ON THE RATIO OF BIOMASS TO TOTAL CARRYING CAPACITY IN HIGH DIMENSIONS

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 subsolution 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.