An instability criterion for activator–inhibitor systems in a two-dimensional ball

Let B be a two-dimensional ball with radius R  . Let (u(x,y),ξ)(u(x,y),ξ) be a nonconstant steady state of the shadow systemut=DuΔu+f(u,ξ)inB×R+andτξt=1|B|∫∫Bg(u,ξ)dxdyinR+,∂νu=0on∂B×R+, where f and g   satisfy the following: fξ(u,ξ)<0fξ(u,ξ)<0, gξ(u,ξ)<0gξ(u,ξ)<0 and there is a function k(ξ)k(ξ) such that gu(u,ξ)=k(ξ)fξ(u,ξ)gu(u,ξ)=k(ξ)fξ(u,ξ). This system includes a special case of the Gierer–Meinhardt system and the FitzHugh–Nagumo system. We show that if Z[Uθ(⋅)]⩾3Z[Uθ(⋅)]⩾3, then (u,ξ)(u,ξ) is unstable for all τ>0τ>0, where U(θ):=u(Rcosθ,Rsinθ)U(θ):=u(Rcosθ,Rsinθ) and Z[w(⋅)]Z[w(⋅)] denotes the cardinal number of the zero level set of w(⋅)∈C0(R/2πZ)w(⋅)∈C0(R/2πZ). The contrapositive of this result is the following: if (u,ξ)(u,ξ) is stable for some τ>0τ>0, then Z[Uθ(⋅)]=2Z[Uθ(⋅)]=2. In the proof of these results, we use a strong continuation property of partial differential operators of second order on the boundary of the domain.