theory.toys – Lotka–Volterra Evolution

Initial Conditions

Prey population: 50 Predator population: 20 Prey trait (z_prey): 0.0 Predator trait (z_pred): 0.0

Ecological Parameters

Prey growth rate (r): 0.50 Base attack rate (α₀): 0.020 Conversion efficiency (e): 0.50 Predator death rate (m): 0.30

Evolutionary Parameters

Trait matching strength (k): 0.50 Selection strength (s): 0.10

Mutation rate (μ): 0.05🔑 Unlock

Simulation

Time span: 200 generations

Eco-Evolutionary Dynamics

When predator-prey dynamics occur on similar timescales to evolution, we get eco-evolutionary feedback. Predator and prey traits coevolve, creating arms races, cycles, or Red Queen dynamics where both species must evolve continuously just to maintain their fitness.

The Model

This extends the classic Lotka-Volterra predator-prey model with evolving traits. The attack rate now depends on trait matching:

α(Δz) = α₀ exp(-k|z_prey - z_pred|²)

When predator and prey traits match (small |Δz|), attack rate is high—the predator is good at catching that type of prey. The parameter k controls how strongly traits affect interactions.

Ecological Dynamics

Prey equation: dN_prey/dt = r N_prey - α(Δz) N_prey N_pred
Predator equation: dN_pred/dt = e α(Δz) N_prey N_pred - m N_pred

These describe population growth, predation (depends on trait matching), and predator death.

Evolutionary Dynamics

Traits evolve based on selection gradients:

Prey selection: Favors higher z to escape predators (increase |Δz|)
Predator selection: Favors z that matches prey (decrease |Δz|)

Mutation adds random drift. The result is a coevolutionary chase where prey evolve to escape and predators evolve to pursue.

Red Queen Hypothesis

In this coevolutionary race, both species must evolve continuously just to maintain their relative fitness—like the Red Queen in Alice in Wonderland, who must run to stay in the same place. Watch how traits chase each other while populations oscillate!

What to Explore

High selection strength (s) → rapid trait evolution and tight coevolutionary coupling
High trait matching (k) → strong eco-evolutionary feedback
Low mutation (μ) → deterministic coevolution; high μ → stochastic wandering
Watch phase space: do populations cycle while traits spiral or chase?