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 [17], 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 [10] 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 [1] proved that the optimal upper bound is 3. Recently, Inoue and Kuto [13] 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 [13] 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 [13] 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.
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