DOI: 10.5061/dryad.7118h7f
title: Data from: Predicting Collapse of Complex Ecological Systems: Quantifying the Stability-Complexity Continuum

description: Collapse, stability, and dynamical shifts between these states are
 hallmarks of ecological systems. A major goal in ecosystem research is to
 identify how limits of these states change with diversity, complexity,
 interaction-topology, and hierarchies. The primary focus has been on
 identifying conditions for a system to shift from strict stability to
 complete collapse. While this boundary is indeed of central importance, it
 is possible that real ecosystems can show a larger variety of responses to
 environmental changes. Here, rather than focusing solely on limits of
 stability or collapse, we quantify and map the full phase space and the
 boundaries between regions with different response characteristics. We
 explore this phase space as biodiversity and complexity are varied for
 interaction webs in which consumer-resource interactions are chosen
 randomly and driven by Generalized-Lotka-Volterra dynamics. The ability to
 pinpoint the location of a system within this phase space and quantify the
 system’s proximity to collapse is made possible via a novel mathematical
 analysis that we develop. Previous work that focused only on collapse
 lacks the context within the overall phase space to be able to predict
 when systems are nearing or are poised to collapse. Moreover, in contrast
 to previous collapse predictions, we account for the fact that dynamics
 often lead to single species extinctions. Allowing and accounting for
 these single-species extinctions reveals more detailed structure of the
 complexity-stability phase space and introduces an intermediate phase
 between stability and collapse - Extinction Continuum – that give a more
 nuanced view of how an ecosystem can respond to internal and external
 changes. With this extended phase space and our construction of predictive
 measures based strictly on observable quantities, real systems can be
 better mapped — than using canonical measures by May or critical slowdown
 – for proximity to collapse and path through phase-space to collapse.
description: ODE simulations standard
 settings, n=1000Data from 10
 simulationsNormal(0,1)-1000_2018124Population dynamics for increasing interaction strength heterogeneity n=20.File contains 5 Mathematica objects from ODE simulations with population size n=20, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-20_2018124Population dynamics for increasing interaction strength heterogeneity n=40.File contains 5 Mathematica objects from ODE simulations with population size n=40, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-40_2018124Population dynamics for increasing interaction strength heterogeneity n=60.File contains 5 Mathematica objects from ODE simulations with population size n=60, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-60_2018124Population dynamics for increasing interaction strength heterogeneity n=100.File contains 5 Mathematica objects from ODE simulations with population size n=100, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-100_2018124Population dynamics for increasing interaction strength heterogeneity n=200.File contains 5 Mathematica objects from ODE simulations with population size n=200, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-200_2018124Population dynamics for increasing interaction strength heterogeneity n=300.File contains 5 Mathematica objects from ODE simulations with population size n=300, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-300_2018124Population dynamics for increasing interaction strength heterogeneity n=400.File contains 5 Mathematica objects from ODE simulations with population size n=400, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-400_2018124Population dynamics for increasing interaction strength heterogeneity n=500.File contains 5 Mathematica objects from ODE simulations with population size n=500, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-500_2018124Population dynamics for increasing interaction strength heterogeneity n=600.File contains 5 Mathematica objects from ODE simulations with population size n=600, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-600_2018124Population dynamics for increasing interaction strength heterogeneity n=700.File contains 5 Mathematica objects from ODE simulations with population size n=700, Normal(0,1) distributed interaction strengths in interaction matrix A of denisty c=0.5, intrinsic growth rates and carrying capacity set to r=K=1. Object 1: sVector (list of interaction strength heterogeneity used in simulations). Object 2: aTotal (list of the random interaction strength matrices used in the 10 runs). Object 3: stabilityTotal (list for all 10 runs of matrices with binary values indicating stability of reduces system (row 1), whole system (row 2) for all values in sVector). Object 4: xFixTotal (list for all 10 runs of fixed point abundances for all species for all values in sVector).Normal(0,1)-700_2018124ODESimulationMathematica notebook containing code for ODE simulations and plotting.