On the influence of cross-diffusion in pattern formation

Abstract

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 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.

Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

“…Here we should refer to a recent numerical result by Breden, Kuehn and Soresina [1], which numerically exhibits the bifurcation diagram of solutions of (1.1) with…”

“…and (5.22) that µ j (d) > 0 for any j ∈ N if d > d (1) . Then, for any d > d (1) , the number σ(d) of negative eigenvalues of I − L(d) is equal to that of negatives of Assume that C < A < B in addition to (3.6). We shall show that (3.2) admits at least one nonconstant solution when d ∈ (d (j+1) , d (j) ) ∩ [ε, ∞) and j is odd.…”

“…An example of profiles of h(u, τ ) in case B < A < C and D(a i , b i , c i , γ) > 0 is shown in Figure 2. In the same setting as [1,Figure 11] for the case C < A < C and D(a i , b i , c i , γ) > 0, profiles of h(u, τ ) are shown in Figure 3. In view of Figure 3, one can see that T is not connected in this case.…”

“…We first show Γ + 1 ⊂ S + 1 , that is, what Γ + 1 does not connects with any other Γ ± j . If not, there exist a solution (d,û,τ ) ( = (d (1) , u * , τ * ) ) of (6.1) and a sequence { (d 2,n , u n , τ n ) } ⊂ Γ + 1 such that lim n→∞ (d 2,n , u n , τ n ) = (d,û,τ ) in R × X andû(x) has a degenerate critical point,…”

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

“…on a disk domain = {(x 1 , x 2 ) : x = x 2 1 + x 2 2 < R}, where = ∂ 2 x 1 + ∂ 2 x 2 is the Laplacian, with Neumann BCs ∂ n u = ∂ n u = 0, where n is the outer normal at x = R. Equations of type (6) (with various nonlinearities, for instance f (u) = νu 2 − u 3 instead of f (u) = νu 3 − u 5 in (6)) are prototypical examples for finite wave number (Turing) pattern formation, and are thus also studied as model problems in [68], over 1D, 2D and 3D boxes (intervals, rectangles and cuboids). The new feature in [81] is the disk domain.…”