Create app.py

app.py

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+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("red"),
+        )
+    )
+    l2 = (
+        alt.Chart(df)
+        .mark_line()
+        .encode(x="time", y="pop_continuous", color=alt.value("blue"))
+    )
+    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"))