The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains

Diffusive and nondiffusive population models

“…where u ∈ R m is a vector-valued function, [76] proved that, in a convex domain, the system (5) with homogeneous Neumann boundary conditions has no stable nonconstant steady state if ∂g i /∂u j > 0 for all i = j (cooperation-diffusion system); the same conclusion holds for m = 2 if ∂g i /∂u j < 0 for i = j (competition-diffusion system). On the other hand, in nonconvex domains, stable nonconstant steady states may exist, see the works [66,67,103] for certain dumbbell-shaped domains.…”

On a Smale theorem and nonhomogeneous equilibria in cooperative systems

“…Note also the work on realization of ODEs in reaction diffusion equations, which among many other results implies the existence of a continuum of equilibria for (3) in the Dirichlet case for Ω ⊆ R m , m > 1 [10]. Regarding item (3) from Theorem 2, in a paper by Kishimoto and Weinberger [9] it is proved that for convex Ω and given a system (3) such that ∂f i /∂u j > 0 for i = j, any nonhomogeneous equilibrium must be linearly unstable. Moreover, in a recent result by Smith, Hirsch, and the author, it is shown that under the same hypotheses the generic bounded solution of (3) converges towards a homogeneous equilibrium [4].…”

A reaction-diffusion system and its shadow system describing harmful algal blooms

“…When ξ is assumed to be a known constant, it is well known that non-constant equilibrium solutions (u ∞ (x; ξ), v ∞ (x; ξ)) of the first two equations of (1.6) and (1.7) are unstable, if Ω is convex ( [12]). However, when ξ is an unknown variable, the situation is drastically changed, that is, there occur stable non-constant equilibrium solutions (u ∞ (x; ξ), v ∞ (x; ξ), ξ), of (1.6) and (1.7) as shown in Figure 2.…”