A general mathematical food web model that predicts a stable green world in the terrestrial ecosystem

Background/Question/Methods

Terrestrial world is generally green and only a small part (< 10%) of plant matter is consumed by herbivore annually, which means the density of herbivore is consistently low and stable. However, the mechanism that causes this pattern has been unclear because of the lack of the food web model that can predict the absolute biomass density of herbivores accompanied by realistic physical units. Therefore, I attempted to establish a simple mathematical food web model that can predict the biomass density of herbivore (primary consumer) h (kg protein/m³) and carnivore (secondary consumer) c (kg protein/m³) from ecological factors such as nutritive values of plant (plant material, producer) n_p (kg protein/m³), herbivore n_h (kg protein/m³), and carnivore n_c (kg protein/m³), searching efficiency (volume) of carnivore S (m³/m³day=/day), eating efficiency (speed) of herbivore e_h (m³/m³day=/day) and carnivore e_c (/day), daily respiratory decrease of herbivore and carnivore biomass, d_h (kg/kg day=/day) and d_c (/day), absorption efficiency of herbivore and carnivore α_h  (ratio) and α_c (ratio), and preying probability of carnivore on herbivore or carnivore, P_hc (ratio) and P_cc(ratio).

Results/Conclusions

A new mathematical food web model was established that predicted a stable convergent equilibrium with biomass densities of herbivore h=n_c{(1-α_c)P_cc(α_he_hn_p–n_hd_h)+2d_cP_hc}/2α_cP_hc²n_hS (kg/m³) and carnivore c=n_c(α_he_hn_p-n_hd_h)/P_hcn_hS (kg/m³). The predicted h is low enough to keep the world green if 1. n_p<n_h,n_c, 2. S>>e_h, and 3. P_hc>P_cc >0. All these conditions are realized in above-ground terrestrial ecosystems, but not in below-ground and aquatic ecosystems, which explains why terrestrial ecosystems (above-ground) are plant-rich while belowground and aquatic ecosystems are animal-rich. The h and c calculated from the model showed surprisingly good agreement with those obtained from empirical observations in forest ecosystems, where both h and c have the order of magnitude of ca. 100 mg (fresh biomass/m² forest), and also in savannah ecosystems. The model predicts that nutritive values of plant n_p, digestibility of plant by herbivore a_h, and eating speed of herbivore e_h positively correlate with h and intensity of herbivory, which theoretically justifies outdoor defensive functions for plant of anti-nutritive or quantitative defenses such as tannins and protease inhibitors, and predicts that c and c/h positively correlate with RGR of herbivore G_h=:(α_he_hn_p-n_hd_h)/n_h. The model further predicts positive correlation between nutrients of litters and their decomposition speeds, conditions of pest outbreaks, etc. Ref.arXiv:1412.5773.