Some counterexamples concerning a model of eradication of invasive species

Abstract

In this note we study the regularity properties of the \emph{Effort Function E(β)}. This can be defined as the minimal L¹-norm of the control α̃(x) ≥ 0 so that the PDE U_t = ΔU + f(U) − α̃(t,x) U, U ∈ [0,1], % α̃≥ 0, admits a traveling profile $U(x - βt)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control α̃(t,x) describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of f(U) and (consequently) on the effort function E(β), the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that E(β) is in general only C¹.

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BibTeX

@misc{bianchini2026some,
  title = {{Some counterexamples concerning a model of eradication of invasive species}},
  author = {Stefano Bianchini and Chiara Trifone},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11604},
  eprint = {2610.11604},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11604},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Some counterexamples concerning a model of eradication of invasive species
AU  - Stefano Bianchini
AU  - Chiara Trifone
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11604
UR  - https://arxiv.org/abs/2610.11604
AB  - In this note we study the regularity properties of the \emph{Effort Function E(β)}. This can be defined as the minimal L^1-norm of the control α̃(x) ≥ 0 so that the PDE U_t = ΔU + f(U) − α̃(t,x) U, U ∈ [0,1], % α̃≥ 0, admits a traveling profile $U(x - βt)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control α̃(t,x) describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of f(U) and (consequently) on the effort function E(β), the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that E(β) is in general only C^1.
ER  -

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