Journal of the
Korean Mathematical Society JKMS

This paper is concerned with a reaction-diffusion logistic model. In \cite{L06}, Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni \cite{HN16} raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li \cite{BHL16} proved that the optimal upper bound is $3$. Recently, Inoue and Kuto \cite{IK20} showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of \cite{IK20} to an arbitrary smooth bounded domain in $\mathbb{R}^n, n \geq 2$. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of \cite{IK20} but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.

Keywords: Logistic model, spatial heterogeneity, total biomass

MSC numbers: 35B09, 35B30, 35Q92

Supported by: This work was supported by 2020 Long-Term KAIST Undergraduate Research Program under the guidance of Professor Jaeyoung Byeon. The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST) (No. NRF-2019R1A5A1028324).