Size dependence of biomass spectra and abundance spectra: the optimal distributions

Based on the hypothesis of self-optimization, we derive four models of biomass spectra and abundance spectra in communities with size-dependent metabolic rates. In Models 1 and 2, the maximum diversity of population abundance in different size classes subject to the constraints of constant mean body mass and constant mean respiration rate is assumed to be the strategy for ecosystems to organize their size structure. In Models 3 and 4, the organizing strategy is defined as the maximum diversity of biomass in different size classes without constraints on mean body mass and subject to the constant mean specific respiration rate of all individuals, i.e. the average specific respiration rate over all individuals of a community or group, which characterizes the mean rate of energy consumption in a community. Models 1 and 2 generate peaked distributions of biomass spectral density whereas Model 3 generates a flat distribution. In Model 4, the distributions of biomass spectral density and of abundance spectral density depend on the Lagrangian multipler (λ2). When λ2 tends to zero or equals zero, the distributions of biomass spectral density and of abundance spectral density correspond to those from Model 3. When λ2 has a large negative value, the biomass spectrum is similar to the empirical flat biomass spectrum organized in logarithmic size intervals. When λ2>0, the biomass spectral density increases with body mass and the distribution of abundance spectral density is an unimodal curve.