TY - JOUR

T1 - On the unboundedness of the ratio of species and resources for the diffusive logistic equation

AU - Inoue, Jumpei

AU - Kuto, Kousuke

N1 - Funding Information:
2020 Mathematics Subject Classification. Primary: 35Q92, 35B30; Secondary: 35B09, 35B40. Key words and phrases. Diffusive logistic equation, elliptic equations, the sub-super solution method, radial solutions, mathematical ecology. The second author is supported by JSPS KAKENHI Grant-in-Aid Grant Number 19K03581. ∗ Corresponding author: Jumpei Inoue.
Publisher Copyright:
© 2021 American Institute of Mathematical Sciences. All rights reserved.

PY - 2021/5

Y1 - 2021/5

N2 - Concerning a class of diffusive logistic equations, Ni [1, Abstract] proposed an optimization problem to consider the supremum of the ratio of the L1 norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is 3 in the one-dimensional case. In [1], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an L1 unbounded sequence of sub-solutions.

AB - Concerning a class of diffusive logistic equations, Ni [1, Abstract] proposed an optimization problem to consider the supremum of the ratio of the L1 norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is 3 in the one-dimensional case. In [1], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an L1 unbounded sequence of sub-solutions.

KW - Diffusive logistic equation

KW - Elliptic equations

KW - Mathematical ecology

KW - Radial solutions

KW - The sub-super solution method

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U2 - 10.3934/dcdsb.2020186

DO - 10.3934/dcdsb.2020186

M3 - Article

AN - SCOPUS:85104596283

SN - 1531-3492

VL - 26

SP - 2441

EP - 2450

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 5

ER -