on the ratio of biomass to total carrying capacity in high dimensions
J. Korean Math. Soc. Published online June 3, 2021
Junyoung Heo and Yeonho Kim
Abstract : This paper is concerned with a reaction-diffusion logistic model. In , 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  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  proved that the optimal upper bound is 3. Recently, Inoue and Kuto  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  to an arbitrary smooth bounded domain in Rn, n ≥ 2. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of  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