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.

#### ODE simulations standard settings, n=1000

Data from 10 simulations

Normal(0,1)-1000_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### Population 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_2018124

#### ODESimulation

Mathematica notebook containing code for ODE simulations and plotting.