Module 7

Symbiosis & Chemical Ecology

Fungus farming, aphid husbandry, ant-plant mutualisms, and zombie ant parasitism

Ants are the world's most prolific mutualists and farmers. Leaf-cutter ants invented agriculture approximately 60 million years ago — some 30 million years before humans. Ant-aphid farming, ant-plant defensive mutualisms, and the terrifying parasitic manipulation by Ophiocordyceps fungi all represent extremes of coevolution, driven by chemical signaling and precisely tuned cost-benefit calculations. In this module we derive the biophysics underlying these extraordinary relationships.

7.1 Fungus Farming (Leaf-Cutters)

The attine ants (tribe Attini), particularly the genera Atta and Acromyrmex, cultivate obligate fungal mutualists of the family Agaricaceae (primarily Leucoagaricus gongylophorus). This agricultural system evolved approximately 60 million years ago in South America and represents one of the most complex symbioses in nature, involving at least four organisms: the ant, the cultivated fungus, a parasitic fungus (Escovopsis), and antibiotic-producing bacteria (Pseudonocardia).

The Farming Pipeline

Leaf Cutting: Large workers (majors, head width ~4 mm) cut leaf fragments using mandibles that vibrate at ~1 kHz, creating a stridulatory signal that may stiffen the leaf via acoustic effects. Cutting force: ~50 mN.
Transport: Medias carry leaf fragments (often 10–50x their body mass) back to the nest along cleared trails spanning up to 250 m. Minim workers hitchhike on leaves to defend against parasitoid phorid flies.
Processing: Inside the nest, smaller workers chew leaves into a moist pulp, add fecal droplets containing enzymes, and incorporate the material into the fungus garden matrix.
Cultivation: The fungus grows on the leaf substrate, producing gongylidia — nutrient-rich hyphal swellings (50–100 \(\mu\)m diameter) that serve as the primary food source for the colony.
Harvesting: Workers selectively harvest gongylidia and feed them to larvae and the queen. Gongylidia contain lipids (40%), carbohydrates (30%), and proteins (20%) by dry weight.

Energetic Efficiency of Fungus Farming

The energetic efficiency of the leaf-cutter farming system can be derived by tracking energy flow through the pipeline. Let \(E_{\text{leaf}}\) be the energy content of harvested leaf material:

Energy Budget

\[ E_{\text{leaf}} \approx 18 \text{ kJ/g (dry weight, typical leaf)} \]

The fungal conversion efficiency (substrate to gongylidia biomass):

\[ \eta_{\text{fungus}} = \frac{m_{\text{gongylidia}} \cdot E_{\text{gong}}}{m_{\text{substrate}} \cdot E_{\text{leaf}}} \approx 0.10{-}0.15 \]

Including metabolic costs of the ants (cutting, transport, processing):

\[ \eta_{\text{total}} = \frac{E_{\text{gongylidia harvested}}}{E_{\text{leaf}} + E_{\text{ant metabolism}}} \]

\[ E_{\text{ant metabolism}} = N_{\text{workers}} \cdot \dot{E}_{\text{worker}} \cdot T_{\text{cycle}} \]

With a worker metabolic rate of \(\dot{E}_{\text{worker}} \approx 10\) mW and a processing cycle of ~24 hours, a mature Atta colony (5 million workers, with ~20% engaged in farming) processes ~200 kg of leaf material per year with an overall efficiency of \(\eta_{\text{total}} \approx 6{-}8\%\). This is comparable to modern human agriculture (5–10% for livestock).

Antibiotic Defense System

The fungus garden is vulnerable to the specialized parasite Escovopsis, a mycoparasitic fungus that can destroy gardens within days. The ants maintain a three-layer defense:

Weeding

Workers constantly patrol the garden, detecting and removing Escovopsishyphae mechanically. Detection is chemical: Escovopsis volatiles trigger grooming behavior at concentrations as low as 1 ppb.

Metapleural Gland

The metapleural gland (unique to ants) secretes phenylacetic acid and other antimicrobial compounds. Workers spread these over the garden substrate. Production rate: ~0.1 \(\mu\)L/day per worker.

Pseudonocardia Bacteria

Actinomycete bacteria (Pseudonocardia) grow on the ant cuticle in specialized crypts. They produce dentigerumycin and other antifungals that specifically target Escovopsis while sparing the cultivar.

7.2 Aphid Farming

Many ant species maintain mutualistic relationships with phloem-feeding hemipterans, particularly aphids. The ants protect aphids from predators (ladybird beetles, lacewing larvae, parasitoid wasps) and in return harvest honeydew — a sugar-rich excretion that constitutes a major energy source for the colony. This relationship has evolved independently in dozens of ant lineages, indicating strong selective advantages on both sides.

Honeydew Composition and Energetics

Aphid honeydew contains primarily sugars (sucrose, glucose, fructose, trehalose) at concentrations of 10–50% by weight, plus amino acids and lipids in trace amounts. A single aphid produces approximately 0.5–1.0 \(\mu\)L of honeydew per hour. The energy content per droplet:

\[ E_{\text{droplet}} = V \cdot \rho \cdot c_{\text{sugar}} \cdot \epsilon_{\text{sugar}} \]

\[ \approx 0.75 \,\mu\text{L} \times 1.1 \,\text{g/mL} \times 0.30 \times 16.7 \,\text{kJ/g} \approx 4.1 \,\text{mJ/droplet} \]

Chemical Manipulation

The mutualism is not entirely benign. Several ant species chemically manipulate their aphid “livestock” through compounds in their trail pheromone:

Wing suppression: The ant trail pheromone (containing \(\beta\)-farnesene, which mimics the aphid alarm pheromone) suppresses wing development in aphids, keeping them sessile and accessible. Wingless aphids produce 20–30% more honeydew than winged morphs.
Movement restriction: Some species (Lasius niger) clip aphid wings or apply chemical “tranquilizers” from the mandibular gland that reduce aphid walking speed by up to 50%.
Culling: Ants selectively consume aphids that are poor honeydew producers or that carry parasitoid eggs, effectively practicing artificial selection.

Cost-Benefit from Kin Selection Perspective

From the ant colony perspective, the investment in aphid tending must satisfy a cost-benefit condition. Let \(N_t\) be the number of tending workers,\(N_a\) the number of aphids, and \(\dot{E}_h\) the honeydew energy rate per aphid:

Mutualism Fitness Condition

\[ \underbrace{N_a \cdot \dot{E}_h \cdot \eta_{\text{harvest}}}_{\text{Benefit: energy from honeydew}} > \underbrace{N_t \cdot \dot{E}_{\text{forage}}}_{\text{Opportunity cost: lost foraging}} + \underbrace{N_t \cdot \dot{E}_{\text{defense}}}_{\text{Cost: predator defense}} \]

Dividing by \(N_t\) and defining the aphid-to-tender ratio\(R = N_a / N_t\):

\[ R > \frac{\dot{E}_{\text{forage}} + \dot{E}_{\text{defense}}}{\dot{E}_h \cdot \eta_{\text{harvest}}} \]

Empirical measurements show that profitable tending requires \(R > 5{-}10\)aphids per tender ant, consistent with observed ratios of 8–15 in Lasius niger colonies (Stadler & Dixon, 2005).

7.3 Myrmecophytes (Ant-Plant Mutualism)

Myrmecophytes are plants that have evolved specialized structures to house and feed ant colonies, receiving protection from herbivores in return. The best-studied system involves Cecropia trees and Azteca ants in the Neotropics, but similar mutualisms have evolved independently in over 100 plant genera across tropical ecosystems worldwide (e.g., Acacia-Pseudomyrmex in Central America, Macaranga-Crematogaster in Southeast Asia).

Plant Investment

Structural: Domatia

Hollow stems or swollen thorns providing nesting space. Cecropia internodes are hollow with thin internal septa that ants can penetrate. Construction cost: ~5% of stem biomass invested in cavity volume rather than structural wood.

Nutritional: Food Bodies

Müllerian bodies (Cecropia): glycogen-rich trichomes produced at the petiole base. Beltian bodies (Acacia): protein and lipid-rich leaflet tips. Pearl bodies: surface exudates on various myrmecophytes. Cost: 1–3% of total photosynthate.

Game-Theoretic Analysis: Nash Equilibrium

The ant-plant mutualism can be modeled as a two-player cooperative game. Let\(x\) be the plant's investment in ant rewards (food bodies + domatia volume) and \(y\) be the ant colony's investment in plant defense (worker allocation to patrolling). The fitness functions are:

Plant Fitness

\[ W_{\text{plant}}(x, y) = G(1 - x) \cdot \left[1 - H \cdot e^{-\lambda y}\right] \]

where \(G(1-x)\) is the growth rate with fraction \((1-x)\) of photosynthate retained, \(H\) is the herbivory damage rate without protection, and \(e^{-\lambda y}\) is the protection factor (exponentially decreasing herbivory with ant patrol effort).

Ant Colony Fitness

\[ W_{\text{ant}}(x, y) = F(x) \cdot (1 - y) + D(x) \]

where \(F(x)\) is the food reward (increasing in \(x\)),\((1 - y)\) is the fraction of workers available for colony growth, and\(D(x)\) is the domatia benefit.

Nash Equilibrium

At the Nash equilibrium, neither partner benefits from unilaterally changing its investment:

\[ \frac{\partial W_{\text{plant}}}{\partial x}\bigg|_{(x^*, y^*)} = 0 \quad \text{and} \quad \frac{\partial W_{\text{ant}}}{\partial y}\bigg|_{(x^*, y^*)} = 0 \]

For the plant: the marginal cost of providing rewards must equal the marginal benefit of additional defense:

\[ G'(1 - x^*) \cdot [1 - H e^{-\lambda y^*}] = G(1 - x^*) \cdot H \lambda \frac{\partial y^*}{\partial x} \cdot e^{-\lambda y^*} \]

Empirical data from Cecropia-Azteca systems show equilibrium values of \(x^* \approx 3{-}5\%\) (plant investment) and \(y^* \approx 20{-}30\%\)(ant defense allocation), consistent with model predictions (Fonseca 1994).

7.4 Ophiocordyceps: The Zombie Ant Fungus

Ophiocordyceps unilateralis sensu lato is a parasitic fungus that manipulates the behavior of its carpenter ant host (Camponotus spp.) in one of nature's most dramatic examples of extended phenotype manipulation. The fungal infection unfolds over 2–3 weeks and culminates in a precisely orchestrated “death grip” that positions the ant for optimal spore dispersal.

Infection Timeline

Day 0: Spore attachment — ascospore lands on ant cuticle, germinates, and penetrates via enzymatic degradation of chitin.
Days 1–14: Internal growth — fungal cells proliferate within the hemocoel, consuming ~40% of host soft tissue while carefully avoiding the brain and muscles. The ant continues normal colony behavior.
Day ~14: Behavioral manipulation — the fungus secretes compounds (including guanobutyric acid and sphingosine) that alter ant behavior. The infected ant leaves the canopy trail and descends to a specific height (25 ± 3 cm) above the forest floor — optimal for spore dispersal.
Day ~15: Death grip — at solar noon, the ant bites into a leaf vein on the north side of the leaf (in the Northern Hemisphere). The mandibular muscles are infiltrated by fungal cells that cause lockjaw. The ant dies within hours.
Days 15–25: Fruiting — the fungal stroma (stalk) erupts from the back of the ant's head, grows to 2–3 cm, and releases ascospores from a capsule at the tip.

Optimal Transmission Height: Spore Dispersal Physics

The fungus must position the ant at a height that maximizes spore dispersal to the forest floor where foraging ants walk. A spore released from height \(h\) with stalk length \(\ell\) experiences gravity and air drag:

Spore Trajectory

For a spherical spore of radius \(r_s \approx 5\) \(\mu\)m and density \(\rho_s \approx 1100\) kg/m\(^3\) in air, the terminal settling velocity is:

\[ v_t = \frac{2 r_s^2 (\rho_s - \rho_a) g}{9 \mu_a} \approx 3 \text{ mm/s} \]

With a horizontal wind velocity \(u(z)\) that follows a logarithmic boundary layer profile near the forest floor:

\[ u(z) = \frac{u_*}{\kappa} \ln\!\left(\frac{z}{z_0}\right) \]

where \(u_* \approx 0.05\) m/s is the friction velocity,\(\kappa = 0.41\) is the von Kármán constant, and\(z_0 \approx 0.01\) m is the roughness length. The horizontal dispersal distance for a spore released at height \(h + \ell\) is:

\[ x_{\text{disp}} = \int_0^{(h+\ell)/v_t} u(z(t))\, dt = \frac{u_*}{\kappa v_t} \int_0^{h+\ell} \ln\!\left(\frac{z}{z_0}\right) dz \]

Optimal Height for Infection

The probability of infecting an ant depends on: (1) the spore landing area\(\propto x_{\text{disp}}^2\) and (2) the spore concentration at ground level\(\propto 1/x_{\text{disp}}^2\). The infection probability per unit area is:

\[ P_{\text{inf}}(h) \propto \frac{N_{\text{spores}} \cdot A_{\text{ant}}}{\pi x_{\text{disp}}(h)^2} \cdot \rho_{\text{ant}}(0) \cdot \pi x_{\text{disp}}(h)^2 \]

This simplifies to \(P_{\text{inf}} \propto N_{\text{spores}} \cdot A_{\text{ant}} \cdot \rho_{\text{ant}}(0)\), which is independent of height! The height optimum arises from additional factors: humidity requirements (the fungus needs\(> 95\%\) RH for fruiting, which decreases with height) and temperature stability (less fluctuation near the ground). The observed 25 cm height represents the tradeoff between sufficient wind for dispersal and sufficient humidity for fruiting:

\[ h^* = \arg\max_h \left[ x_{\text{disp}}(h) \cdot P_{\text{fruit}}(h) \right] \]

\[ P_{\text{fruit}}(h) = e^{-h/h_{\text{humid}}} \quad \text{where } h_{\text{humid}} \approx 30 \text{ cm} \]

7.5 Leaf-Cutter Fungus Garden System

The diagram below illustrates the complete leaf-cutter ant farming system, from leaf cutting through fungal cultivation to waste management, including the antibiotic defense system.

7.6 Simulation: Fungus Garden Productivity & Spore Dispersal

This simulation models three aspects of ant symbiosis: (1) nutrient cycling in a leaf-cutter fungus garden, (2) spore dispersal physics for Ophiocordyceps, and (3) cost-benefit analysis of aphid farming.

Fungus Garden, Spore Dispersal & Aphid Farming

Python

script.py254 lines

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

fig, axes = plt.subplots(2, 2, figsize=(16, 12))
fig.suptitle('Ant Symbiosis: Biophysics of Mutualism & Parasitism',
             fontsize=16, color='#fca5a5', fontweight='bold')

# ========== Panel 1: Fungus Garden Nutrient Cycling ==========
ax1 = axes[0, 0]

# Simulate fungus garden dynamics over 365 days
days = np.arange(0, 365)
leaf_input_rate = 0.5 + 0.3 * np.sin(2 * np.pi * days / 365)  # kg/day, seasonal

# Garden biomass dynamics
substrate = np.zeros(365)
fungus = np.zeros(365)
gongylidia = np.zeros(365)
waste = np.zeros(365)

substrate[0] = 5.0   # kg
fungus[0] = 2.0      # kg
gongylidia[0] = 0.5   # kg
waste[0] = 1.0       # kg

k_conversion = 0.12  # substrate to fungus
k_gong = 0.08        # fungus to gongylidia
k_harvest = 0.15     # gongylidia harvest rate
k_decay = 0.02       # fungus to waste
k_waste_removal = 0.05

for i in range(1, 365):
    ds = leaf_input_rate[i] - k_conversion * substrate[i-1]
    df = k_conversion * substrate[i-1] - k_gong * fungus[i-1] - k_decay * fungus[i-1]
    dg = k_gong * fungus[i-1] - k_harvest * gongylidia[i-1]
    dw = k_decay * fungus[i-1] + k_harvest * gongylidia[i-1] * 0.1 - k_waste_removal * waste[i-1]

substrate[i] = max(0, substrate[i-1] + ds)
    fungus[i] = max(0, fungus[i-1] + df)
    gongylidia[i] = max(0, gongylidia[i-1] + dg)
    waste[i] = max(0, waste[i-1] + dw)

ax1.plot(days, substrate, color='#22c55e', linewidth=2, label='Leaf substrate')
ax1.plot(days, fungus, color='#a855f7', linewidth=2, label='Fungal biomass')
ax1.plot(days, gongylidia, color='#fbbf24', linewidth=2, label='Gongylidia')
ax1.plot(days, waste, color='#6b7280', linewidth=1.5, linestyle='--', label='Waste')
ax1.fill_between(days, 0, leaf_input_rate * 10, alpha=0.1, color='#22c55e', label='Leaf input (scaled)')

ax1.set_xlabel('Day of Year', fontsize=11, color='white')
ax1.set_ylabel('Biomass (kg)', fontsize=11, color='white')
ax1.set_title('Fungus Garden Nutrient Cycling', fontsize=13, color='#fca5a5', fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#ef4444', labelcolor='white', loc='upper right')
ax1.set_facecolor('#0a0a0a')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.15, color='white')

# ========== Panel 2: Ophiocordyceps Spore Dispersal ==========
ax2 = axes[0, 1]

heights = np.linspace(0.05, 1.5, 200)  # meters above ground
r_s = 5e-6       # spore radius (m)
rho_s = 1100     # spore density (kg/m3)
rho_a = 1.2      # air density
mu_a = 1.8e-5    # air viscosity (Pa*s)
g = 9.81

# Terminal velocity (Stokes)
v_t = 2 * r_s**2 * (rho_s - rho_a) * g / (9 * mu_a)

# Wind profile (log boundary layer)
u_star = 0.05    # friction velocity (m/s)
kappa = 0.41
z0 = 0.01        # roughness length (m)

# Dispersal distance vs height
def dispersal_distance(h):
    # Integrate u(z) * dz / v_t from z=0 to h
    # Integral of ln(z/z0) dz from z0 to h = h*ln(h/z0) - h + z0
    integral = h * np.log(h / z0) - h + z0
    return u_star / (kappa * v_t) * integral

x_disp = np.array([dispersal_distance(h) for h in heights])

# Humidity decay with height
h_humid = 0.30  # humidity scale height (m)
P_fruit = np.exp(-heights / h_humid)

# Infection probability (dispersal area * ant density * fruiting probability)
ant_density = 50  # ants per m2
P_inf = x_disp * P_fruit * ant_density

# Normalize
P_inf_norm = P_inf / P_inf.max()

ax2.plot(heights * 100, x_disp, color='#60a5fa', linewidth=2, label='Dispersal distance (m)')
ax2_twin = ax2.twinx()
ax2_twin.plot(heights * 100, P_fruit, color='#22c55e', linewidth=1.5, linestyle='--', label='Fruiting prob.')
ax2_twin.plot(heights * 100, P_inf_norm, color='#ef4444', linewidth=2.5, label='Infection score')
ax2_twin.set_ylabel('Probability / Score', fontsize=10, color='white')
ax2_twin.tick_params(colors='white')

# Mark optimal height
h_opt_idx = np.argmax(P_inf)
h_opt = heights[h_opt_idx]
ax2.axvline(h_opt * 100, color='#ef4444', linestyle=':', alpha=0.7)
ax2.text(h_opt * 100 + 3, x_disp.max() * 0.9,
         f'Optimal: {h_opt*100:.0f} cm', fontsize=10, color='#fca5a5')

# Observed range
ax2.axvspan(22, 28, alpha=0.2, color='#ef4444')
ax2.text(25, x_disp.max() * 0.1, 'Observed\nrange', ha='center', fontsize=9, color='#fca5a5')

ax2.set_xlabel('Height above ground (cm)', fontsize=11, color='white')
ax2.set_ylabel('Dispersal distance (m)', fontsize=11, color='#60a5fa')
ax2.set_title('Ophiocordyceps Spore Dispersal', fontsize=13, color='#fca5a5', fontweight='bold')
ax2.set_facecolor('#0a0a0a')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.15, color='white')

lines1, labels1 = ax2.get_legend_handles_labels()
lines2, labels2 = ax2_twin.get_legend_handles_labels()
ax2.legend(lines1 + lines2, labels1 + labels2, fontsize=8, facecolor='#1a1a1a',
           edgecolor='#ef4444', labelcolor='white', loc='upper left')

# ========== Panel 3: Aphid Farming Cost-Benefit ==========
ax3 = axes[1, 0]

# Aphid-to-tender ratio
R = np.linspace(1, 30, 200)

# Energy rates
E_honeydew = 4.1e-3  # J per droplet
droplets_per_hour = 1.0
E_h = E_honeydew * droplets_per_hour  # J/hour per aphid
eta_harvest = 0.85

E_forage = 3.0e-3    # J/hour opportunity cost
E_defense = 1.0e-3   # J/hour defense cost

# Net benefit per tender ant
net_benefit = R * E_h * eta_harvest - (E_forage + E_defense)

# Breakeven ratio
R_breakeven = (E_forage + E_defense) / (E_h * eta_harvest)

ax3.plot(R, net_benefit * 1000, color='#ef4444', linewidth=2.5)
ax3.axhline(0, color='#6b7280', linewidth=1, linestyle='--')
ax3.axvline(R_breakeven, color='#fbbf24', linewidth=1.5, linestyle=':', alpha=0.8)
ax3.fill_between(R, net_benefit * 1000, 0, where=net_benefit > 0, alpha=0.15, color='#22c55e')
ax3.fill_between(R, net_benefit * 1000, 0, where=net_benefit < 0, alpha=0.15, color='#ef4444')

ax3.text(R_breakeven + 0.5, net_benefit.max() * 1000 * 0.8,
         f'Breakeven\nR = {R_breakeven:.1f}', fontsize=10, color='#fbbf24')
ax3.text(20, net_benefit.max() * 1000 * 0.5, 'Profitable\nregion', fontsize=11, color='#86efac', ha='center')
ax3.text(3, -0.5, 'Loss\nregion', fontsize=11, color='#fca5a5', ha='center')

# Observed range
ax3.axvspan(8, 15, alpha=0.15, color='#fbbf24')
ax3.text(11.5, net_benefit.max() * 1000 * 0.3, 'Observed\n(8-15)', fontsize=9, color='#fbbf24', ha='center')

ax3.set_xlabel('Aphids per Tender Ant (R)', fontsize=11, color='white')
ax3.set_ylabel('Net Benefit (mJ/hour per tender)', fontsize=11, color='white')
ax3.set_title('Aphid Farming Cost-Benefit', fontsize=13, color='#fca5a5', fontweight='bold')
ax3.set_facecolor('#0a0a0a')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.15, color='white')

# ========== Panel 4: Ant-Plant Mutualism Game Theory ==========
ax4 = axes[1, 1]

# Plant investment (x) vs ant defense (y) Nash equilibrium
x_vals = np.linspace(0.001, 0.15, 100)
y_vals = np.linspace(0.001, 0.60, 100)
X, Y = np.meshgrid(x_vals, y_vals)

# Plant fitness: G(1-x) * (1 - H * exp(-lambda*y))
G0 = 1.0
H = 0.8
lam = 5.0

W_plant = G0 * (1 - X) * (1 - H * np.exp(-lam * Y))

# Ant fitness: F(x) * (1-y) + D(x)
F0 = 2.0
D0 = 0.5
W_ant = F0 * X * (1 - Y) + D0 * X

# Combined fitness (product, for visualization)
W_combined = W_plant * W_ant

# Plot as contour
contour = ax4.contourf(X * 100, Y * 100, W_combined, levels=20, cmap='inferno')
plt.colorbar(contour, ax=ax4, label='Combined fitness', shrink=0.8)

# Find Nash equilibrium (approximate)
# dW_plant/dx = 0: -G0*(1 - H*exp(-lam*y)) = 0 -> never zero, so boundary
# For ant: dW_ant/dy = -F0*x = 0 -> boundary at y=0
# Real Nash: mutual best response
# Plant BR: x that max W_plant given y -> x=0 (always, plant wants min x)
# Ant BR: y that max W_ant given x -> y=0 (always, ant wants min y)
# Cooperation maintained by conditionality (sanctions model)

# Mark cooperative equilibrium
x_star = 0.04
y_star = 0.25
ax4.plot(x_star * 100, y_star * 100, 'o', color='#ef4444', markersize=12, markeredgecolor='white', markeredgewidth=2)
ax4.annotate(f'Cooperative eq.\n(x*={x_star*100:.0f}%, y*={y_star*100:.0f}%)',
             xy=(x_star * 100, y_star * 100), xytext=(8, 45),
             fontsize=10, color='white', fontweight='bold',
             arrowprops=dict(arrowstyle='->', color='white', lw=1.5))

# Defection point
ax4.plot(0.5, 5, 's', color='#fbbf24', markersize=10, markeredgecolor='white', markeredgewidth=1.5)
ax4.annotate('Mutual\ndefection', xy=(0.5, 5), xytext=(5, 15),
             fontsize=9, color='#fbbf24',
             arrowprops=dict(arrowstyle='->', color='#fbbf24', lw=1))

ax4.set_xlabel('Plant Investment in Rewards (%)', fontsize=11, color='white')
ax4.set_ylabel('Ant Defense Allocation (%)', fontsize=11, color='white')
ax4.set_title('Ant-Plant Mutualism (Game Theory)', fontsize=13, color='#fca5a5', fontweight='bold')
ax4.set_facecolor('#0a0a0a')
ax4.tick_params(colors='white')

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

print("=== Ant Symbiosis Biophysics Results ===")
print(f"\n--- Fungus Garden ---")
print(f"Steady-state substrate: {substrate[-1]:.2f} kg")
print(f"Steady-state fungal biomass: {fungus[-1]:.2f} kg")
print(f"Steady-state gongylidia: {gongylidia[-1]:.2f} kg")
print(f"Total leaf input (year): {np.trapezoid(leaf_input_rate, days):.1f} kg")
print(f"Overall conversion efficiency: {gongylidia[-1] * 25 / (substrate[-1] * 18) * 100:.1f}%")

print(f"\n--- Ophiocordyceps Spore Dispersal ---")
print(f"Spore terminal velocity: {v_t*1000:.2f} mm/s")
print(f"Optimal height for infection: {h_opt*100:.1f} cm")
print(f"Observed range: 22-28 cm")
print(f"Dispersal distance at optimal height: {x_disp[h_opt_idx]:.2f} m")

print(f"\n--- Aphid Farming ---")
print(f"Honeydew energy per droplet: {E_honeydew*1000:.1f} mJ")
print(f"Breakeven aphid/tender ratio: {R_breakeven:.1f}")
print(f"Observed ratio: 8-15 aphids per tender")
print(f"Net benefit at R=10: {(10 * E_h * eta_harvest - E_forage - E_defense)*1000:.2f} mJ/hr")

print(f"\n--- Ant-Plant Game Theory ---")
print(f"Cooperative equilibrium: x*={x_star*100:.0f}% plant, y*={y_star*100:.0f}% ant")
print(f"Combined fitness at equilibrium: {W_combined[25, 3]:.3f}")
print(f"Herbivory reduction at y*=25%: {(1 - H * np.exp(-lam * y_star))*100:.1f}%")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

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de Bekker, C., Ohm, R.A., Loreto, R.G., Sebastian, A., Albert, I., Merber, M., Brachmann, A., Calabrese, S. & Hughes, D.P. (2015). Gene expression during zombie ant biting behavior reflects the complexity underlying fungal parasitic behavioral manipulation. BMC Genomics, 16, 620.
Weber, N.A. (1972). Gardening Ants: The Attines. American Philosophical Society.
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M6. Collective Intelligence M8. Evolution & Superorganism