Fig 1.
Functional redundancy as low-rank interspecific interactions.
(A) A three-level food web, in which lower levels contain multiple species that duplicate one other’s role. (B) The interspecific interaction matrix decomposes into a full-rank group-level component, and a low-rank assignment matrix mapping species to groups. Small-amplitude variations among taxa within each group restore the rank, but lead to high condition number. The exact form of the interaction matrix is given by Eq 3.
Fig 2.
Long-lived transients in ill-conditioned ecosystems.
(A) Equilibration of a random food web without (top) and with (bottom) a pair of functionally-redundant species. Long-lived transients appear in the latter case. (B) Settling time versus condition number
for 104 random communities; dashed line shows the scaling expected for an iterative linear program solver.
Fig 3.
Slow manifolds form a complex optimization landscape.
(A) An embedding of 103 trajectories with different initial conditions in an ill-conditioned ecosystem (). The global equilibrium is marked with a star, and the corresponding solutions of the degenerate (
) case are overlaid (blue). (B) Time that ill-conditioned trajectories for different random ecosystems (colored by condition number) spend near the former (
) solutions, versus the solution’s Morse instability index. (C) Projection of a single trajectory onto the right singular vectors associated with the largest (red) and smallest (blue) singular values.
Fig 4.
Singular value decomposition of a species interaction matrix.
A schematic of singular value decomposition of the hierarchical species interaction matrix shown in Fig 1. The left and right sets of singular vectors U,V isolate groups of species that are functionally redundant, while the diagonal elements in the singular value matrix
encode the hierarchy of timescales that emerge due to functional redundancy.
Fig 5.
Transient chaos due to slow manifold scattering.
(A) The "pachinko" mechanism for ill-conditioned dynamics, in which slow manifolds temporarily disperse neighboring trajectories that later reunite at the global equilibrium. (B) Caustics in the Fast Lyapunov Indicator () versus initial conditions on a two-dimensional slice through the N-dimensional space of initial species densities.
Fig 6.
Selection for diversity produces ill-conditioning.
(A) The condition number versus generation for 103 replicate random ecosystems of N = 103 species evolved to have high steady-state diversity, defined as the number of coexisting species at equilibrium. (B) The restricted condition number
of the interaction matrix of species that survive at steady-state, versus the overall steady-state diversity. Dashed line indicates expected condition number for a random
matrix with normally-distributed elements.
Fig 7.
Community response to perturbations reveals the slow manifold.
(A) A set of 100 experiments in which a random pulse perturbation is applied to an equilibrated ecosystem, and the dynamics are allowed to return to steady state. The full N = 200 dimensional dynamics are projected onto the fastest and slowest directions associated with the singular vectors of the interaction matrix A. The slow manifold (blue points) is numerically detected by repeating the experiment with in Eq 1. (B) The Pearson correlation (inner product) between the velocity vector of the dynamics at each timepoint, and the fast and slow manifolds. Error bars represent standard errors over 100 replicate communities.
Table 1.
Mathematical symbols and their descriptions.
Fig 8.
Complex transients in an alternative ecosystem model.
(A) The condition number as an ecosystem model transitions from noninvasible global stability to multistability. Settling time versus condition number
for 104 random communities. (B) Caustics in the Fast Lyapunov Indicator (
) versus initial conditions on a two-dimensional slice through the N-dimensional space of initial species densities.
Fig 9.
The effect of network connectivity on scaling of transient dynamics.
Settling time versus condition number
for
random communities; dashed line shows the scaling expected for an iterative linear program solver. Colors indicate communities with different values of the connectance (
).
Fig 10.
The effect of network connectivity on transient dynamics.
Caustics in the Fast Lyapunov Indicator () versus initial conditions on a two-dimensional slice through the N-dimensional space of initial species densities. Panels correspond to three levels of network connectance (
).
Fig 11.
The effect of slow manifold dimension on scaling of transient dynamics.
Settling time versus condition number
for
random communities; dashed line shows the scaling expected for an iterative linear program solver. Colors indicate communities with different slow manifold dimensionalities (M/N).
Fig 12.
The effect of slow manifold dimension on transient dynamics.
Caustics in the Fast Lyapunov Indicator () versus initial conditions on a two-dimensional slice through the N-dimensional space of initial species densities. Panels correspond to three values of the slow manifold dimension (M/N).