Mathematical Model

ideasim implements a Piecewise Deterministic Markov Process (PDMP) for ideological competition. The continuous dynamics follow a replicator-mutator equation with latent-driven immigration.

Continuous Dynamics

The manifest adoption share \(x_i\) evolves according to:

\[\frac{dx_i}{dt} = x_i (f_i - \bar{f}) + \rho (y_i - x_i Y)\]

where:

\(f_i\) is the fitness of ideology \(i\) (environmental response + competitive interaction + institutional capture + latent reservoir).
\(\bar{f} = \sum_j x_j f_j\) is the mean population fitness.
\(\rho\) controls the latent-to-manifest coupling strength.
\(y_i\) is the latent sympathy for ideology \(i\).
\(Y = \sum_j y_j\) is the total latent sympathy.

Latent Sympathy

Latent sympathy evolves via:

\[\frac{dy_i}{dt} = \alpha_i x_i - \beta_i y_i + \sigma_i (1 - x_i) + \delta_i S_i\]

where:

\(\alpha_i\) is the generation rate from active adherents.
\(\beta_i\) is the natural decay rate.
\(\sigma_i\) is background sympathy generation.
\(\delta_i S_i\) is the martyrdom effect (suppression creates sympathy).

Stochastic Events

Overlaying the continuous dynamics, stochastic events occur as a Poisson process:

Hybridization: Two parent ideologies blend to create a new hybrid.
Charismatic bursts: Sudden surges in adoption of a particular ideology.
Parameter mutation: Gradual drift in ideology parameters over time.

For the complete formal treatment, see the paper/ directory in the repository.