Fig 1.
Conceptual diagrams of internal metabolism, resource flow in the model community, and example consumer-resource dynamics.
(A) A schematic of the internal metabolism in our model for one consumer N1. This consumer depletes two resources—R1 is limiting for consumer N1 while R2 is not. The consumption of resource R1 goes fully towards towards the consumer’s internal requirements. Some portion (θ) of this consumption is leaked out into the environment (as resource R3 in the diagram), while the remaining consumed resource R1 goes towards biomass. Because resource R2 is not limiting, it is consumed in excess of consumer N1’s requirements for growth. This excess consumption is recycled back into the environment (as resource R4). The remaining resource then goes to biomass producing processes, exactly as the entirety of the limiting resource intake. Therefore, some of the required consumption of resource R2 is leaked and the rest goes to biomass. (B) A schematic of the consumer-resource model in Eq (1). Consumers (Ni) deplete the resource (Rj) at rates Cij (all blue arrows), but only some of the consumption goes to consumer growth. Instead, some consumed resource is leaked back into the environment as new resources at rates Pji or (green arrows). Resources are externally supplied at rates ρi (gray arrows) and consumers undergo density independent mortality at rate ηk (red arrows). (C) The dynamics of the consumers and resources in Eq (1) when there is a stable equilibrium and when there is not. Consumers grow according to Liebig’s law as described in the text. When there is no stable equilibrium, the consumer abundances undergo large fluctuations, reaching low abundances.
Table 1.
Variables in Eq 1 and in our stability criteria with their meanings.
Fig 2.
Example consumption and production structures.
Each panel is a visualization of the different consumption and production patterns which satisfy our analytical criteria. Darker colors indicate larger values. (A) The tradeoff parameterization of the consumption matrix C. C is symmetric with identical row (or column) sums and diagonal coefficients all set to Cd. (B) The constant parameterization of the production matrix P. Each entry off-diagonal entry is set to , while the diagonal entries are set to 0. (C) The circulant parameterization of the consumption matrix. Each row of the matrix is a permuted version of the previous row and the diagonal entries are all set to Cd. (D) The circulant parameterization of the production matrix. Each row of the matrix is a permuted version of the previous row and the diagonal entries are set to 0.
Fig 3.
Analytical stability criteria predict empirical values.
(A-C) We plot the value of the theoretical bound for Cd in the second stability criterion against the smallest Cd value at which the fixed point becomes stable numerically for 100 realizations of the random matrices comprising C, P and while varying the standard deviation (colors) and mean (shapes) of the off-diagonal elements of the consumption matrix. The different panels are for the three different parameterizations of the consumption and production matrices ((A) is the tradeoff parameterization, (B) is the circulant parameterization and (C) is the unstructured parameterization). Parameters: S = 15, n = r = 1, θ = 0.5, ϵii = 0.05 and ϵij = 1 for i ≠ j in all panels. For each of the different matrix parameterizations, we first sample the consumption coefficients from uniform distributions with mean given by the average consumption value (1, 3, or 5 in these simulations) and the specified standard deviations so that the coefficients of variation in consumption coefficients vary from
to
. Then, we impose the constraints for the tradeoff and circulant parameterizations afterwards. (D-F) Histograms of the differences between the predicted and observed values in panels A-C in each of the three parameterizations when the average consumption coefficient is 5 and the coefficient of variation in consumption coefficients is
.
Fig 4.
Unstructured networks violate theoretical stability criteria at low consumer abundances.
We plot the value of the theoretical stability bound for Cd averaged over 100 replicates (solid lines) as the consumer abundance n varies over three orders of magnitude for the three different matrix parameterizations (panels and colors). We also plot the average smallest Cd value for which the system first becomes stable numerically (shapes) along with error bars showing one standard deviation above and below the mean. In panels (A-B), the theoretical bound accurately predicts the dependence of Cd on n, while for the unstructured case (panel (C)), the theoretical predictions fail at low consumer abundance. Parameters: S = 15, r = 1, θ = 0.5, ϵii = 0.05 and ϵij = 1 for i ≠ j. The off-diagonal elements of the consumption matrices are sampled from uniform distributions on [0, 2] before the parameterizations are imposed.
Fig 5.
Low consumer abundances induce instability for more general resource inflows and interaction networks.
We plot the probability of finding a feasible and stable fixed point in 25 replicates across a range of Cd values and resource inflows ρ with only one externally supplied nutrient for three different matrix parameterizations. We also enforce that the fixed point is realized under a Liebig’s law growth rule, where each consumer grows on the most limiting nutrient of the resources. (A) The tradeoff matrix parameterization does not show any dependence on the resource inflow ρ, as in Fig 4. (B) The unstructured case does not have any feasible and stable fixed points at low resource inflow. (C) The banded matrix parameterization has a consumption matrix with non-zero values on the upper and lower bands of the matrix, displaced from the diagonal by one index. It also has a constant production matrix, as described previously. It does not show any dependence on the resource inflow. Parameters: S = 15, ηi = 1, θ = 0.9, ϵii = 0.05 and ϵij = 1 for i ≠ j. Consumption coefficients sampled from uniform distributions on [0.5, 1.5] before the constraints are imposed.
Fig 6.
Stable and unstable consumer dynamics for varying resource inflows.
We plot the consumer dynamics for two different consumption networks—one where the coefficients are sampled randomly from a distribution with correlations across the diagonal (labeled Correlated) and the other according to the tradeoff consumption network described in the text (labeled Tradeoff)—for three different magnitudes of one externally supplied resource (Low Inflow: ρ1 = 0.001, Medium Inflow: ρ1 = 0.1 and High Inflow: ρ1 = 1). The S = 15 consumers’ growth rates are determined by the most limiting resources following Liebig’s law. All other resources are not supplied (ρi = 0 for i = 2, …, S). In each case, the production network is given by the constant parameterization where all other resources are produced from a given incoming resource. For the tradeoff network, the equilibrium is stable regardless of resource inflow levels, while the correlated network becomes unstable when the resource inflow is low. Parameters: S = 10, Cd = 10, ηi = 1, θ = 0.9, ϵii = 0.05 and ϵij = 1 for i ≠ j. Consumption coefficients sampled from uniform distributions on [0.5, 1.5] before the constraints are imposed.