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

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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

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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);

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