Multiple patterns formation for an aggregation/diffusion predator-prey system

Figure 1.  A possible example of a stationary solution to (3) with $ N_\rho = N_\eta = 3 $ is plotted as described in Definition 1.1

Figure 2.  Example of mixed stationary state. Note that by symmetry $ L_\rho=-R_\rho $ and $ L_\eta=-R_\eta $

Figure 3.  An example of a separated stationary state

Figure 4.  In this figure, a mixed steady state is plotted by using initial data given by (70), $ \alpha = 0.1 $, $ \theta = 0.4 $. Number of particles are chosen equal to number of cells in the finite volume method, which is $ N = 71 $

Figure 5.  A separated steady state is presented in this figure. Initial data are given by (71). The parameters are $ \alpha = 0.2 $ and $ \theta = 0.4 $ with $ N = 91 $

Figure 6.  This figure shows how from the initial densities $ \rho_0,\eta_0 $ and $ \theta $, as in Figure 4, a transition between mixed and a sort of separated steady state appears by choosing the value of $ \alpha = 6 $. This large value of $ \alpha $ suggests an unstable behavior in the profile, see Remark 3.1

Figure 7.  A steady state of four bumps is showed in this figure starting from initial data as in (72) with $ \alpha = 0.05 $ and $ \theta = 0.3 $. The number of particles $ N = 181 $, which is the same as number of cells

Figure 8.  Starting from initial data as in (73) with $ \alpha = 1 $ and $ \theta = 0.3 $, we get a five bumps steady state

Figure 9.  This last figure shows a possible existence of traveling waves by choosing initial data as in (74), $ \alpha = 1 $ and $ \theta = 0.2 $. The first two plots are performed by particles method, while the third one is done by finite volume method. Here we fix $ N = 101 $