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Article Contents

# On the influence of cross-diffusion in pattern formation

• * Corresponding author: Maxime Breden

MB and CK have been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CS has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie grant agreement No. 754462. Support by INdAM-GNFM is gratefully acknowledged by CS

• In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software $\mathtt{pde2path}$. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

Mathematics Subject Classification: Primary: 35Q92, 92D25; Secondary: 35K59, 65P30.

 Citation:

• 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 $d_{22}$

Table 1.  The parameter sets used in the numerical simulations presented in this section

 $r_1$ $r_2$ $a_1$ $a_2$ $b_1$ $b_2$ regime α β 5 2 3 3 1 1 weak competition + - [5,24,27] 2 5 1 1 0.5 3 strong competition + 0 [27] 15/2 16/7 4 2 6 1 weak competition - + 5 5 2 3 5 4 strong competition + +

Table 2.  Convergence of the first three bifurcation points, numerically detected, to the predicted limited values $\alpha/(v_*\lambda_k)$ when $d_{12}$ increases ($d_{21} = 0$)

 $d_{12}$ $d^1_{ \text{bif}}$ $d^2_{ \text{bif}}$ $d^3_{ \text{bif}}$ 3 0.0328 0.0205 0.0113 10 0.0762 0.0273 0.0131 100 0.1190 0.0311 0.0139 1000 0.1258 0.0315 0.0140 $d^k_{ \text{bif},\infty}$ 0.1267 0.0317 0.0141

Figures(15)

Tables(2)