We analyze a simplistic model for run-and-tumble dynamics, motivated by observations of complex spatio-temporal patterns in colonies of myxobacteria. In our model, agents run with fixed speed either left or right, and agents turn with a density-dependent nonlinear turning rate, in addition to diffusive Brownian motion. We show how a very simple nonlinearity in the turning rate can mediate the formation of self-organized stationary clusters and fronts. Phenomenologically, we demonstrate the formation of barriers, where high concentrations of agents at the boundary of a cluster, moving towards the center of a cluster, prevent the agents caught in the cluster from escaping. Mathematically, we analyze stationary solutions in a four-dimensional ODE with a conserved quantity and a reversibility symmetry, using a combination of bifurcation methods, geometric arguments, and numerical continuation. We also present numerical results on the temporal stability of the solutions found here.
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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 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 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 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 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
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Homoclinic trajectories in phase space (black), their projections onto the
The Hamiltonian potential,
Left column, top to bottom: Potentials
Maxima and minima of homoclinic orbits as a function of the background state (left). Plots of density profiles
Plot of the boundaries of
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
Maxima of clusters and minima of gaps in the continuation, plotted against the background state, for sample values of
Heteroclinic profiles plotted as
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
Spectra of clusters and gaps as functions of the background
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
Instability of cluster boundaries (left), showing space-time plots for