On the influence of cross-diffusion in pattern formation

Figure 1.  Sign of the quantities $ \alpha $ and $ \beta $ in the weak competition regime (2), depending on the value of $ r_1/r_2 $

Figure 2.  Sign of the quantities $ \alpha $ and $ \beta $ in the strong competition casek__ge (3), depending on the value of $ r_1/r_2 $

Figure 3.  Bifurcation diagrams represented with different quantities in the weak competition case (first parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $). (A) "Usual" bifurcation diagram with respect to $ v(0) $. (B) Bifurcation diagram with respect to $ ||u||_{L^2} $

Figure 4.  Different types of stable solutions on the blue and red branches of the bifurcation diagram 3. The densities of $ u $ and $ v $ on the domain are denoted with black and blue lines respectively

Figure 5.  Bifurcation diagrams for different values of the cross-diffusion coefficient $ d_{21} $ in the weak competition case (first parameter set in Table 1, with $ d_{12} = 3 $). Notice that the range of values of $ d $ for which non-trivial solutions exist is getting smaller and smaller when $ d_{21} $ increases (especially compared to Figure 3b)

Figure 6.  Different types of stable solutions coexisting with the homogeneous one. Species $ u $ and $ v $ on the domain are denoted with black and blue lines respectively. Figures (A) and (C) refer to $ d_{21} = 0.045 $ and correspond to solutions on the pastel blue and red branches of the bifurcation diagram in Figure 5a. Figures (B) and (D) refer to $ d_{21} = 0.055 $ and correspond to solutions on the pastel blue and pastel red branches of the bifurcation diagram in Figure 5b

Figure 7.  Bifurcation diagrams for different values of the cross-diffusion coefficient $ d_{21} $ in the weak competition case (first parameter set in Table 1, with $ d_{12} = 3 $). We start in Figure 7a with a zoom in of Figure 3b, and then slowly increase $ d_{21} $

Figure 8.  Solutions belonging to different branches when $ d_{12} = 1000 $. The species $ u $ and $ v $ are denoted with black and blue lines respectively, while the red line corresponds to $ uv $. As in the bifurcation diagrams, thin lines indicate unstable solutions, while thicker lines corresponds to stable ones. The red dotted line represents the product $ uv $

Figure 9.  Bifurcation diagrams for different values of the cross-diffusion parameter $ d_{21} $ in the strong competition case (second parameter set in Table 1, with $ d_{12} = 3 $)

Figure 10.  Evolution of the first bifurcation branch for increasing values of $ d_{21}\in [0,1000] $ (darker the blue, greater the value), corresponding to the second parameter set in Table 1, with $ d_{12} = 3 $

Figure 11.  Bifurcation diagram and some solutions (third parameter set in Table 1 and $ d_{12}\; = \; d_{21} = 100 $). (A) Bifurcation diagram with respect to the parameter $ d $. Yellow points indicate the positions on the branches at which solutions are shown. (B)–(D) Solutions $ u(x),\;v(x) $ for different values of $ d $ (black lines correspond to species $ u $, while blue lines to species $ v $)

Figure 12.  Evolution of the first bifurcation branch for increasing values of $ d_{12}\in [0,3] $ (darker the blue, greater the value), corresponding to the fourth parameter set in Table 1, with $ d_{21} = 0 $

Figure 13.  Bifurcation diagrams with bifurcation parameter $ r_1 $. (A) Weak competition case with $ d = 0.005 $ and the other parameter values as in the first parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $. (B) Strong competition case with $ d = 0.05 $ and the other parameter values as in the second parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $

Figure 14.  Stable non-homogeneous solutions appearing beyond the usual weak or strong competitions regimes (species $ u $ and $ v $ are denoted with black and blue lines respectively). Weak competition case ($ d = 0.005 $, the other parameter values as in the first parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $): (A), (C) stable solutions with $ r_1 = 7.5 $, (E) stable solution with $ r_1 = 15 $. Strong competition case ($ d = 0.05 $, the other parameter values as in the second parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $): (B), (D) stable solutions with $ r_1 = 10 $, (F) stable solution with $ r_1 = 18 $. The colors of the branches refer to Figure 13

Figure 15.  Bifurcation diagrams with bifurcation parameter $ d $ obtained with the first parameter set in Table 1 (weak competition case), with $ d_{12} = 3 $, $ d_{21} = 0 $, $ d_{11} = 0 $ and different values of the self-diffusion coefficient $ d_{22} $