Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics

Figure 1.  Fixing $ \beta^{(1)} = \beta^{(2)} $, $ p = 2 $ and $ \mathfrak{m}^{(i)}_{\iota\kappa} = \rho^{(i)}_\iota $, we illustrate the energy landscapes (heat map), stationary states (light blue circles), their corresponding energy level sets (light blue lines), and dynamics (quivers) for different cross-interaction, while staying in the regime $ D^{(11)}, D^{(22)}<0 $, $ D^{(11)}D^{(22)}>\left(D^{(12)}\right)^2 $

Figure 2.  Fixing $ \beta^{(1)} = \beta^{(2)} $, $ p = 2 $ and $ \mathfrak{m}^{(i)}_{\iota\kappa} = \rho^{(i)}_\iota $, we display the energy landscapes (heat map), stationary states (light blue circles), their corresponding energy level sets (light blue lines), and dynamics (quivers) in the different regimes $ D^{(11)}D^{(22)} \lesseqqgtr \left(D^{(12)}\right)^2 $ for repulsive (top row) and attractive (bottom row) self-interactions

Figure 3.  Fixing $ \beta^{(1)} = \beta^{(2)} $, $ p = 2 $ and $ \mathfrak{m}^{(i)}_{\iota\kappa} = \rho^{(i)}_\iota $, we illustrate the energy landscapes (heat map), stationary states (light blue circles), their corresponding energy level sets (light blue lines), and dynamics (quivers), for self-interactions of opposite signs and varying strength, while the cross-interactions are fixed and repulsive

Figure 4.  Fixing $ p = 2 $ and $ \mathfrak{m}^{(i)}_{\iota\kappa} = \rho^{(i)}_\iota $, $ D^{(11)} = D^{(22)} = 1 $ and $ D^{(12)} = 0 $, we display the energy landscapes (heat map), stationary states (light blue circles), and dynamics (quivers) for varying $ \beta^{(1)} $ and $ \beta^{(2)} $

Figure 5.  Fixing $ \mathfrak{m}^{(i)}_{\iota\kappa} = \rho^{(i)}_\iota $, $ D^{(11)} = D^{(22)} = 1 $, $ D^{(12)} = 0 $ and $ \beta^{(1)} = \beta^{(2)} $, we display the energy landscapes (heat map), stationary states (light blue circles), and dynamics (quivers) for varying $ p $

Figure 6.  We display the dynamics of the non-symmetric ($ \varepsilon \in (0, 1) $) four-point space with the initial mass of $ \rho^{(2)} $ concentrated on $ x_4 $ (green, bottom row) and the initial mass of $ \rho^{(1)} $ concentrated on $ x_1 $ (red, top row). Only if the cross-repulsion strength is strong enough, $ \alpha>1 $, can the second species drive the first one out of the node it is inhabiting. Against any intuition based on the associated particle system, this is not the case if the repelling cross-interaction is weak

Figure 7.  In the symmetric setting, two species that are initially concentrated on neighbouring vertices of the four-point space will drive out one another in the case of sufficiently strong cross-repulsion, $ \alpha>1 $. Only if the cross-repulsion strength is strong enough, $ \alpha>1 $, can the second species drive the first one out of the node it is inhabiting and vice versa. This phenomenon is quite counter-intuitive as both species are self-attractive. Nonetheless, the mass of both species is evenly split. Against any intuition based on the associated particle system, this is not the case if the repelling cross-interaction is weak

Figure 8.  In the situation with symmetric nodes and non-symmetric cross-repulsion (i.e., $ \beta \ne 1 $), one of the two species which are initially concentrated on neighboring edges of the four-point space will drive out the other in the case of sufficiently strong cross-repulsion, $ \alpha>1 $. Again, against particle system intuition, this is not the case if the repelling cross-interaction is weak

Figure 9.  Short-range repulsive, long-range attractive self-interactions and repulsive interactions with a sensing radius lead to segregation (right panels) of the initially randomly distributed species (left panels). The two species are color-coded in red and green, respectively. In the top row, only nearest neighbors are connected, while the graph is fully connected in the bottom row. A higher connectivity leads to a faster convergence to the steady state

Figure 10.  We display the evolution of both species (red and green, respectively) for attractive self- and cross-interactions, the mobility $ m(r, s) = r $ and $ p\in\{1.65, 2, 5\} $. In all cases, the simulation starts from the same random initial data (left panel). For all $ p $ we observe aggregation of both species to a single vertex as time progresses. Comparing different $ p $, we see that an increase of $ p $ slows down aggregation speed

Figure 11.  Starting from random initial data (left panel), we display the evolution of both species (red and green, respectively) for attractive self- and cross-interactions, the mobility $ m(r, s) = r(1-s) $, $ \mu_\iota = 1/20 $ for all $ \iota\in\{1, \ldots, 100\} $ and $ p = 2 $. We observe that the aggregation is confined by the upper bound $ \rho^{(i)}_\iota \le \mu_\iota $ for $ i = 1, 2 $, which results in the formation of a patch of populated vertices

Figure 12.  We display the evolution of both species (red and green, respectively) for attractive self- and cross-interactions, the mobility $ m(r, s) = r(1-s) $, $ \mu_\iota = 1/8 $ for all $ \iota\in\{1, \ldots, 100\} $ and $ p = 2 $ on a random graph. In the bottom row, the graph is fully connected while, in the top row, vertices are connected if and only if their distance from each other is at most $ 25\% $ of the side length of the populated area. Starting from the same random initial data (left panels), in both case we observe the convergence of the dynamics to a stationary state with $ 8 $ populated vertices, which are close to each other. However, on the fully connected graph, this stationary state is reached much quicker than on the partially connected graph. Furthermore, the stationary states differ from each other