{ "cells": [ { "cell_type": "markdown", "id": "3d0a47e3-7c72-43fc-8563-d44cf886a24b", "metadata": {}, "source": [ "---\n", "title: \"A2\"\n", "subtitle: \"Systems of IVPs\"\n", "date: 2026-02-02\n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: A2.ipynb\n", "---\n", "\n", ":::{.callout-note}\n", "\n", "* Complete the following and submit to Canvas before Feb 11, 23:59\n", "* Late work will recieve 0%,\n", "* Each assignment is worth the same, \n", "* Please get in contact with the TA/instructor in plenty of time if you need help,\n", "* Before submitting your work, make sure to check everything runs as expected. Click **Kernel > Restart Kernel and Run All Cells**.\n", "* Feel free to add more cells to experiment or test your answers,\n", "* I encourage you to discuss the course material and assignment questions with your classmates. However, unless otherwise explicitly stated on the assignment, you must complete and write up your solutions on your own,\n", "* The use of GenAI is prohibited as outlined in the course syllabus. If I suspect you of cheating, you may be asked to complete a written or oral exam on the content of this assignment.\n", "\n", ":::" ] }, { "cell_type": "code", "execution_count": 1, "id": "5b86f5a6", "metadata": {}, "outputs": [], "source": [ "#| echo: false\n", "using Plots, LaTeXStrings" ] }, { "cell_type": "markdown", "id": "93f519cf", "metadata": {}, "source": [ "::: {.box}\n", "\n", "***Name:***\n", "\n", "***Approx. time spent on this assignment:***\n", "\n", "***Did you do the suggested reading?*** Yes/No\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "1cdd630f", "metadata": {}, "source": [ "
\n", "\n", "In classes, we considered the following problem: seek $u:[0,T] \\to \\mathbb R$ such that \n", "\n", "\\begin{align}\n", " u(0) &= u_0 \\nonumber\\\\\n", " u'(t) &= f\\big( t, u(t) \\big). \\tag{IVP}\n", "\\end{align}\n", "\n", "This equation governs the dynamics of a single unknown. In practice, there are often multiple unknowns that interact over time and one wishes to model the dynamics as the system evolves. \n", "\n", "## A: Lotka-Volterra equations\n", "\n", "For example, one may consider the so-called *Lotka–Volterra predator–prey model* describing the evolution of a biological system involving two species: a prey population of size $x(t)$ at time $t$, and a predator population of size $y(t)$ at time $t$. These populations are modelled by the following system of equations: $x(0) = x_0, y(0) = y_0$ for some initial data $(x_0,y_0)$ and \n", "\n", "\\begin{align}\n", " x'(t) &= A x(t) - B x(t) y(t) \\nonumber\\\\\n", " y'(t) &= -C y(t) + D x(t) y(t) \\tag{LV}\n", "\\end{align}\n", "\n", "for some constants $A, B, C, D \\geq 0$\n", "\n", "* $A$ describes the growth rate of the prey population (this growth is proportional to the current size of the population), \n", "* $B$ describes the rate at which the predators eat the prey (which is larger if there are more predators), \n", "* $C$ is the death rate of the predators, \n", "* $D$ is the rate at which predators increase due to availability of prey.\n", "\n", "***Exercise 1.*** Write $(\\text{LV})$ as a problem of the form: seek $u : [0,T] \\to \\mathbb R^2$ such that \n", "\n", "\\begin{align}\n", " u(0) &= u_0 \\nonumber\\\\\n", " u'(t) &= f\\big(t, u(t)\\big)\n", "\\end{align}\n", "\n", "for some $f: [0,T] \\times \\mathbb R \\to \\mathbb R^2$.\n", "\n", "*Hint*: You will want to define \n", "\n", "\\begin{align}\n", " u(t) &= \\begin{pmatrix}\n", " x(t) \\\\ y(t)\n", " \\end{pmatrix}. \\nonumber\n", "\\end{align}\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "
\n", "\n", "Notice that you may apply the numerical schemes from lectures to this system of IVPs. For example, the midpoint method reads:\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "95addb7b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "midpoint (generic function with 1 method)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function midpoint( u0, f, T, n )\n", " h = T/n \n", " t = 0:h:T \n", " u = fill(float(u0), n+1)\n", " u[1] = u0\n", " for j = 1:n\n", " u[j+1] = u[j] + h * f( t[j] + h/2, u[j] + (h/2)*f(t[j], u[j]) )\n", " end\n", " return u\n", "end " ] }, { "cell_type": "markdown", "id": "6fbb9927", "metadata": {}, "source": [ "Here, is the numerical solution of $(\\text{VP})$ with a particular choice of parameters $A, B, C, D$, initial condition $u_0$, and mesh $\\{ t_j \\}$:" ] }, { "cell_type": "code", "execution_count": 3, "id": "ea44a047", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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"\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A, B, C, D = 1., 1., .5, 1.\n", "u0 = [1.,.01]\n", "f(t, u) = [A*u[1]-B*u[1]*u[2], -C*u[2]+D*u[1]*u[2]]\n", "\n", "T, n = 50., 500\n", "u = midpoint( u0, f, T, n)\n", "u = [U[j] for U ∈ u, j ∈ 1:2]\n", "plot(0:T/n:T, u, \n", " label=[\"prey\" \"predator\"], l=(1, :black), m=2)" ] }, { "cell_type": "markdown", "id": "dce3b735", "metadata": {}, "source": [ "
\n", "\n", "***Exercise 2.*** Explain this plot with reference to the physical system that is being modelled. \n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "78262995", "metadata": {}, "source": [ "Another way to see the dynamics is to plot $(x(t), y(t))$ in *phase space*:" ] }, { "cell_type": "code", "execution_count": 4, "id": "c4972f4c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\LV.gif\")" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "name": "stderr", "output_type": "stream", "text": [ "\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mSaved animation to c:\\Users\\math5\\Math 5485\\Math5486\\pics\\LV.gif\n" ] } ], "source": [ "X, Y = [], []\n", "anim = @animate for t ∈ 1:n+1\n", " x,y = u[t,:]\n", " plot(X, Y,\n", " title=\"LV numerical solution\",\n", " xlabel=L\"x(t)\", ylabel=L\"y(t)\", legend=false)\n", " push!(X, x)\n", " push!(Y, y)\n", " scatter!([x], [y], m=(5, :red))\n", "end\n", "mp4(anim, \"pics/LV.gif\")" ] }, { "cell_type": "markdown", "id": "8225ac9f", "metadata": {}, "source": [ "
\n", "\n", "***Exercise 3.*** Play around with the parameters in this model ($A, B, C, D$). What do you notice about the numerical solution? \n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "2c8bf065", "metadata": {}, "source": [ "***Exercise 4.*** What happens when $(x_0, y_0) = (C/D, A/B)$? Why?\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "e0f8eb3e", "metadata": {}, "source": [ "## B. Higher order IVPs\n", "\n", "Consider the follwing initial value problem: seek $\\theta: [0,1] \\to \\mathbb R$ such that\n", "\n", "\\begin{align}\n", " \\theta(0) &= \\theta_0 \\nonumber\\\\\n", " \\theta'(0) &= 0 \\nonumber\\\\\n", " \\theta''(t) &= - \\gamma \\theta'(t) -\\frac{g}{L} \\sin \\theta(t). \\nonumber\n", "\\end{align}\n", "\n", "This is a *second order IVP* modelling the motion of a pendulum of length $L$ and with damping $\\gamma > 0$. Here, $\\theta$ is the angle made with the vertical position and $g \\approx 9.81 m/s^2$ is the gravitational constant. See the beginning of Chapter 5 of @Burden for a picture.\n", "\n", "***Exercise 5.*** Show that $\\theta(t) = 0$ is a solution when $\\theta_0 = 0$. Explain why this makes sense with reference to the physical system we are modelling.\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "d4817ca0", "metadata": {}, "source": [ "***Exercise 6.*** Write down the system of differential equations governing the variable\n", "\n", "\\begin{align}\n", " u(t) = \\begin{pmatrix}\n", " u_1(t) \\\\ u_2(t)\n", " \\end{pmatrix} = \\begin{pmatrix}\n", " \\theta(t) \\\\ \\theta'(t)\n", " \\end{pmatrix}.\n", "\\end{align}\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "5bdc059d", "metadata": {}, "source": [ "***Exercise 7.*** Use the ```midpoint``` method to implement the numerical solution to this system of equations. Plot your solution and comment on the dependence of the solution on $\\gamma$. *Hint:* You can use the code samples given above. " ] }, { "cell_type": "code", "execution_count": 5, "id": "6ec4775a", "metadata": {}, "outputs": [], "source": [ "# Your answer here" ] }, { "cell_type": "markdown", "id": "f08091dc", "metadata": {}, "source": [ "## C. Reading \n", "\n", "Read: Sections 5.1 - 5.4 of @Burden" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.11", "language": "julia", "name": "julia-1.11" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }