{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Explicit Euler for the Predator-Prey-System\n", "\n", "With the Predator-Prey-System (also known as Lotka-Volterra-System) one can describe the dynamics of biological systems where two species interact.\n", "One species as a predator and the other as prey. \n", "It consists of the pair of equations:\n", "$$\n", "\\frac{\\partial x}{\\partial t} = \\alpha x - \\beta xy\n", "$$\n", "$$\n", "\\frac{\\partial y}{\\partial t} = -\\gamma y + \\delta xy\n", "$$\n", "\n", "Hereby stands:\n", " * t is the time \n", " * x is the number or density of prey (rabbits)\n", " * y is the number or density of predators (foxes)\n", " * $\\alpha>0$ is the growth rate of rabbits\n", " * $\\beta>0$ is the catch rate of the rabbits\n", " * $\\gamma>0$ is the death rate of foxes without having prey\n", " * $\\delta>0$ is the growth rate of foxes, having prey\n", " \n", "Together with a initial condition, for example $x(0)=10, y(0)=5$ this can be solved with an explicit Euler scheme.\n", " \n", "More details to such systems and other population models can be found in:\n", "https://www-user.tu-chemnitz.de/~uro/publications/Kandler-Unger--Population_dispersal_via_diffusion-reaction_equations.pdf\n", "\n", "See Chapter 5.1, page 33 for predator prey systems.\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Task\n", "\n", "Implement the Euler Scheme for the problem above, play around with the parameters $\\alpha$,$\\beta$,$\\gamma$,$\\delta$ and try to understand their influence.\n" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "alpha = 1.1\n", "beta = 0.5\n", "gamma = 0.1\n", "delta = 0.2\n", "\n", "dt = 0.001\n", "D = 1000\n", "N = int(D/dt)\n", "\n", "from numpy import zeros, linspace\n", "t = linspace(0, N*dt, N+1)\n", "X = zeros(N+1)\n", "Y = zeros(N+1)\n", "\n", "X[0]=10\n", "Y[0]=5\n", "\n", "# from here as TODO\n", "\n", "# Plot the graphs X(t), Y(t)\n", "# as TODO\n", "import matplotlib.pyplot as plt\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.9" } }, "nbformat": 4, "nbformat_minor": 4 }