Effect of changing model parameters on emergent competition and the relative strength of top-down versus bottom-up control.

The last column refers to related hypothesis or observations summarized in Table 2. The symbols indicate increase ↑, or decrease ↓. The table is also valid with symbols flipped (↑ replaced by ↓, and vice versa). See S2 and S3 Figs for corresponding numerical simulations.</p

Top-down and bottom-up control in toy model.

In this section, we analyze the special case below where there is only 1 species with intra-species competition on each level. (PDF)</p

Fig 4 -

(a) as a function of energy influx to primary producers k and death rate of carnivores u for four different ratios of regional species pool, r1, r2 ∈ {0.4, 1.2}, indicated by the pie chart, with σc = 0.5 and σd = 0.5, and (b) as a function of the species trait diversity, σd and σc, for four different ratios of regional species pools with environmental parameters k = 4.2, u = 2.6 and k = 4.2, u = 1.</p

System size affects cavity solution convergence.

(a)Histograms of the steady state reached by dynamics of a system with MX = 50 species of carnivores, MN = 56 herbivores and MR = 62 plants, with k = 4, m = 1, u = 1, σc = σd = 0.5, μc = μd = 1, ηX = 1, ηN = 1, σk = σm = σu = 0.1, and the distribution predicted by our cavity solution. Note that a black dot correspond the finite extinction probability (instead of probability density) predicted by cavity solution, while a black dash correspond to the probability density(b) The average square deviation of the single system statistics from cavity solution as a function of MX while keeping fixed ratios r1 = r2 = 0.9, averaged from 200 sample systems. (PDF)</p

Existing observations and hypothesis in trophic ecology that relates to the model behavior in Fig 4 and S2 and S3 Figs.

The first column refers to the related model behavior summarized in Table 1.</p

Plot of effective competition coefficients, (a) , (b) and (c) under the same conditions as Fig 4.

Plot of effective competition coefficients, (a) , (b) and (c) under the same conditions as Fig 4.</p

Fig 1 -

(a) Schematic of food web interaction with three-level trophic structure, colored corresponding to the model equation in (e). (b) Simulated dynamics of a system with MX = 50 species of carnivores, MN = 56 herbivores and MR = 62 plants with k = 4, m = 1, u = 1, σc = σd = 0.5, μc = μd = 1, ηX = 0.8, ηN = 0.6, σk = σm = σu = 0.1. (c) Histograms of the steady-state distributions reached by simulated dynamics of 200 systems of the same condition in (b) and the distributions predicted by our cavity solutions. Note that the peaks at zero correspond to delta functions for extinct species. For visual clarity, they are not shown to full scale, but also agree with the cavity predictions (see S4 Fig). (d) Schematic of the coarse-grained view of the three-level trophic structure, colored corresponding to equations in (f). (e) Equations of the three-level trophic structure model corresponding to (a). (f) Effective mean-field (TAP) equations for steady-states have additional emergent competition and random variation terms proportional to () and (), respectively. In 1(a), the icon of the wolf is adapted from “Creative-Tail-Animal-wolf” by Creative Tail licensed under CC BY 4.0, and all other icons are adapted from cliparts from Openclipart licened under CC0 1.0 DEED.</p

Demonstration of bottom-up control and top-down control analogous to Fig 2 for special case with single species on each level.

(a) Exact biomass at each trophic level as a function of u1 with k1 = 5, m1 = 1, ηN = ηX = 0.9, DR = 1, DX = 5, DC = 5.c11 = 4, d11 = 4 (left), and as a function of k1 with u1 = 2 (right). (b) Same as (a) except DR = 3, DX = 3, d11 = 9. (PDF)</p

Plots of effective quantities versus model parameters.

Plot of , , , and versus r1 ∈ [0.4, 1.2], r2 ∈ [0.4, 1.2], σd ∈ [1, 2.5], σc, u ∈ [1, 5], and k ∈ [1, 5], obtained from evaluating the cavity solutions with all 6 parameters varied simultaneously, each with 6 possible values in the range. Each trajectory correspond to varying the parameter on it’s x-axis, while fixing the other 5 parameters. These plots arranged in table directly maps to the result in Table 1. (PDF)</p

Fig 2 -

(a) Bottom-up control. Increasing the total energy energy influx k to primary producers in the bottom trophic level increases the average biomass 〈N〉 of herbivores in the middle trophic level. (b) Top-down control. Increasing the death rate u of predators in the top trophic level increases the biomass of the middle trophic level. (c) Average biomass at each trophic level obtained from cavity solutions as a function of u with k = 1, r1 = 0.2, r2 = 1.2, σc = 0.5, σd = 0.5, (left) and as a function of k with u = 3 (right). (d) Same as (c) except r1 = 1.3, r2 = 0.3. This difference in parameters is an example of the herbivore species richness increase described in Table I. In 2(a)-(b), the icon of the wolf is adapted from “Creative-Tail-Animal-wolf” by Creative Tail licensed under CC BY 4.0, and all other icons are adapted from cliparts from Openclipart licened under CC0 1.0 DEED.</p