, and heteroclinics, respectively. Right column: Associated phase portraits

Figure 3.  Maxima and minima of homoclinic orbits as a function of the background state (left). Plots of density profiles $ u $ (blue, left-traveling), $ v $ (red, right-traveling), and $ \rho = u+v $ (black) for $ \gamma = 0.15 $, $ c = -0.5,-0.234,0.2 $ from top to bottom (right). Note that concentrations of inward traveling populations peak at the boundary of high-density regions

Figure 5.  Plot of the boundaries of $B$ and $G$ in the $\rho$-$\rho'$ plane for $\mu = 1$

Figure 6.  Cluster and gap solutions, with associated phase portraits. Individual plots show the families of solutions as the background state is varied. Different plots correspond to different values of the parameter $ \gamma $. Shown is the actual computational domain, grid sizes vary in $ dx = 0.01\ldots 0.025 $

Figure 7.  Maxima of clusters and minima of gaps in the continuation, plotted against the background state, for sample values of $ \gamma $

Figure 8.  Heteroclinic profiles plotted as $ \gamma $ varies from $ \gamma = 1/13 $ to $ \gamma = 1/6639 $. Plots of $ \rho\sqrt{\gamma/6} $, $ \log(\rho\sqrt{\gamma/6}) $, and $ \rho'/\rho $ exhibit the asymptotically simple structure of the heteroclinic. Bottom left shows the actual computational domain, grid size is $ dx = 0.088 $

Figure 9.  Real part of the spectrum of the heteroclinics (left) and the two background concentrations of the heteroclinics in black (right); numerically the maximum and minimum of $ \rho(x) $, and the region where the corresponding constant solutions are stable in pink. Computations here use grid size from the previous heteroclinic continuation

Figure 10.  Spectra of clusters and gaps as functions of the background $ \rho_\infty $, for various values of $ \gamma $. Note that eigenvalues with positive real parts exist for gaps and clusters with large or small $ \rho_\infty $, that is, near small-amplitude or heteroclinic limit, respectively

Figure 11.  Cluster instability (left) and gap instability (right). Time evolution of perturbation of a stationary profile in the direction of the unstable eigenvector. Shown are space-time plots for $ u $ and $ v $ (top row), shape of the most unstable eigenfunction (middle row), and snapshots of time evolution for $ u $ (red), $ v $ (blue), and $ u+v $ (black). Parameter values are $ \gamma = 1/64 $, $ \rho_\infty = 1.3115 $ (left) and $ \rho_\infty = 18.3805 $ (right)

Figure 12.  Instability of cluster boundaries (left), showing space-time plots for $ u $ and $ v $, top row, profile of the leading eigenfunction, and time snapshots, for $ \gamma = 1/13.81 $. Stable clusters ($ \gamma = 1/64 $, $ \rho = 1.3072 $) and cluster boundaries ($ \gamma = 1/21.41 $) on the right, with leading eigenfunction corresponding to mass change (cluster) and translation (cluster boundary), respectively