Researcher: Jason Holt Institution: Independent Research Status: Active - Phase 4 Complete Last Updated: December 2025
The Amazon rainforest generates much of its own rainfall through moisture recycling - trees release water vapor that falls as rain downwind, sustaining more forest. When we cut down trees, we don't just lose that patch of forest; we break the invisible moisture highways that keep neighboring forests alive.
This research uses computational models to understand a troubling phenomenon: deforestation may create a "one-way valve" where forest loss becomes progressively easier while recovery becomes progressively harder.
┌─────────────────────────────────────────────────────────────────┐
│ KEY DISCOVERIES │
├─────────────────────────────────────────────────────────────────┤
│ │
│ 1. CRITICAL CONNECTIVITY THRESHOLD: 50-75% │
│ • Below 50% connectivity → recovery essentially impossible │
│ • Sharp phase transition, NOT gradual decline │
│ • This is MUCH higher than expected (was 10-25%) │
│ │
│ 2. KEYSTONE EDGES ONLY MATTER WHEN NETWORK IS INTACT │
│ • At 100% connectivity: keystones provide +14% recovery │
│ • At <50% connectivity: keystones provide NO benefit │
│ • Must maintain connectivity FIRST, then protect keystones │
│ │
│ 3. RESTORATION IS POSSIBLE (with sufficient connectivity) │
│ • recovery ≈ 39% + (74% × intervention_effort) │
│ • But only if network connectivity > 50% │
│ │
└─────────────────────────────────────────────────────────────────┘
When we model the Amazon as a network of 50 connected forest cells, breaking connections (simulating deforestation) creates increasingly asymmetric dynamics:
| Connectivity | Tip vs Recovery | What It Means |
|---|---|---|
| 100% intact | 1.005 (balanced) | Healthy forest can recover as easily as it degrades |
| 50% intact | 1.025 (+2.5%) | First signs of asymmetry emerge |
| 25% intact | 1.077 (+7.7%) | Clear bias toward degradation |
| 10% intact | 1.148 (+14.8%) | Tipping is now significantly easier than recovery |
Tip/Recovery Ratio vs Forest Connectivity
1.15 ┤ ●
│ ╱
1.10 ┤ ● ╱
│ ╱ ╱
1.05 ┤ ● ╱ ╱
│ ● ╱ ╱ ╱
1.00 ┤──●────●──────●────────────────────── Balanced
│
0.95 ┼───┴───┴──────┴────┴────┴────┴────┴─
100% 90% 75% 50% 25% 10% 0%
Forest Connectivity
Why does this matter?
Each percentage of deforestation makes the next percentage easier. This creates a feedback loop where degradation accelerates - the very definition of a "tipping cascade."
Earlier in our research, we thought ecosystems might be completely trapped once tipped. Our refined models show this isn't true:
| Intervention Level | Recovery Rate | Interpretation |
|---|---|---|
| None (passive) | 38.6% | Nature can partially heal itself |
| Moderate (f=0.1) | 52% | Crosses 50% threshold |
| Significant (f=0.3) | 73% | Most cells recover |
| Major (f=0.5) | 88% | Near-complete restoration |
Recovery Fraction vs Intervention Effort
90% ┤ ●
│ ● ╱
75% ┤ ● ╱ ╱
│ ● ╱ ╱ ╱
60% ┤ ● ╱ ╱ ╱ ╱
│ ● ╱ ╱ ╱ ╱ ╱
45% ┤ ● ╱ ╱ ╱ ╱ ╱ ╱
│ ●═══════════════════════ 50% line
30% ┤
│
15% ┼──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴─────
0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.5
Intervention Strength (|f|)
The formula: recovery ≈ 0.74 × intervention + 0.39
This linear relationship is good news for conservation: partial effort yields partial results. There's no threshold below which intervention is "wasted."
We tested two deforestation patterns:
- Random clearing (as often happens with smallholder agriculture)
- Targeted clearing of the most connected forest patches (as with strategic development)
Surprising result: Random clearing is actually MORE damaging to recovery capacity:
| Destruction Pattern | Asymmetry at 10% | Recovery Difficulty |
|---|---|---|
| Random | 17.4% | Higher |
| Targeted (hub removal) | 12.3% | Lower |
Interpretation: The most connected forest patches support both degradation and recovery. Removing them reduces overall activity but preserves distributed recovery pathways. Random clearing may preferentially destroy recovery-supporting connections while leaving cascade-promoting pathways intact.
We identified keystone edges - specific moisture recycling connections whose removal disproportionately hurts recovery. Only 6% of edges are true keystones, while 82% of edge removals actually improve recovery (by reducing cascade propagation).
But here's the critical finding: Keystones only matter when the network is mostly intact.
| Connectivity Level | Keystone Preserve | Random | Keystone Remove |
|---|---|---|---|
| 100% | 63.5% | 49.1% | 42.4% |
| 75% | 27.7% | 25.9% | 15.1% |
| 50% | 7.7% | 9.8% | 3.4% |
| 25% | 5.0% | 7.7% | 4.0% |
| 10% | 1.9% | 2.3% | 1.2% |
Key insight: At <50% connectivity, ALL strategies perform poorly. The network has crossed a percolation threshold below which recovery signals cannot propagate, regardless of which edges remain.
Recovery vs Connectivity (Percolation Threshold)
60% ┤ ●──Preserve
│ /
50% ┤ ● Random
│ /
40% ┤ ● /
│ /
30% ┤ ●────/─ Remove
│ /
20% ┤ /
│ CRITICAL /
10% ┤════════════════════ZONE═/════════════════════
│ ●──●──●──●──●──●──●──●/
0% ┼──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴─────
1% 5% 10% 15% 20% 25% 50% 75% 100%
Network Connectivity
graph TD
A[Deforestation] --> B[Reduced Connectivity]
B --> C[Asymmetric Dynamics]
C --> D[Easier Tipping]
D --> E[More Forest Loss]
E --> A
C --> F[Harder Recovery]
F --> G[Less Regeneration]
G --> E
style A fill:#ff6b6b
style E fill:#ff6b6b
style D fill:#ffa94d
style F fill:#ffa94d
Our models show restoration IS possible:
- ~39% passive recovery: Some areas will regenerate naturally
- Linear scaling: Each unit of effort produces proportional results
- No "point of no return": Even heavily degraded systems can recover with sufficient intervention
- Prevention is ~15% more efficient than cure: But restoration IS achievable
Each forest cell is modeled as a bistable system - it can exist stably in either a "forested" or "tipped" state, with an energy barrier between them:
Energy Landscape for a Forest Cell
│ Barrier
│ /\
E │ / \
n │ / \
e │ ____/ \____
r │ / \
g │ / \
y │──●────────────────────●──
│Forest Tipped
│State State
└────────────────────────────
Forest Health (x)
Cells are connected through moisture recycling - based on real data from Wunderling et al. (2022) showing how Amazon rainfall depends on upwind evapotranspiration.
We use Lévy stable noise to model climate variability:
| Parameter α | Noise Type | Physical Meaning |
|---|---|---|
| α = 2.0 | Gaussian | Normal rainfall variability |
| α = 1.5 | Lévy | Includes extreme droughts, fires |
| α < 1.5 | Heavy Lévy | Very extreme events |
Key insight: Ecosystems tip during extreme events (modeled by Lévy noise) but must recover under normal conditions (Gaussian noise). This inherent asymmetry explains much of the observed hysteresis.
| # | Name | Question | Key Finding |
|---|---|---|---|
| 8 | Network Fragmentation | Does connectivity loss create asymmetry? | Yes - 14.8% at 10% retention |
| 9 | Recovery Dynamics | What mechanism causes hysteresis? | Coupling degradation is DOMINANT (0% recovery) |
| 10 | Alpha Sweep | Does noise type affect recovery? | Noise amplitude matters more than type |
| 10b | 2D Parameter Sweep | Can any passive condition enable recovery? | Maximum ~4% passive recovery |
| 10c | Restoration Forcing | Does active intervention help? | Yes - linear relationship, 88% achievable |
| 11 | Fragmentation × Forcing | Does fragmentation affect forcing? | Forcing 2.4× more effective in fragmented networks |
| 12 | Keystone Edge Analysis ⭐ | Which edges are most critical? | Only 6% are keystones; 82% of removals IMPROVE recovery |
| 12b | Keystone Protection Test | Can keystones alone sustain recovery? | No - keystones necessary but not sufficient |
| 12c | Connectivity Threshold ⭐ | What's the minimum connectivity? | 50-75% threshold - sharp phase transition |
| Network State | Recovery | Strategy |
|---|---|---|
| >75% intact | 26-64% | Protect keystone edges (+14% benefit) |
| 50-75% intact | 10-26% | CRITICAL ZONE - prevent fragmentation |
| 25-50% intact | <10% | Network structure irrelevant - needs forcing |
| <25% intact | <5% | Passive recovery impossible |
Key insight: The connectivity threshold (~50-75%) is much higher than expected. Below this threshold, keystones provide NO benefit - the network has lost its ability to propagate recovery signals.
| Mechanism | Recovery | Effect |
|---|---|---|
| Symmetric baseline | 47% | Normal (validates solver) |
| Barrier asymmetry only | 15% | 68% reduction |
| Coupling degradation only | 0% | 100% reduction |
| Combined (realistic) | 6% | 87% reduction |
Implication: Restoring connectivity (reforestation corridors) may be more effective than direct forcing (planting isolated trees).
| # | Name | Question |
|---|---|---|
| 13 | Recovery Trajectories | How does recovery propagate through the network? |
| 14 | Localized vs Distributed Forcing | Is targeted restoration more efficient? |
| 15 | Coupling Restoration ⭐ | Is restoring connectivity more effective than direct forcing? |
| 16 | Restoration Sequencing | Force first then restore coupling, or vice versa? |
┌─────────────────────────────────────────────────────────────┐
│ EnergyConstrainedNetwork │
├─────────────────────────────────────────────────────────────┤
│ │
│ ┌─────────────┐ Coupling ┌─────────────┐ │
│ │ Forest Cell │ ←──────────────→ │ Forest Cell │ │
│ │ (Cusp) │ (Moisture │ (Cusp) │ │
│ │ │ Recycling) │ │ │
│ └─────────────┘ └─────────────┘ │
│ ↕ ↕ │
│ ┌─────────────┐ ┌─────────────┐ │
│ │ Lévy │ │ Gaussian │ │
│ │ Noise │ │ Noise │ │
│ │ (Extremes) │ │ (Normal) │ │
│ └─────────────┘ └─────────────┘ │
│ │
│ Simulation: Euler-Maruyama SDE solver with soft │
│ reflection boundaries at ±2 │
│ │
└─────────────────────────────────────────────────────────────┘
from energy_constrained import (
# Core elements
EnergyConstrainedNetwork, # Network container
EnergyConstrainedCusp, # Bistable elements
EnergyDependentCusp, # Energy-dependent thresholds
HumanSettlementElement, # Urban infrastructure stability
# Coupling types
GradientDrivenCoupling, # Thermodynamic coupling
EnergyMediatedCoupling, # κ(E) dependent coupling
MultiLayerCoupling, # Cross-layer connections
# Multi-layer networks
MultiLayerNetwork, # 3-layer socio-ecological network
create_three_layer_network, # Factory for climate/biosphere/human
create_amazon_settlement_network, # Amazon + cities
# Simulation & Analysis
run_two_phase_experiment, # Cascade → Recovery simulation
EnergyAnalyzer, # Entropy & tipping analysis
get_dask_client, # Parallel execution
# Sensitivity analysis
run_sobol_analysis, # Global sensitivity
quick_sensitivity_scan # Quick parameter importance
)The research project's core hypothesis is that human systems are dissipative structures requiring continuous energy throughput. When energy supply is disrupted, the system becomes easier to "tip" into a degraded state.
┌───────────────────────────────────────────────────────────────┐
│ THREE-LAYER NETWORK │
├───────────────────────────────────────────────────────────────┤
│ │
│ LAYER 1: CLIMATE │
│ ┌─────┐ ┌─────┐ ┌──────┐ ┌───────────┐ │
│ │ GIS │◄─►│AMOC │◄─►│Amazon│◄─►│Permafrost │ │
│ └──┬──┘ └──┬──┘ └──┬───┘ └─────┬─────┘ │
│ │ │ │ │ │
│ ▼ ▼ ▼ ▼ │
│ LAYER 2: BIOSPHERE │
│ ┌─────────┐ ┌─────────┐ ┌──────────┐ ┌──────────┐ │
│ │Ecosystem│ │Ecosystem│ │ Moisture │ │ Carbon │ │
│ │Services │ │Services │ │Recycling │ │ Storage │ │
│ └────┬────┘ └────┬────┘ └────┬─────┘ └────┬─────┘ │
│ │ │ │ │ │
│ ▼ ▼ ▼ ▼ │
│ LAYER 3: HUMAN SETTLEMENTS │
│ ┌──────┐ ┌────────┐ ┌─────┐ ┌───────┐ ┌─────┐ │
│ │Bogota│ │Sao Paulo│ │Dhaka│ │Karachi│ │Lagos│ │
│ └──────┘ └────────┘ └─────┘ └───────┘ └─────┘ │
│ │
│ Coupling types: │
│ Climate → Biosphere: Environmental stress │
│ Biosphere → Human: Ecosystem services (stabilizing) │
│ Human → Biosphere: Resource extraction (destabilizing) │
│ │
└───────────────────────────────────────────────────────────────┘
Energy-Dependent Dynamics:
The key innovation is that tipping thresholds depend on energy availability:
c(E) = c_base × (E / E_nominal)^sensitivity
When E = E_nominal: Normal operation
When E < E_nominal: Threshold decreases → easier to tip
When E < E_critical: System tips regardless of forcing
This captures how cities and ecosystems become more fragile when their energy/resource supply is disrupted.
- Network topology: Amazon moisture recycling data from Wunderling et al. (2022)
- Climate patterns: ERA5 reanalysis data (2003 dry season)
- Model framework: Based on PyCascades from PIK Potsdam
This research runs on a Kubernetes (k3s) cluster optimized for scientific computing:
| Service | Purpose | Access |
|---|---|---|
| JupyterLab | Interactive notebooks | localhost:30888 |
| Dask Dashboard | Parallel computing monitor | localhost:30787 |
| MLflow | Experiment tracking | localhost:30505 |
Compute: 14 Dask workers (1.5 CPU, 1GB RAM each) with optimized scatter-based task distribution.
cascades/
├── src/energy_constrained/ # Custom thermodynamic module
│ ├── elements.py # Bistable cusp elements
│ │ ├── EnergyConstrainedCusp # Base energy-tracking element
│ │ ├── EnergyDependentCusp # Energy-dependent thresholds
│ │ └── HumanSettlementElement # Urban infrastructure stability
│ │
│ ├── couplings.py # Energy-aware coupling functions
│ │ ├── GradientDrivenCoupling # Thermodynamic gradient flow
│ │ ├── EnergyMediatedCoupling # κ(E) = κ⁰·f(E_src, E_tgt)
│ │ ├── EnergySupplyCoupling # Infrastructure supply links
│ │ └── MultiLayerCoupling # Cross-layer connections
│ │
│ ├── network.py # Network containers
│ │ ├── EnergyConstrainedNetwork # Base energy network
│ │ ├── MultiLayerNetwork # 3-layer socio-ecological
│ │ └── create_three_layer_network # Factory function
│ │
│ ├── solvers.py # SDE solver with Lévy noise
│ ├── analysis.py # Energy & tipping analysis
│ ├── sensitivity.py # Sobol sensitivity analysis
│ └── dask_utils.py # Parallel execution
│
├── notebooks/
│ ├── 06_network_fragmentation.ipynb # Exp 8: Network fragmentation
│ ├── 07_recovery_dynamics.ipynb # Exp 9: Recovery dynamics
│ ├── 08_alpha_sweep.ipynb # Exp 10: Alpha sweep
│ ├── 09_restoration_sequencing.ipynb # Exp 16: Restoration sequencing
│ ├── 10_network_size_validation.ipynb # Network validation
│ ├── 11_keystone_protection.ipynb # Exp 12b: Keystone protection test
│ └── 12_connectivity_threshold.ipynb # Exp 12c: Connectivity threshold ⭐
│
├── docs/
│ ├── phase4_results.md # Detailed experiment results
│ └── phase4_research_plan.md # Experimental design
│
├── data/amazon/ # Moisture recycling network data
└── external/pycascades/ # PIK framework (dependency)
-
Wunderling, N., et al. (2022) - "Recurrent droughts increase risk of cascading tipping events by outpacing adaptive capacities in the Amazon rainforest" - PNAS - [Dataset source]
-
Wunderling, N., et al. (2021) - "Interacting tipping elements increase risk of climate domino effects under global warming" - Earth System Dynamics - [PyCascades framework]
-
Lenton, T.M., et al. (2008) - "Tipping elements in the Earth's climate system" - PNAS - [Foundational tipping points theory]
-
Zemp, D.C., et al. (2017) - "Self-amplified Amazon forest loss due to vegetation-atmosphere feedbacks" - Nature Communications - [Moisture recycling dynamics]
This is active research. If you're interested in collaborating on:
- Model validation against observed data
- Extended network analysis
- Policy applications
- Code improvements
Please open an issue or reach out directly.
Research code is available under MIT License. Data from external sources retains original licensing.
┌─────────────────────────────────────────────────────────────────┐
│ POLICY IMPLICATIONS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ 1. CRITICAL 50% CONNECTIVITY THRESHOLD ⭐ NEW │
│ Below 50% connectivity, passive recovery is IMPOSSIBLE │
│ This is a sharp threshold, not gradual decline │
│ │
│ 2. KEYSTONES ONLY WORK IN INTACT NETWORKS │
│ At >75% connectivity: keystones provide +14% recovery │
│ At <50% connectivity: keystones provide ZERO benefit │
│ → Must maintain overall connectivity FIRST │
│ │
│ 3. CONNECTIVITY MATTERS MORE THAN AREA │
│ Preserving forest corridors is as important as │
│ preserving total forest area │
│ │
│ 4. RESTORATION WORKS (if connectivity maintained) │
│ Active intervention follows linear returns: │
│ Double the effort → Double the recovery │
│ But ONLY if network connectivity > 50% │
│ │
│ 5. CONSERVATION PRIORITIES: │
│ • >75% intact: Protect keystone connections │
│ • 50-75%: CRITICAL - prevent ANY further loss │
│ • <50%: Passive recovery impossible - needs forcing │
│ │
└─────────────────────────────────────────────────────────────────┘
This research is ongoing. Results are preliminary and subject to refinement as experiments continue.
Contact: [Open an issue on this repository]
Understanding how ecosystems tip - and how they can recover