Module 6 | Climate Change & Biodiversity

Extinction Dynamics

Mass extinction rates, species–area relationships, extinction debt, ecological tipping points, population viability analysis, and climate velocity — the quantitative framework for understanding biodiversity collapse

The Sixth Mass Extinction

Earth has experienced five major mass extinctions over the past 540 million years — events that each eliminated more than 75% of species within geologically brief intervals. Today, extinction rates are estimated at 100–1,000 times the background rate, leading many biologists to conclude we have entered a sixth mass extinction driven primarily by human activities: habitat destruction, overexploitation, pollution, invasive species, and climate change.

This module derives the quantitative tools ecologists use to measure, model, and project extinctions — from the fossil record's background rate through species–area curves, network collapse models, population viability analysis, and climate velocity comparisons.

1. Background Extinction Rate from the Fossil Record

The background extinction rate is the “normal” rate of species loss in the absence of mass extinction events. It is measured in units of E/MSY (extinctions per million species-years).

Derivation: Background Extinction Rate

The fossil record shows that the average lifespan of a marine invertebrate species is approximately $L_{\text{sp}} \approx 5\text{--}10$ million years. If species go extinct at a constant rate, the per-species extinction rate is:

$\mu = \frac{1}{L_{\text{sp}}}$

For $L_{\text{sp}} = 10$ Myr, $\mu = 0.1$ extinctions per species per million years, or 0.1 E/MSY. For a cohort of $S_0$ species:

$E_{\text{background}} = \mu \cdot S_0 = \frac{S_0}{L_{\text{sp}}}$

With an estimated $S_0 \approx 8$ million eukaryotic species alive today, the background rate predicts roughly $8 \times 10^6 / 10^7 = 0.8$ extinctions per year. De Vos et al. (2015) estimated 0.1 E/MSY as a conservative upper bound on background rates, while Pimm et al. (1995) used $\sim 1$ E/MSY.

Current observed rate: Among well-studied vertebrate groups, documented extinctions since 1500 CE yield rates of $\sim 25\text{--}100$ E/MSY for mammals and $\sim 40$ E/MSY for birds. Including estimated undocumented extinctions, Ceballos et al. (2015) found rates $\sim 100$ E/MSY — approximately 1,000 times background.

The Big Five Mass Extinctions

The five major mass extinctions and their estimated species losses:

• End-Ordovician (~443 Ma): ~86% species lost. Glaciation and sea-level drop.
• Late Devonian (~372 Ma): ~75% species lost. Ocean anoxia and cooling.
• End-Permian (~252 Ma): ~96% species lost. Siberian Traps volcanism, ocean acidification, $\sim 10$°C warming.
• End-Triassic (~201 Ma): ~80% species lost. Central Atlantic Magmatic Province volcanism.
• End-Cretaceous (~66 Ma): ~76% species lost. Chicxulub asteroid impact + Deccan Traps volcanism.

The end-Permian extinction is the closest analogue to current climate change — it was driven by massive CO$_2$ release from volcanic activity, causing ocean acidification and warming of $\sim 10$°C. The rate of current CO$_2$ emissions ($\sim 10$ Gt C/yr) is approximately 10 times faster than during the end-Permian (Zeebe et al., 2016).

2. Species–Area Relationship and Extinction Debt

The species–area relationship (SAR) is one of the oldest and most robust patterns in ecology. First described by Arrhenius (1921), it relates the number of species $S$ found in an area $A$ by a power law:

Derivation: Extinction from Habitat Loss

$S = cA^z$

where $c$ is a taxon-specific constant and $z$ is the power-law exponent, typically $z \approx 0.15\text{--}0.35$ (canonical value $z \approx 0.25$for island biogeography; Preston, 1962). If a fraction $f = \Delta A / A$ of habitat is destroyed, the remaining area is $A' = A(1 - f)$, giving:

$S' = c[A(1-f)]^z = cA^z(1-f)^z = S(1-f)^z$

The fraction of species eventually lost is:

$\frac{\Delta S}{S} = 1 - (1-f)^z$

For $z = 0.25$ and $f = 0.5$ (50% habitat loss):

$\frac{\Delta S}{S} = 1 - (0.5)^{0.25} = 1 - 0.841 \approx 0.159$

So losing 50% of habitat eventually leads to ~16% species loss. This is the concept of extinction debt — species committed to extinction but not yet gone, because population decline and local extinction take time (decades to centuries).

Thomas et al. (2004) applied this framework to climate-driven habitat loss. Using climate envelope models for 1,103 species across six regions, they projected the area of suitable climate shrinkage under IPCC warming scenarios. Their central estimate: 15–37% of species committed to extinction by 2050 under mid-range warming ($\sim 2$°C), with the range depending on dispersal assumptions (no dispersal vs. unlimited dispersal).

Extinction Debt: Relaxation Time

The timescale over which extinction debt is “paid” follows an exponential relaxation:

$S(t) = S_{\text{eq}} + (S_0 - S_{\text{eq}}) \, e^{-t/\tau}$

where $S_0$ is the initial species richness, $S_{\text{eq}} = cA'^z$ is the new equilibrium, and $\tau$ is the relaxation time. Diamond (1972) estimated$\tau \sim 50\text{--}10{,}000$ years for birds on land-bridge islands, depending on island size and species generation times. For tropical trees, Wearn et al. (2012) estimated relaxation times of centuries. This means many species alive today are already “committed” to extinction — they are the “living dead.”

3. Tipping Points and Cascading Extinctions

Species do not go extinct in isolation. Ecological communities are networks of interacting species, and the loss of one species can trigger cascading extinctions through the network. This is especially critical in mutualistic networks (pollination, seed dispersal) and food webs.

Derivation: Network Robustness

Consider a bipartite mutualistic network with $S_A$ species in guild A (e.g., plants) and $S_P$ species in guild P (e.g., pollinators). A species goes secondarily extinct when it loses all its mutualistic partners. The robustness$R$ of the network is defined as (Burgos et al., 2007):

$R = 1 - \frac{1}{S_P} \sum_{t=1}^{S_A} S_{\text{ext}}(t)$

where $S_{\text{ext}}(t)$ is the cumulative number of secondary extinctions after removing $t$ species from guild A. $R$ ranges from 0 (immediate total collapse) to 1 (no secondary extinctions). The integral is normalized so that $R = 0.5$corresponds to a linear decline.

Nested networks (where specialist species interact with subsets of the partners of generalists) are more robust than random networks. Bascompte et al. (2003) showed that nestedness is a universal property of plant–pollinator networks. However, even nested networks exhibit abrupt collapse past a critical threshold of primary extinctions.

Keystone Species and Trophic Cascades

Keystone species are those whose removal causes disproportionate effects relative to their abundance. The concept (Paine, 1969) arose from experiments removing the sea star Pisaster ochraceus from intertidal communities, which led to competitive dominance by mussels and loss of 7 out of 15 species.

The interaction strength between species $i$and $j$ in a food web is quantified by the community matrix element:

$a_{ij} = \frac{\partial f_i}{\partial N_j}\bigg|_{N^*}$

where $f_i$ is the per capita growth rate of species $i$ and$N^*$ is the equilibrium. A species is a keystone when$\sum_j |a_{ij}|$ is large relative to its abundance. Climate change can disrupt keystones: for example, warming oceans threaten krill populations in Antarctica, which underpin the entire Southern Ocean food web.

4. Population Viability Analysis (PVA)

The IUCN Red List classifies species into threat categories based on quantitative criteria including population size, rate of decline, and geographic range. Population Viability Analysis (PVA) provides the mathematical framework for assessing extinction risk.

Derivation: Quasi-extinction Probability from a Birth–Death Process

Consider a population of size $N$ subject to stochastic births (rate $b$per individual per time) and deaths (rate $d$ per individual per time). The probability that the population eventually reaches zero starting from size $N_0$ is (Feller, 1968):

$P_{\text{ext}} = \begin{cases} \left(\dfrac{d}{b}\right)^{N_0} & \text{if } b > d \\[10pt] 1 & \text{if } d \geq b \end{cases}$

When $b > d$ (growing population), there is still a non-zero extinction probability due to demographic stochasticity. For example, with $d/b = 0.9$ and$N_0 = 50$: $P_{\text{ext}} = 0.9^{50} \approx 0.005$. But for$N_0 = 10$: $P_{\text{ext}} = 0.9^{10} \approx 0.35$. This dramatic sensitivity to population size drives the concept of minimum viable population (MVP).

Minimum Viable Population (MVP)

The MVP is the smallest population size that gives a specified probability of persistence over a given time horizon. Shaffer (1981) defined it as the population with $\geq 99\%$ probability of persisting for 1,000 years.

For a population with environmental stochasticity (variance $\sigma_e^2$ in growth rate $r$), the mean time to extinction from the diffusion approximation is:

$T_{\text{ext}} \approx \frac{2}{\sigma_e^2}\left[\ln(K) \cdot \ln\!\left(\frac{K}{N_0}\right) + \text{Li}_2\!\left(1 - \frac{N_0}{K}\right)\right]$

A simpler approximation when $N_0 \approx K$ (at carrying capacity):

$T_{\text{ext}} \approx \frac{K}{2\sigma_e^2} \quad \text{(Lande, 1993)}$

To achieve $T_{\text{ext}} \geq 1{,}000$ years with typical $\sigma_e^2 \approx 0.1$, we need $K \geq 200$. Traill et al. (2007) meta-analyzed 212 species and found a median MVP of ~4,169 individuals, consistent with the genetic “50/500 rule” (50 for short-term inbreeding avoidance, 500 for long-term evolutionary potential; Franklin, 1980).

5. Climate Velocity vs. Dispersal Speed

Species that cannot track their shifting climate envelope face extinction. The critical comparison is between the climate velocity (the speed at which isotherms move across the landscape) and the dispersal capacityof species.

Derivation: Climate Velocity

Climate velocity is defined as (Loarie et al., 2009):

$v_{\text{climate}} = \frac{\partial T / \partial t}{|\nabla_{\text{spatial}} T|}$

where $\partial T / \partial t$ is the temporal rate of warming (°C/yr) and$|\nabla_{\text{spatial}} T|$ is the spatial temperature gradient (°C/km). In mountainous regions, the large spatial gradient means isotherms shift slowly (species need only move uphill ~10 m/yr). On flat terrain (tropics, oceans), climate velocity can exceed 1 km/yr because $|\nabla T|$ is small.

Extinction risk is then:

$\text{Risk} = P(v_{\text{dispersal}} < v_{\text{climate}})$

Loarie et al. (2009) found global mean climate velocities of $0.42$ km/yr for current warming trends, but up to $\sim 1.3$ km/yr in flat biomes like flooded grasslands. Many species have maximum dispersal rates well below these velocities:

• Trees: 0.05–0.5 km/yr (Clark, 1998)
• Herbaceous plants: 0.01–0.1 km/yr
• Small mammals: 0.1–1 km/yr
• Amphibians: 0.01–0.1 km/yr
• Birds: 1–10 km/yr

Pecl et al. (2017) documented range shifts averaging 6.1 km/decade poleward and 6.1 m/decade upslope — rates insufficient to track projected warming under RCP8.5, which implies climate velocities of 4–8 km/yr in many regions.

6. Mass Extinction Timeline

7. Current vs. Background Extinction Rates

8. Computational Laboratory

Extinction Dynamics: Species-Area Curves, Network Collapse, PVA & Climate Velocity

Python

script.py336 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ===== 1. SPECIES-AREA EXTINCTION PROJECTIONS =====
print("=" * 65)
print("SPECIES-AREA RELATIONSHIP & EXTINCTION PROJECTIONS")
print("=" * 65)

# SAR parameters
z_values = [0.15, 0.20, 0.25, 0.30, 0.35]
habitat_loss_fractions = np.linspace(0, 0.95, 200)

print(f"\nSpecies-Area Relationship: S = c * A^z")
print(f"Fraction of species lost = 1 - (1 - f)^z, where f = fraction habitat lost\n")
print(f"{'Habitat loss':>14} {'z=0.15':>10} {'z=0.20':>10} {'z=0.25':>10} {'z=0.30':>10} {'z=0.35':>10}")
print("-" * 68)

for f_val in [0.1, 0.2, 0.3, 0.5, 0.7, 0.9]:
    row = f"  {f_val*100:5.0f}%       "
    for z in z_values:
        sp_loss = 1 - (1 - f_val)**z
        row += f"{sp_loss*100:9.1f}%"
    print(row)

# Thomas et al. (2004) projections
print(f"\nThomas et al. (2004) Climate-driven Extinction Projections:")
print(f"  Minimum warming (~1.5C): 15% species committed to extinction")
print(f"  Mid-range warming (~2C):  24% committed")
print(f"  Maximum warming (~3C):   37% committed")

# ===== 2. NETWORK ROBUSTNESS UNDER CASCADING EXTINCTIONS =====
print(f"\n{'=' * 65}")
print("CASCADING EXTINCTIONS IN MUTUALISTIC NETWORKS")
print("=" * 65)

np.random.seed(42)

def create_nested_network(n_plants, n_pollinators, connectance=0.3):
    """Create a nested bipartite network."""
    matrix = np.zeros((n_plants, n_pollinators))
    for i in range(n_plants):
        for j in range(n_pollinators):
            # Nestedness: generalists interact with more partners
            p_link = connectance * (1 + 2*np.exp(-(i+j)/(n_plants+n_pollinators)*3))
            if np.random.random() < min(p_link, 0.95):
                matrix[i, j] = 1
    # Ensure every species has at least one partner
    for i in range(n_plants):
        if matrix[i,:].sum() == 0:
            matrix[i, np.random.randint(n_pollinators)] = 1
    for j in range(n_pollinators):
        if matrix[:,j].sum() == 0:
            matrix[np.random.randint(n_plants), j] = 1
    return matrix

def simulate_cascade(matrix, removal_order):
    """Remove plants in given order, track pollinator extinctions."""
    n_plants, n_pollinators = matrix.shape
    surviving_pollinators = np.ones(n_pollinators, dtype=bool)
    current_matrix = matrix.copy()
    secondary_extinctions = []

for step, plant_idx in enumerate(removal_order):
        current_matrix[plant_idx, :] = 0  # Remove plant
        # Check which pollinators lost all partners
        for j in range(n_pollinators):
            if surviving_pollinators[j] and current_matrix[:, j].sum() == 0:
                surviving_pollinators[j] = False
        ext_count = n_pollinators - surviving_pollinators.sum()
        secondary_extinctions.append(ext_count)

return np.array(secondary_extinctions)

n_plants, n_pollinators = 50, 80
network = create_nested_network(n_plants, n_pollinators)

print(f"\nNetwork: {n_plants} plants, {n_pollinators} pollinators")
print(f"Total links: {int(network.sum())}")
print(f"Connectance: {network.sum()/(n_plants*n_pollinators):.3f}")

# Three removal strategies
# Random removal
random_order = np.random.permutation(n_plants)
ext_random = simulate_cascade(network, random_order)

# Most-connected first (targeted attack)
connectivity = network.sum(axis=1)
targeted_order = np.argsort(-connectivity)
ext_targeted = simulate_cascade(network, targeted_order)

# Least-connected first
reverse_order = np.argsort(connectivity)
ext_reverse = simulate_cascade(network, reverse_order)

# Calculate robustness
frac_removed = np.arange(1, n_plants+1) / n_plants
R_random = 1 - np.trapezoid(ext_random / n_pollinators, frac_removed)
R_targeted = 1 - np.trapezoid(ext_targeted / n_pollinators, frac_removed)
R_reverse = 1 - np.trapezoid(ext_reverse / n_pollinators, frac_removed)

print(f"\nRobustness R (area under survival curve):")
print(f"  Random removal:           R = {R_random:.3f}")
print(f"  Most-connected first:     R = {R_targeted:.3f}")
print(f"  Least-connected first:    R = {R_reverse:.3f}")
print(f"\n  -> Targeted removal of generalists causes fastest collapse")

# ===== 3. POPULATION VIABILITY ANALYSIS =====
print(f"\n{'=' * 65}")
print("POPULATION VIABILITY ANALYSIS (PVA)")
print("=" * 65)

def pva_simulation(N0, r_mean, sigma_d, sigma_e, K, n_years, n_sims):
    """Run PVA with demographic and environmental stochasticity."""
    extinctions = 0
    extinction_times = []
    final_sizes = []

for sim in range(n_sims):
        N = float(N0)
        extinct = False
        for t in range(n_years):
            if N <= 0:
                extinct = True
                extinction_times.append(t)
                break
            # Environmental stochasticity
            r_env = np.random.normal(r_mean, sigma_e)
            # Density dependence (logistic)
            r_dd = r_env * (1 - N / K)
            # Demographic stochasticity
            if N < 1000:
                # Each individual: birth or death
                births = np.random.poisson(max(0, (r_dd + 1)) * N) if N > 0 else 0
                deaths = np.random.poisson(max(0, 1.0) * N) if N > 0 else 0
                N = max(0, N + births - deaths)
            else:
                N = max(0, N * np.exp(r_dd + np.random.normal(0, sigma_d/np.sqrt(N))))
            N = min(N, K * 2)  # Cap at 2K

if extinct:
            extinctions += 1
        else:
            final_sizes.append(N)

p_ext = extinctions / n_sims
    mean_time = np.mean(extinction_times) if extinction_times else n_years
    return p_ext, mean_time, final_sizes

# Test different initial population sizes
N0_values = [10, 20, 50, 100, 200, 500, 1000, 5000]
n_sims = 500
n_years = 100

print(f"\nBirth-death extinction probability: P_ext = (d/b)^N0")
print(f"With d/b = 0.9:")
for N0 in [5, 10, 20, 50, 100]:
    p = 0.9**N0
    print(f"  N0 = {N0:>4d}:  P_ext = {p:.6f}")

print(f"\nStochastic PVA simulations ({n_sims} runs, {n_years} years):")
print(f"  r_mean=0.05, sigma_e=0.15, sigma_d=0.10, K=1000\n")
print(f"{'N0':>8} {'P(extinction)':>16} {'Mean T_ext':>12}")
print("-" * 40)

pva_results = []
for N0 in N0_values:
    p_ext, mean_t, finals = pva_simulation(N0, 0.05, 0.10, 0.15, 1000, n_years, n_sims)
    pva_results.append((N0, p_ext, mean_t))
    t_str = f"{mean_t:.0f} yr" if p_ext > 0.01 else ">100 yr"
    print(f"  {N0:>6d} {p_ext:>15.3f} {t_str:>12}")

# ===== 4. CLIMATE VELOCITY VS DISPERSAL SPEED =====
print(f"\n{'=' * 65}")
print("CLIMATE VELOCITY VS DISPERSAL SPEED")
print("=" * 65)

# Climate velocity by biome
biomes = [
    ("Tropical forest", 0.08, 0.42),
    ("Temperate forest", 0.06, 0.35),
    ("Boreal forest", 0.03, 0.52),
    ("Tropical grassland", 0.02, 1.26),
    ("Temperate grassland", 0.04, 0.59),
    ("Desert", 0.01, 0.71),
    ("Tundra", 0.05, 0.67),
    ("Montane", 0.12, 0.11),
    ("Flooded grassland", 0.015, 1.30),
]

print(f"\nClimate velocity = (dT/dt) / |grad_spatial T|")
print(f"Under current warming trend (~0.02 C/yr):\n")
print(f"{'Biome':<25} {'Spatial grad':>14} {'Climate vel':>14}")
print(f"{'':25} {'(C/km)':>14} {'(km/yr)':>14}")
print("-" * 55)
for name, grad, vel in biomes:
    print(f"  {name:<23} {grad:>12.3f} {vel:>12.2f}")

# Dispersal capacities
species_groups = [
    ("Trees (wind-dispersed)", 0.05, 0.5),
    ("Trees (animal-dispersed)", 0.1, 2.0),
    ("Herbaceous plants", 0.01, 0.1),
    ("Small mammals", 0.1, 1.0),
    ("Large mammals", 0.5, 5.0),
    ("Amphibians", 0.01, 0.1),
    ("Reptiles", 0.05, 0.5),
    ("Birds", 1.0, 10.0),
    ("Insects (flying)", 0.5, 5.0),
]

print(f"\nDispersal capacities:")
print(f"{'Species group':<28} {'Min':>8} {'Max':>8} (km/yr)")
print("-" * 50)
for name, d_min, d_max in species_groups:
    print(f"  {name:<26} {d_min:>7.2f} {d_max:>7.1f}")

# Extinction risk calculation
print(f"\nExtinction risk = P(v_dispersal < v_climate)")
print(f"For flat tropical grassland (v_climate = 1.26 km/yr):")
for name, d_min, d_max in species_groups:
    # Assume log-normal distribution of dispersal
    d_mean = (d_min + d_max) / 2
    d_std = (d_max - d_min) / 4
    v_climate = 1.26
    # Fraction that can't keep up
    from scipy.stats import norm
    if d_std > 0:
        p_risk = norm.cdf(v_climate, d_mean, d_std)
    else:
        p_risk = 1.0 if d_mean < v_climate else 0.0
    print(f"  {name:<28} Risk = {p_risk:.2f}")

# ===== PLOTTING =====
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.suptitle("Extinction Dynamics: Quantitative Analysis", fontsize=16, fontweight='bold', color='#67e8f9')
fig.patch.set_facecolor('#0a0a0a')

# --- Plot 1: Species-Area Extinction Projections ---
ax1 = axes[0, 0]
ax1.set_facecolor('#111111')
colors_z = ['#22d3ee', '#06b6d4', '#0891b2', '#0e7490', '#155e75']
for z, color in zip(z_values, colors_z):
    species_loss = 1 - (1 - habitat_loss_fractions)**z
    ax1.plot(habitat_loss_fractions * 100, species_loss * 100, color=color, linewidth=2, label=f'z = {z:.2f}')

# Thomas et al. projections
ax1.axhspan(15, 37, alpha=0.15, color='red', label='Thomas et al. 2050')
ax1.axvline(50, color='#ef444480', linestyle='--', linewidth=1)
ax1.set_xlabel('Habitat Loss (%)', color='#d1d5db', fontsize=11)
ax1.set_ylabel('Species Committed to Extinction (%)', color='#d1d5db', fontsize=11)
ax1.set_title('Species-Area Extinction Projections', color='#67e8f9', fontsize=13, fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#333', labelcolor='#d1d5db')
ax1.set_xlim(0, 100)
ax1.set_ylim(0, 100)
ax1.tick_params(colors='#9ca3af')
ax1.grid(True, alpha=0.15, color='#333')
for spine in ax1.spines.values():
    spine.set_color('#333')

# --- Plot 2: Network Robustness ---
ax2 = axes[0, 1]
ax2.set_facecolor('#111111')
frac_primary = np.arange(1, n_plants+1) / n_plants
ax2.plot(frac_primary * 100, ext_random / n_pollinators * 100, color='#22d3ee', linewidth=2, label=f'Random (R={R_random:.2f})')
ax2.plot(frac_primary * 100, ext_targeted / n_pollinators * 100, color='#ef4444', linewidth=2, label=f'Targeted (R={R_targeted:.2f})')
ax2.plot(frac_primary * 100, ext_reverse / n_pollinators * 100, color='#22c55e', linewidth=2, label=f'Least-connected (R={R_reverse:.2f})')
ax2.plot([0, 100], [0, 100], '--', color='#666', linewidth=1, label='Linear decline')
ax2.set_xlabel('Primary Extinctions (% plants removed)', color='#d1d5db', fontsize=11)
ax2.set_ylabel('Secondary Extinctions (% pollinators lost)', color='#d1d5db', fontsize=11)
ax2.set_title('Cascading Extinctions in Mutualistic Network', color='#67e8f9', fontsize=13, fontweight='bold')
ax2.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#333', labelcolor='#d1d5db')
ax2.set_xlim(0, 100)
ax2.set_ylim(0, 100)
ax2.tick_params(colors='#9ca3af')
ax2.grid(True, alpha=0.15, color='#333')
for spine in ax2.spines.values():
    spine.set_color('#333')

# --- Plot 3: PVA - Extinction probability vs initial population size ---
ax3 = axes[1, 0]
ax3.set_facecolor('#111111')
N0s = [r[0] for r in pva_results]
p_exts = [r[1] for r in pva_results]
ax3.semilogx(N0s, p_exts, 'o-', color='#22d3ee', linewidth=2, markersize=8, markerfacecolor='#06b6d4')
ax3.axhline(0.01, color='#22c55e', linestyle='--', linewidth=1, label='1% threshold (IUCN CR)')
ax3.axhline(0.10, color='#f97316', linestyle='--', linewidth=1, label='10% threshold (IUCN EN)')
ax3.axvline(500, color='#a855f7', linestyle=':', linewidth=1, label='50/500 rule: N=500')
ax3.set_xlabel('Initial Population Size (N0)', color='#d1d5db', fontsize=11)
ax3.set_ylabel('P(extinction in 100 yr)', color='#d1d5db', fontsize=11)
ax3.set_title('Population Viability Analysis', color='#67e8f9', fontsize=13, fontweight='bold')
ax3.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#333', labelcolor='#d1d5db', loc='upper right')
ax3.set_ylim(-0.02, 1.02)
ax3.tick_params(colors='#9ca3af')
ax3.grid(True, alpha=0.15, color='#333')
for spine in ax3.spines.values():
    spine.set_color('#333')

# --- Plot 4: Climate Velocity vs Dispersal ---
ax4 = axes[1, 1]
ax4.set_facecolor('#111111')
biome_names = [b[0] for b in biomes]
biome_vels = [b[2] for b in biomes]
biome_colors = ['#22d3ee' if v < 0.5 else '#f97316' if v < 1.0 else '#ef4444' for v in biome_vels]
y_pos = np.arange(len(biomes))
ax4.barh(y_pos, biome_vels, color=biome_colors, alpha=0.7, height=0.6)
ax4.set_yticks(y_pos)
ax4.set_yticklabels(biome_names, fontsize=9, color='#d1d5db')
ax4.set_xlabel('Climate Velocity (km/yr)', color='#d1d5db', fontsize=11)
ax4.set_title('Climate Velocity by Biome', color='#67e8f9', fontsize=13, fontweight='bold')

# Add dispersal reference lines
ax4.axvline(0.1, color='#ef444480', linestyle='--', linewidth=1)
ax4.text(0.12, len(biomes)-0.5, 'Trees', color='#ef4444', fontsize=8)
ax4.axvline(1.0, color='#22c55e80', linestyle='--', linewidth=1)
ax4.text(1.05, len(biomes)-0.5, 'Small mammals', color='#22c55e', fontsize=8)
ax4.tick_params(colors='#9ca3af')
ax4.grid(True, alpha=0.15, color='#333', axis='x')
for spine in ax4.spines.values():
    spine.set_color('#333')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()

print(f"\n{'=' * 65}")
print("SUMMARY")
print("=" * 65)
print("PLOTS:")
print("  Top-left: Species-area extinction projections for different z values")
print("  Top-right: Cascading extinctions under random vs targeted removal")
print("  Bottom-left: Population viability analysis - extinction probability")
print("  Bottom-right: Climate velocity by biome with dispersal thresholds")
print("=" * 65)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Module Summary

• Background extinction rate is ~0.1 E/MSY; current rates are 100–1,000 times higher, qualifying as a mass extinction event comparable to the Big Five.
• Species–area relationship $S = cA^z$ predicts that 50% habitat loss eventually eliminates ~16% of species ($z = 0.25$); Thomas et al. (2004) projected 15–37% species committed to extinction by 2050.
• Cascading extinctions in mutualistic networks show that targeted removal of highly connected species causes rapid collapse; network robustness $R$ quantifies resistance to cascading failure.
• Population Viability Analysis shows extinction probability $P_{\text{ext}} = (d/b)^{N_0}$ is extremely sensitive to population size, motivating MVP concepts and the 50/500 rule.
• Climate velocity $v = (\partial T/\partial t)/|\nabla T|$ exceeds dispersal capacity for many taxa, especially in flat biomes; species that cannot track their climate envelope face extinction.

Key References

• Arrhenius, O. (1921). Species and area. J. Ecology, 9(1), 95–99.
• Bascompte, J. et al. (2003). The nested assembly of plant–animal mutualistic networks. PNAS, 100(16), 9383–9387.
• Burgos, E. et al. (2007). Why nestedness in mutualistic networks? J. Theor. Biol., 249(2), 307–313.
• Ceballos, G. et al. (2015). Accelerated modern human–induced species losses. Science Advances, 1(5), e1400253.
• De Vos, J.M. et al. (2015). Estimating the normal background rate of species extinction. Conservation Biology, 29(2), 452–462.
• Lande, R. (1993). Risks of population extinction from demographic and environmental stochasticity. Am. Nat., 142(6), 911–927.
• Loarie, S.R. et al. (2009). The velocity of climate change. Nature, 462(7276), 1052–1055.
• Pimm, S.L. et al. (1995). The future of biodiversity. Science, 269(5222), 347–350.
• Thomas, C.D. et al. (2004). Extinction risk from climate change. Nature, 427(6970), 145–148.
• Traill, L.W. et al. (2007). Minimum viable population size: A meta-analysis of 30 years of published estimates. Biol. Conserv., 139(1–2), 159–166.
• Zeebe, R.E. et al. (2016). Anthropogenic carbon release rate unprecedented during the past 66 million years. Nature Geoscience, 9(4), 325–329.

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