Introduction (1 week)
Introduction to Math. Modeling. Non-uniqueness of
math model.
The Greatest model - Universe (from Ptolemy to Big Bang)
HW Problem: Divide the class into working groups
Populations dynamics 1. Single species (1 weeks)
Classical models, Discrete and continuous
description. Equations with delay
Leslie model of reproduction and dynamics of World population.
Populations dynamics 2. Interacting species. (3 weeks)
Lotka-Volterra model, variations, interaction of
several species.
Application to markets
Dynamics of epidemics
Evolutionary games
Migration; description by PDE.
Traffic flow (3 weeks)
Continuum model (Lighthill & Whitham and
Richards),. Shock waves. Variations of the model
Discrete model. Cellular automata. Rule 184
Games and Fairness (2 weeks)
Cooperative games: Fair shares (Shapley values)
Application to voting systems.
Unstable motion. (2 weeks)
Bouncing box
Avalanche
Nonlinear Waves (2 weeks)
Discrete Waves: Domino chain
Chain of masses the with bistable edges.
Metamaterials (3 weeks)
Auxetic materials
Morphing structures
Design of metamaterials
HW1 (due August 27)
1. Suggest an algorithm of forming groups of three for each
project so that each student works with maximal number of
classmates.
2. Learn to plot solutions of differential and difference
equations using Maple or other platforms.
a) u'(t)= a u(t)- b u(t)^2, u(0)= 1, t > 0;
b) u(n)= u(n-1) [1- u(n-2)], u(0)=0, u(1)=.1, n=0, 1, 2, ...
100.
c) Read from
A Mathematical Introduction to Population Dynamics
by Howie Weiss (Georgia Tech)
http://www.math.epn.edu.ec/emalca2010/files/material/Biomatematica/CUP101310.pdf
http://www.math.epn.edu.ec/emalca2010/files/material/Biomatematica/CUP101310.pdf
ch 2 2.1-2.6
Maple code:
Solve and plot solutions to Difference eqns:
with(plots):
x[1] = 0.01;
for i to 100 do
x[i+1] := (1+(1-(1/10)*x[i])*.15)*x[i]
end do;
pointplot({seq([n, x[n]], n = 1 .. 50)})
x[1] := 1; x[2] := 1;
for i from 2 to 100 do
x[i+1] := (1+(1-(1/2)*x[i-1])*.5)*x[i]
end do;
pointplot({seq([n, x[n]], n = 1 .. 20)});
Plot solutions to Differential equations
with(DEtools);
LG := diff(y(t), t) = k*y(t)*(K-y(t));
K := 40; k := 0.01;
ivs := [y(0) = 1, y(0) = 10, y(0) = 50];
DEplot(LG, y(t), t = 0 .. 50, ivs);
DEplot(LG, y(t), t = 0 .. 50, y = 0 .. 60, ivs, arrows =
medium, linecolor = black);
Or, you may first solve the
differential equation
and then plot the solution:
de := dsolve({diff(z(t), t) = a*(1-c*z(t))*z(t), z(0) = zo});
p := eval(z(t), de); p0 := eval(p, {a = 1.3, c = .2, zo =
.1});
plot(p0, t = 0 .. 8);