Ordinary Differential Equations in 1D

Initial value problems, nonlinear rate laws, and a reusable solver workflow

Build Once, Reuse Everywhere

The first session sets the numerical pattern for the whole lab: define a model, choose initial data, integrate forward, and interpret what the trajectory means.

Mathematical core

IVP

\[ \dot y = f(t, y), \qquad y(t_0) = y_0 \]

Computational habit

Use the same solve_ivp() pipeline in epidemic models, chemical kinetics, and population dynamics.

Student payoff

You leave this session with a template that you will reuse in later ODE, synchronization, and PDE topics.

Session Arc

Solver workflow

Write the right-hand side, define t_span, choose y0, and inspect the output carefully.

SIR model

See how even a small coupled system still fits the same initial value problem pipeline.

Nonlinear response

Study saturation and threshold effects in Michaelis-Menten style kinetics and the spruce budworm model.

The Reusable Pipeline

Step 1

Write a function fun(t, y) that returns the rate law.

Step 2

Choose parameters, a time interval, and an initial state.

Step 3

Integrate with solve_ivp() and sample the solution on a useful time grid.

Step 4

Plot the trajectory and ask which parameters or initial conditions changed the outcome.

\[ \dot y = f(t, y), \qquad y(t_0) = y_0 \]

What We Need in Python

scipy.integrate.solve_ivp

The main numerical integrator for initial value problems in the course.

numpy

For initial conditions, parameter arrays, time grids, and vectorized post-processing.

matplotlib.pyplot

For time series, parameter comparisons, and phase-line style interpretation.

Three Models, One Solver Pattern

SIR epidemic model

\[ \dot S = -\beta S I, \quad \dot I = \beta S I - \gamma I, \quad \dot R = \gamma I \]

Track growth, peak prevalence, and the role of \(R_0 = \beta / \gamma\).

Saturating rate law

\[ v(S) = \frac{V_{\max} S}{K_m + S} \]

Use this to discuss why many rates stop growing linearly at large substrate levels.

Spruce budworm

\[ \dot x = r x \left(1 - \frac{x}{k}\right) - \frac{x^2}{1 + x^2} \]

This is the cleanest example of threshold behavior and bistability in the session.

Bistability Is a Modeling Signal

Spruce budworm takeaways

The balance between logistic growth and saturating predation creates multiple equilibria.
Small perturbations can push the system toward a low-population or high-population regime.
The same ODE solver works, but the interpretation now matters as much as the integration.

Use the full module page when you want the phase-line analysis and parameter exploration.

Minimal Code Pattern

import numpy as np
from scipy.integrate import solve_ivp


def rhs(t, y, rate):
    return -rate * y


t_eval = np.linspace(0.0, 10.0, 200)
sol = solve_ivp(rhs, (0.0, 10.0), y0=[2.0], t_eval=t_eval, args=(0.5,))

Keep this pattern in mind. Most later solver-based sessions are variations on this same structure.

Full Module Pages

Use the deck for the lecture flow, then move to the module pages for derivations, code walkthroughs, and exercises.

Module overview Compartmental models: SIR and SIS Michaelis-Menten kinetics Spruce budworm Practice: Budworm app