Stability criteria for the consumption and exchange of essential resources
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.