Numerical Methods in EGTtools

In many cases, the complexity of the problem—whether due to a large number of strategies or a large population size—makes numerical simulations necessary. egttools implements several methods for estimating key indicators that characterize a population’s stochastic evolutionary dynamics. These methods are based on the analytical approaches described here and are available in the egttools.numerical.PairwiseMoranNumerical class.

To use these methods, you need to provide: - A game object, which must inherit from the egttools.games.AbstractGame class, - The population size, - A cache size to control memory usage.

Note

The cache parameter determines the size of the fitness cache in memory (RAM). A larger cache speeds up simulations by avoiding redundant calculations but also increases memory consumption. Choose a value appropriate for the available system memory.

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Methods Overview

  • `estimate_fixation_probability(index_invading_strategy, index_resident_strategy, nb_runs, nb_generations, beta)` Estimates the probability that a mutant adopting strategy index_invading_strategy fixates in a resident population adopting strategy index_resident_strategy. - nb_runs: Number of independent simulations (higher values yield better estimations). - nb_generations: Maximum number of generations per simulation. - beta: Intensity of selection.

  • `estimate_stationary_distribution(nb_runs, nb_generations, transitory, beta, mu)` Estimates the stationary distribution over all possible population states. - transitory: Number of generations discarded at the beginning (burn-in). - mu: Mutation rate.

  • `estimate_stationary_distribution_sparse(nb_runs, nb_generations, transitory, beta, mu)` Same as estimate_stationary_distribution, but returns a sparse matrix, suitable for very large state spaces.

  • `estimate_strategy_distribution(nb_runs, nb_generations, transitory, beta, mu)` For very large systems where the number of states exceeds what can be stored in a 64-bit integer, this method estimates the distribution of strategies instead of full population states.

  • `evolve(nb_generations, beta, mu, init_state)` Runs a single simulation for nb_generations generations and returns only the final population state. - init_state: A numpy.ndarray representing the initial counts of each strategy.

  • `run_with_mutation(nb_generations, transient, beta, mu, init_state)` Runs a simulation with mutation. After an initial transient phase (burn-in), returns the sequence of visited states. - beta: Intensity of selection. - mu: Mutation rate. - init_state: Initial population configuration (numpy.ndarray).

  • `run_without_mutation(nb_generations, transient, beta, init_state)` Runs a simulation without mutation (mu = 0). After an initial transient phase, returns the sequence of visited states. - beta: Intensity of selection. - init_state: Initial population configuration (numpy.ndarray).

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Tutorials

Estimate Fixation Probabilities

Estimate the probability that a rare mutant strategy successfully invades and takes over a resident population. Useful for studying invasion dynamics and evolutionary stability.

Estimate Stationary Distributions

Warning

Only use this method if the number of possible states fits within a 64-bit integer (int64_t). For very large systems, use estimate_strategy_distribution instead.

Estimate the long-term probability distribution over all possible population states. This provides insight into stable evolutionary states under mutation-selection balance.

Estimate Strategy Distributions

For extremely large systems, where tracking full states is infeasible, estimate the distribution of strategies across the population.

Run a Single Simulation

Use evolve when you are only interested in the final state of the population after a fixed number of generations.

Run Full Trajectories

Use run_with_mutation or run_without_mutation to obtain the full trajectory of population states over time. Useful for studying dynamic evolutionary paths.

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Note

Currently, egttools.numerical supports simulations only in well-mixed populations. Planned future extensions: - Version 0.14.0: Support for structured populations (e.g., networks), - Version 0.15.0: Support for multi-level selection models.