Tripartite population model

import numpy as np

def model(
    time: float,
    variables: list[float], 
):
    Public, Cheater, Private = variables
    r_p = 0.4
    eta = 0.0001
    nu = 0.00001
    r_m = 0.2
    gamma = 0.0001
    alpha = 0.0002
    beta = 0.0001
    dPdt = (((r_p * Public) - (alpha * Public * Cheater)) - (beta * Public * Private)) - (eta * Public * Public)
    dCdt = (alpha * Public * Cheater) - (nu * Cheater * Cheater)
    dMdt = ((r_m * Private) - (beta * Public * Private)) - (gamma * Private * Private)
    dPublicdt = +dPdt
    dCheaterdt = +dCdt
    dPrivatedt = +dMdt
    return [dPublicdt, dCheaterdt, dPrivatedt]

def all_derived(
    time: float,
    variables: list[float], 
):
    Public, Cheater, Private = variables
    r_p = 0.4
    eta = 0.0001
    nu = 0.00001
    r_m = 0.2
    gamma = 0.0001
    alpha = 0.0002
    beta = 0.0001
    dPdt = (((r_p * Public) - (alpha * Public * Cheater)) - (beta * Public * Private)) - (eta * Public * Public)
    dCdt = (alpha * Public * Cheater) - (nu * Cheater * Cheater)
    dMdt = ((r_m * Private) - (beta * Public * Private)) - (gamma * Private * Private)
    return [dPdt, dCdt, dMdt]

derived = all_derived
y0 = {"Public": 1, "Cheater": 1, "Private": 1}
    

\begin{align*}
      \frac{d P}{dt} &= r_P \cdot P - \alpha \cdot P \cdot C - \beta \cdot P \cdot M - \eta \cdot P \cdot P\\ 
\frac{d C}{dt} &= \alpha \cdot P \cdot C - \nu \cdot C \cdot C\\ 
\frac{d M}{dt} &= r_M \cdot M - \beta \cdot P \cdot M - \gamma \cdot M \cdot M
    \end{align*}