Complex System Simulation Course Project
MSc Computational Science | University of Amsterdam & VU Amsterdam
January 2026
Species-Area Relationships (SAR) describe how biodiversity scales with habitat size, a fundamental pattern in ecology often exhibiting power-law behavior:
Building on the baseline model by García Martín & Goldenfeld (2006) [1], we extend classical SAR theory by integrating:
- Habitat loss via Extinction-Area Relationships (EAR)
- Resource competition through spatial resource allocation dynamics
Our simulations reveal that while power-law SAR emerges elegantly from spatial clustering and lognormal abundance, it is fragile under realistic ecological pressures with important implications for conservation biology.
Hypothesis: Power-law emerges from the combination of spatial clustering (fractal branching patterns) and lognormal species abundance distribution.
Approach: Implement fractal tree generation with clustering parameter α, compute proximity-based species detection, validate against theoretical predictions.
Hypothesis: Classical SAR gives wrong estimates for extinctions by ignoring the number of individuals necessary for a species to survive and rarity. Extinction-Area Relationship (EAR) models, which account for critical abundance thresholds, provide better predictions (Kitzes & Hart, 2014) [2].
Approach: Compare SAR vs. EAR extinction predictions under varying habitat loss scenarios, using minimum viable population sizes (
Hypothesis: Power-law SAR persists but with modified parameters as interspecific competition reshapes spatial distributions and abundance patterns.
Approach: Implement grid-based resource competition model with gamma-distributed resources, competition radii, and survival thresholds.
Following García Martín & Goldenfeld (2006) [1], we generate species distributions using self-similar fractal trees:
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Spatial clustering: Individuals of each species cluster according to a branching process controlled by parameter
$\alpha$ $(0.5 \leq \alpha$ \leq 1$)- Each tree starts at a random position within diameter d
- Branches bifurcate with probability determined by
$\alpha$ - Branch length decreases geometrically:
$l_n = l_0 · 1.5^{-(n-1)}$ - Branching angle varies:
$\theta ± \delta_{max}$ , where$\delta_{max}$ depends on iteration depth
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Species-Area Relationship: Computed via proximity function approach
$S_C(R) = \sum F_s(R) = ∫ h(R,c) p(c) dc$ Where:-
$F_s(R)$ : Probability that species s is within distance R of a random sample point -
$h(R,c)$ : Detection probability for species with cover c -
$p(c)$ : Lognormal abundance distribution
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Implementation: Sample N random points in region
$\Omega$ , compute minimum distance to each species, count species within radius$R$ , average over samples.
Expected result:
Classical SAR assumes species go extinct when all individuals are removed. EAR improves this by modeling extinction probability based on critical abundance:
-
Critical abundance (
$n_c$ ): Minimum viable population size below which extinction occurs (Traill et al., 2014) [3]. -
Extinction probability: For species with
$n_0$ initial individuals:$\epsilon(a|n_c, n_0) = \frac{q^{n_c +1} - 1}{q^{n_0 +1} - 1}$ . -
Parameter estimation: Solve for q using numerical root-finding (bisection method) to match observed extinction patterns:
$a n_0 = \frac{q}{1-q}-\frac{(n_0 + 1) q^{n_0+1}}{1-q^{n_0+1}}$ , with$a$ being the fractional area loss.
Expected result: EAR predicts that the extinction probability increases faster as the habitat decreases than classical SAR, with the extinction probability increasing as the critical abundace becomes larger.
Grid-based spatial competition model:
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Resource distribution: Resources spread on grid according to gamma distribution
$\Gamma(k, \theta)$ -
Species competition: Each species has competition coefficient
$\beta_s$ -
Competition radius: Species within distance
$r_{comp}$ compete for resources - Survival rule: Individuals with resources below threshold T die
- SAR computation: Calculate S(A) for surviving populations across different grid sizes
Expected result: Power-law persists but with altered exponent z and modified abundance distributions.
plots.py contains all configuration for plotting the final data.
habitat.py contains any code related to extinction.
main.py is the main entry point of the project, and handles the species generation alongside the needed analysis for the core SAR analysis and resource competition.
AI-usage.md is the markdown which describes when and how we used AI assistance.
Team 6 CSS is the presentation file, that can also be found at this link
The following packages are required to run the modelling:
- numpy
- matplotlib
- scikit-learn
- scipy
- Power-law SAR successfully reproduced:
$S(A) \sim A^{0.36}, (R^2 = 1.00)$ - Clustering (fractal branching) is necessary condition for scale-free behavior
- Confirms García Martín & Goldenfeld (2006) theoretical framework
Habitat loss disrupts power-law
- EAR models provide more accurate extinction predictions
- Measures are fully based on area loss, initial number of individuals and critical abundance.
Resource competition modifies equilibria
- Species interactions reshape spatial patterns
- Power-law persists but with altered parameters
[1] García Martín, H., & Goldenfeld, N. (2006). On the origin and robustness of power-law species–area relationships in ecology. Proceedings of the National Academy of Sciences, 103(27), 10310–10315. https://doi.org/10.1073/pnas.0510605103
[2] Kitzes, J., & Harte, J. (2013). Beyond the species-area relationship: improving macroecological extinction estimates. Methods in Ecology and Evolution, 5(1), 1–8. https://doi.org/10.1111/2041-210x.12130
[3] Traill, L. W., Bradshaw, C. J. A., & Brook, B. W. (2007). Minimum viable population size: A meta-analysis of 30 years of published estimates. Biological Conservation, 139(1-2), 159–166. https://doi.org/10.1016/j.biocon.2007.06.011
This project utilized AI as a coding assistant, mainly for documentation structure and debugging. AI strategy is available at AI-usage.md.