Protection Zones in Periodic-Parabolic Problems

Abstract

Abstract This paper characterizes whether or not Σ ∞ ≡ lim λ ↑ ∞ ⁡ σ ⁢ [ 𝒫 + λ ⁢ m ⁢ ( x , t ) , 𝔅 , Q T ] \Sigma_{\infty}\equiv\lim_{\lambda\uparrow\infty}\sigma[\mathcal{P}+\lambda m(% x,t),\mathfrak{B},Q_{T}] is finite, where m ⪈ 0 {m\gneq 0} is T-periodic and σ ⁢ [ 𝒫 + λ ⁢ m ⁢ ( x , t ) , 𝔅 , Q T ] {\sigma[\mathcal{P}+\lambda m(x,t),\mathfrak{B},Q_{T}]} stands for the principal eigenvalue of the parabolic operator 𝒫 + λ ⁢ m ⁢ ( x , t ) {\mathcal{P}+\lambda m(x,t)} in Q T ≡ Ω × [ 0 , T ] {Q_{T}\equiv\Omega\times[0,T]} subject to a general boundary operator of mixed type, 𝔅 {\mathfrak{B}} , on ∂ ⁡ Ω × [ 0 , T ] {\partial\Omega\times[0,T]} . Then this result is applied to discuss the nature of the territorial refuges in periodic competitive environments.