app.py

import streamlit as st
import numpy as np
from tools import integrate
from dynamical_system import exp_discrete, exp_continuous
from scipy.integrate import odeint
import pandas as pd
import altair as alt

st.title("Discrete vs continuous population")

st.header("Exponential growth population")

st.markdown(
    r"""
Discrete time model

$N_{t+1} = N_t + R N_t$

Continuous time model

$\frac{dN}{dt} = r N$


"""
)

st.write("")

init = 0.2  # initial population density
time_series = np.arange(0, 101, 1)

r = st.sidebar.number_input("r", 1e-3, 0.2, value=0.05, step=1e-4, format="%.4f")
R = st.sidebar.number_input("R", 1e-3, 0.2, value=0.05, step=1e-4, format="%.4f")
st.sidebar.write(
    r"The two models give similar results when $R = e^r - 1 = $", np.exp(r) - 1
)

# simulation for continuous time model
pop_continuous = odeint(exp_continuous, init, time_series, args=(r,), tfirst=True)

# simulation for discrete time model
t_discrete, pop_discrete = integrate(exp_discrete, init, time_series[-1], R)

df = pd.DataFrame(
    dict(
        time=t_discrete, pop_discrete=pop_discrete, pop_continuous=pop_continuous[:, 0]
    )
)


def make_plot(scale, title):
    tt = alt.TitleParams(title, anchor="middle")
    l1 = (
        alt.Chart(df, title=tt)
        .mark_line()
        .encode(
            x="time",
            y=alt.Y(
                "pop_discrete",
                axis=alt.Axis(title="Population density"),
                scale=alt.Scale(type=scale),
            ),
            color=alt.value("#1f77b4"),
        )
    )
    l2 = (
        alt.Chart(df)
        .mark_line()
        .encode(x="time", y="pop_continuous", color=alt.value("#ff7f0e"))
    )
    return l1 + l2


col1, col2 = st.columns(2, gap="large")

with col1:
    st.altair_chart(make_plot("linear", "Normal scale"))
with col2:
    st.altair_chart(make_plot("log", "Log scale"))