One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.
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Figure 1. Asymptotic profiles of the single interior spike (top left) and double-boundary spike (top right) with the coupled steady chemical concentrations (in the bottom) given by (1.5) and (1.6) with $ M = 2 $ and $ \chi = 20 $. We shall prove that both the single interior spike and the double-boundary spike are unstable
Figure 2. Left: Illustration of the solution loop through a phase plane; Right: a solution with four similar profiles on the right. The maximum of $ v(x) $ are achieved at $ A $ and $ C $, and then minimum at $ B $ and $ D $. Each closed loop on the phase plane leads to a solution with a complete period, and it can be extended repeatedly. However, each loop intersects with the $ v $-axis only twice and this suggests that the solution $ v(x) $ must be similar-profiled
Figure 3. Leading profiles of the eigenpair $ (\phi^d,\psi^d) $ of (2.2) over $ (0,\pi) $ with $ \chi = 5,10,20 $ and 100. Each function is normalized such that $ \Vert \phi^d\Vert_{L^2} = \Vert \psi^d\Vert_{L^2} = 1 $. These eigenfunctions are odd about the middle point $ x = \frac{L}{2} $ and they drive the insability of the double-boundary spike $ (u^d,v^d) $ in (1.5)
Figure 4. Leading profiles of the eigenpair $ (\phi^i,\psi^i) $ of (2.16) over $ (0,\pi) $ with $ \chi = 5,10,20 $ and 100. Similar as in Figure 3, we normalize each eigenfunction such that $ \Vert \phi^i\Vert_{L^2} = \Vert \psi^i\Vert_{L^2} = 1 $. These eigenfunctions are odd about the middle point $ x = \frac{L}{2} $ and they drive the instability of the single interior spike $ (u^i,v^i) $ in (1.6)
Figure 5. Staircased hierarchy of free energies of the constant solution (3.3) and the multi-spikes (3.5). Theorem 1.3 finds that when the chemotaxis rate is large, the single boundary spike is the least energy solution among all the steady states, whereas the constant solution has the largest energy; moreover, the energy increases as the number of spikes. In light of the energy dissipation (1.10), the hierarchy indicates that the single boundary spike is the most stable and the constant solution is most unstable when the chemotaxis rate is large. Accordingly, the spatial-temporal dynamics of (1.1) are dominated by the single boundary spike but for symmetric initial data. These findings will be demonstrated and verified in our coming simulations
Figure 6. This figure demonstrates the instability of the double-boundary spike $ (u^d,v^d) $. Here we choose $ \kappa = 20 $ and the initial data $ (u_0,v_0) = (u^d,v^d) + (0.2e^{-2x^2},0) $ to be small perturbations from the double-boundary spike, with the initial cell density $ u_0 $ slightly tilted to the left end. Top: Very quickly the cellular density develops the double-boundary spike tilted to the left. This asymmetric profile preserves for an extremely long time to $ t\approx 280 $, when a transition occurs and the small spike on the right disappears within a very short period. Similar dynamics are observed in the chemical concentration $ v(x,t) $ on the right. Bottom: The contour plot of cellular density on the left and the decay of free energy (1.9) along the solution trajectory on the right. In particular, one observes that the energy decays extremely slowly for a very long time and then a phase transition occurs at time $ t\approx 280 $ as described earlier
Figure 7. This figure demonstrates the instability of the interior spike $ (u^i,v^i) $. Here we choose $ \kappa = 20 $ and the initial data $ (u_0,v_0) = (u^i,v^i) + (0.2e^{-2x^2},0) $ to be small perturbations from the single interior spike, with the initial cell density $ u_0 $ slightly tilted to the left end. Top: Initially the cell density spike decays with a linear rate, and it stabilizes to a single interior spike at time $ t\approx 1 $. Similar to the double-boundary spike in Figure 6, this single interior spike is asymmetric and slightly tilted to the left. Very slowly it shifts to the left end until time $ t\approx 11 $ when a transition occurs quickly and this interior spike eventually stabilizes at the left endpoint. Snapshots capture similar dynamics observed in the chemical concentration $ v(x,t) $ on the top right. Bottom: The energy evolution captures the meta-stability of the interior spike and the occurrence of a phase transition at time $ t\approx11 $. Needless to say, the convergence to the single boundary spike is much faster than that of the double-boundary spike in Figure 6
Figure 8. Exponential stability of the double-boundary spike $ u^d $ and the interior spike $ u^i $ in the symmetric class. Here we choose the initial data to be symmetric perturbations of these spikes in both simulations. In strong contrast to the dynamics in Figure 6 and Figure 7 that take exponentially long to converge to an equilibrium, we observe exponential decay to the steady states $ u^d $ and $ u^i $ in the plots, at least for $ t\in(0.5,4.5) $, and the exponential convergence rates are $ \lambda^d\approx -4.8786 $ and $ \lambda^i = \approx -4.8771 $ for the double-boundary spike and the single interior spike
Figure 9. Left: Dynamics and free energy evolution out of $ (\mathbb U^i_4,\mathbb V^i_4) $ with an odd perturbation tilted to the right end. The two interior spikes sense the chemical gradient, attract and move towards each other, and merge into a single interior spike at time $ t\approx 12 $. This interior spike is tilted and shifts to the right end very slowly until it stabilizes on the boundary at time $ t\approx 45 $ through a phase transition. Two phase transitions are observed in the energy evolution at the bottom. Right: Dynamics and free energy evolution out of $ (\mathbb U^i_4,\mathbb V^i_4) $ with an even perturbation. They attract and merge each other very quickly, stay symmetric, and finally stabilize into the single interior spike
Figure 10. Instability of multi-spikes given in (1.7) and (1.8) with $ n = 5,6 $ and 7 presented on the left, the middle, and the right. Top. Local dynamics of (1.1) out of these multi-spikes are presented here and they demonstrate the dissipation and merging of these multi-spikes which occur in a relatively short period. Middle. Global dynamics out of these multi-spikes readily demonstrate their instability, as well as the stability of the single and double-boundary spike. Bottom. Energy evolution here showcases several phase transitions of (1.1). They suggest the rich spatial-temporal dynamics of this relatively simple system
Figure 11. Spatial-temporal dynamics of (1.1) out of initial data randomly perturbed from $ (\bar u,\bar v) = (\frac{2}{\pi},\frac{2}{\pi}) $ with $ \chi = 20 $. In each column, we report the local, global dynamics of (1.1), and the evolution of the free energy. These results suggest that the single boundary spike the global attractor in these settings. The center of initial cellular density from the left to the right is $ \bar x_1\approx 1.5630 $. $ \bar x_2\approx 1.5531 $, $ \bar x_3\approx 1.5794 $, respectively. These simulations, together with others we have conducted but not reported here, tend to suggest the asymmetry of the initial data is preserved: initial data tilted to the left will converge to the decreasing single boundary, and initial data tilted to the right will converge to the increasing single boundary
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Asymptotic profiles of the single interior spike (top left) and double-boundary spike (top right) with the coupled steady chemical concentrations (in the bottom) given by (1.5) and (1.6) with
Left: Illustration of the solution loop through a phase plane; Right: a solution with four similar profiles on the right. The maximum of
Leading profiles of the eigenpair
Leading profiles of the eigenpair
Staircased hierarchy of free energies of the constant solution (3.3) and the multi-spikes (3.5). Theorem 1.3 finds that when the chemotaxis rate is large, the single boundary spike is the least energy solution among all the steady states, whereas the constant solution has the largest energy; moreover, the energy increases as the number of spikes. In light of the energy dissipation (1.10), the hierarchy indicates that the single boundary spike is the most stable and the constant solution is most unstable when the chemotaxis rate is large. Accordingly, the spatial-temporal dynamics of (1.1) are dominated by the single boundary spike but for symmetric initial data. These findings will be demonstrated and verified in our coming simulations
This figure demonstrates the instability of the double-boundary spike
This figure demonstrates the instability of the interior spike
Exponential stability of the double-boundary spike
Left: Dynamics and free energy evolution out of
Instability of multi-spikes given in (1.7) and (1.8) with
Spatial-temporal dynamics of (1.1) out of initial data randomly perturbed from