dlvt 2.0.1

Dynamic Leadership Vitality Theory: A dynamical systems model of executive sustainability

Dynamic Leadership Vitality Theory (DLVT) — Python Package

Author: William Bendinelli Paper: Dynamic Leadership Vitality Theory: A Formal Model of the Zombie-Leader Equilibrium Target Journal: The Leadership Quarterly (2026)

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Overview

This package implements the Dynamic Leadership Vitality Theory (DLVT) model — a two-dimensional autonomous dynamical system that formalizes how leaders who accumulate career capital simultaneously generate the organizational complexity that drains the vitality they need to deploy it.

The model predicts a single robust attractor: a zombie-leader equilibrium at $(V^{} \approx 4.7,; C^{} \approx 32)$ in which the leader retains authority and reputation but permanently lacks the energetic bandwidth for strategic work. The zombie equilibrium is globally asymptotically stable (proved via Bendixson–Dulac + Poincaré–Bendixson on a trapping rectangle).

Three outcome regimes emerge from the parameter space:

• Sustainable — the leader reaches equilibrium with $V^{*} > V_{\mathrm{strategic}}$
• Zombie — the leader converges to $V^{*} < V_{\mathrm{strategic}}$ (the default attractor)
• Collapse-Prone — unsustainable trajectory where burnout is inevitable

Quick Start

Installation

git clone https://github.com/wbendinelli/dlvt.git
cd dlvt
pip install -e .

Requirements: Python ≥ 3.8, NumPy ≥ 1.24, SciPy ≥ 1.10, Matplotlib ≥ 3.7

Basic Usage

from dlvt import make_params, simulate
from dlvt.analysis import find_interior_equilibria, carrying_capacity

# Create parameter set (defaults match Table 1 of the paper)
p = make_params()

# Simulate from initial state (V₀=8.0, C₀=5.0) over T=120 time units
t, V, C, O, I, G = simulate(p, V0=8.0, C0=5.0, T=120)
# V — vitality, C — career capital, O — complexity, I — impact, G — depletion ratio

# Find equilibria and classify stability
equilibria = find_interior_equilibria(p)
for eq in equilibria:
    status = "stable" if eq['stable'] else "unstable"
    print(f"V*={eq['V']:.2f}, C*={eq['C']:.2f} ({status})")

# Maximum sustainable career capital (Proposition 3)
C_max = carrying_capacity(p)
print(f"Carrying capacity: C*_max = {C_max:.2f}")

Scenario Analysis

from dlvt import make_params, simulate
from dlvt.analysis import classify_regime

# What happens if we increase recovery support?
p_high_R = make_params(R=5.0)
regime = classify_regime(p_high_R)
print(f"Regime with R=5.0: {regime}")

# What if complexity coupling is halved?
p_low_beta = make_params(beta=0.125)
t, V, C, O, I, G = simulate(p_low_beta, V0=8.0, C0=5.0, T=200)
print(f"Final V={V[-1]:.2f}, C={C[-1]:.2f}")

The Dynamical System

The DLVT model is governed by two coupled ODEs:

$$\frac{dV}{dt} = R \left(1 - \frac{V}{V_{\max}}\right) - \delta \cdot O^{\gamma} \cdot \frac{V}{V + \varepsilon}$$

$$\frac{dC}{dt} = \alpha \cdot I - \mu \cdot C$$

with derived quantities:

• Organisational Complexity: $O(t) = O_0 + \beta \cdot C(t)^{\eta}$
• Leadership Impact: $I(t) = C \cdot V ,/, (1 + \phi \cdot O)$ — energy-gated effectiveness
• Depletion Ratio: $\Gamma(t) = \delta \cdot O^{\gamma} ,/, R$ — net drain when $\Gamma > 1$

The first equation balances bounded vitality recovery against complexity-driven drain. The second models career capital accumulation through impact feedback minus depreciation. The smooth barrier $V/(V+\varepsilon)$ ensures positive invariance of the state space.

Parameter Table

All parameters from Table 1 of the paper (baseline calibration, $C_0 = 5.0$):

Symbol	Key	Default	Description
$R$	R	3.0	Vitality recovery rate
$V_{\max}$	Vmax	10.0	Maximum vitality capacity
$\delta$	delta	0.02	Energetic cost coefficient
$\gamma$	gamma	2.0	Complexity exponent in drain
$O_0$	O0	1.0	Baseline organisational complexity
$\beta$	beta	0.25	Capital–complexity coupling
$\eta$	eta	1.0	Capital scaling exponent
$\alpha$	alpha	0.1	Capital accumulation rate
$\phi$	phi	0.15	Complexity–impact suppression
$\mu$	mu	0.2	Capital depreciation rate
$\varepsilon$	eps	0.1	Smooth barrier regularisation

Key Theoretical Results

Theorem 1 (Finite-Time Depletion): Under exogenous complexity growth that creates a persistent deficit ($\Gamma > 1$), vitality reaches the $\varepsilon$-neighborhood of zero in finite time.

Theorem 2 (Global Asymptotic Stability): The zombie equilibrium $(V^{}, C^{})$ is the unique interior attractor. Proved via Bendixson–Dulac (no closed orbits), trapping rectangle $[0, V_{\max}] \times [0, C_{\max}]$, and Poincaré–Bendixson.

Lemma 2 (Scope Absorption): $V^{}(\beta) \equiv V^{}_0$ for all $\beta$ in the structurally interior region; the invariant $\beta C^{*} \approx 8.008$ is preserved. Scope reduction alone does not raise equilibrium vitality.

Proposition 3 (Carrying Capacity): $C^{*}_{\max} = \left[\left(R/\delta\right)^{1/\gamma} - O_0\right]^{1/\eta} / \beta$

API Reference

dlvt.model — Core ODE System

from dlvt import make_params, simulate, complexity, impact

Function	Description
make_params(**overrides)	Create parameter dict with keyword overrides
complexity(C, p)	$O = O_0 + \beta C^{\eta}$
impact(V, C, O, p)	$I = CV/(1 + \phi O)$
dlvt_system(t, y, p)	ODE right-hand side for scipy.integrate.solve_ivp
dlvt_exogenous(t, y, p, C_func)	Variant with exogenous career capital path
simulate(p, V0, C0, T, max_step)	Integrate ODE; returns (t, V, C, O, I, G)

dlvt.analysis — Equilibrium Theory & Stability

from dlvt.analysis import (
    find_interior_equilibria, carrying_capacity,
    jacobian_eigenvalues, is_zombie, classify_regime, regime_map
)

Function	Description
find_interior_equilibria(p)	Find all $(V^{}, C^{}) > 0$ fixed points
carrying_capacity(p)	Maximum sustainable capital $C^{*}_{\max}$
jacobian_eigenvalues(V, C, p)	Eigenvalue structure and stability class
is_zombie(V, p)	Check $V^{*} < V_{\mathrm{strategic}}$
classify_regime(p)	Returns 'sustainable', 'zombie', or 'collapse-prone'
regime_map(betas, deltas)	$(\beta, \delta)$ regime classification grid

dlvt.figures — Publication Figure Generation

from dlvt.figures import fig1, fig2, fig3, fig4, fig5, fig6, fig7

fig3(output_dir='figures/')  # Phase portrait

Figure	Content
Fig 1	Temporal evolution of $V(t)$, $C(t)$, $O(t)$, $\Gamma(t)$
Fig 2	Three outcome scenarios (sustainable, zombie, collapse)
Fig 3	Phase portrait with nullclines and equilibria
Fig 4	Bifurcation diagrams: $C^{}$ and $V^{}$ vs $\beta$, $R$
Fig 5	Impact comparison: DLVT vs Human Capital Theory (Becker)
Fig 6	Carrying capacity heatmap $C^{*}_{\max}(\beta, R)$
Fig 7	Regime map in $(\beta, \delta)$ space

Reproducing All Paper Figures

python3 scripts/run_all_figures.py             # All 10 figures (PDF + PNG)
python3 scripts/run_all_figures.py --fig 1-7   # Core only
python3 scripts/run_all_figures.py --fig 8-10  # Extended analysis only

Extended figures (standalone scripts):

python3 scripts/fig8_bifurcation_hysteresis.py    # Hysteresis detection
python3 scripts/fig9_robustness.py                # Structural robustness
python3 scripts/fig10_intervention_comparison.py   # Recovery vs redesign

Running the Tests

pytest tests/ -q          # 38 tests, all must pass
pytest tests/ -v --tb=short  # verbose with tracebacks

The test suite pins every numerical claim in the paper (equilibrium values, carrying capacity, scope-absorption invariant, eigenvalue signs, basin-of-attraction convergence) so that any code change that breaks paper–code consistency is caught immediately.

Project Structure

dlvt/
├── dlvt/                     # Core package
│   ├── __init__.py           # Public API exports
│   ├── model.py              # ODE system, parameters, integration
│   ├── analysis.py           # Equilibria, stability, bifurcation, regimes
│   └── figures.py            # Publication figures 1–7
├── tests/                    # 38 pinning tests
│   ├── test_model.py         # Model numerics, simulation, positive invariance
│   └── test_analysis.py      # Equilibria, carrying capacity, regimes
├── scripts/                  # Figure generation (8–10) and robustness grid
├── figures/                  # Generated output (PDF + PNG)
├── pyproject.toml            # Package metadata and dependencies
├── setup.py                  # Legacy installer
└── LICENSE                   # MIT

Citation

@article{bendinelli2026dlvt,
  title   = {Dynamic Leadership Vitality Theory:
             A Formal Model of the Zombie-Leader Equilibrium},
  author  = {Bendinelli, William},
  year    = {2026},
  note    = {Manuscript submitted to The Leadership Quarterly}
}

License

MIT License — see LICENSE for details.

Copyright 2026 William Bendinelli

Contributing

See CONTRIBUTING.md. Issues, feature requests, and pull requests are welcome.