Modeling Template you can actually use (equations + runnable doc via PythonTeX)

What’s inside

* Lotka–Volterra: ∂x/∂t = αx − βxy, ∂y/∂t = γxy − δy; fixed point x\* = δ/γ, y\* = α/β. Compute Jacobian, eigenvalues, phase portrait, limit cycles.
* SIR: ∂S/∂t = −βSI/N, ∂I/∂t = βSI/N − γI, ∂R/∂t = γI; R₀ = β/γ; check peak time, final size, herd threshold 1 − 1/R₀.
* Monte Carlo: I ≈ (b − a)/N · Σᵢ₌₁ᴺ f(Xᵢ) with error ∼ N^({−1/2};) random walk Xₙ = Σᵢ₌₁ⁿ Sᵢ, E(|Xₙ|) ∼ √n. Add variance reduction (antithetic, control variates).
* Agent-based flocking: vᵢ^({t+1}) = vᵢ^(t) \+ F\_sep + F\_align + F\_coh; periodic boundaries for space.

How it runs (PythonTeX)

* Equations and code live in one .tex file.
* Simulations run at compile time; figures update automatically.
* Parameter sweeps are straightforward (e.g., α ∈ \[0.1, 2.0\]).

Minimal workflow

1. Write the ODEs/PDEs with ∂, ∇.
2. Implement the solver (e.g., SciPy) in a PythonTeX block.
3. Compute equilibria and local stability.
4. Produce phase portraits, time series, sensitivity plots.
5. Tweak α, β, γ and recompile to refresh results.

When to use

* Teaching or research where ∂x/∂t = f(x, parameters) and you want theory, code, and figures to stay in sync.

Links

* Template (.tex): [https://cocalc.com/share/public\_paths/156bf9ab56c7fb123f87dd8cefae68641998cda9/main.tex](https://cocalc.com/share/public_paths/156bf9ab56c7fb123f87dd8cefae68641998cda9/main.tex)
* PDF: [https://cocalc.com/share/public\_paths/156bf9ab56c7fb123f87dd8cefae68641998cda9/main.pdf](https://cocalc.com/share/public_paths/156bf9ab56c7fb123f87dd8cefae68641998cda9/main.pdf)

If you think another model class would be neat (e.g., SDEs, bifurcation continuation), say which equations and outputs you want to see next..

Mathematical Modeling Template: From ∂x/∂t Theory to Computational Implementation