Introduction (1 week)
Introduction to Math. Modeling. Non-uniqueness of
math model.
The Greatest model - Universe (from Ptolemy to Big Bang)
Problem: Divide the class into working groups
Populations dynamics 1. Single species (2 weeks)
Classical models, Equations with delay
Leslie model of reproduction
Populations dynamics 2. Interacting species. (2 weeks)
Lotka-Volterra model, variations, several species
Dynamics of epidemics
Fairness and evolution (2 weeks)
Cooperative games: Fair shares
Evolutionary games
Traffic flow (3 weeks)
Continuum model. Shock waves. Dissipation
Discrete model. Cellular automata. Rule 184
Unstable motion. (2 weeks)
Bouncing box
Avalanche
Repeated Motion. (2 weeks)
Discrete Waves: Domino chain
Chain with bistable links
Material models: Metamaterials (2 weeks)
Auxetic materials
Morphing structures
HW1
1. Suggest an algorithm of forming groups of three for each
project so that each student works with maximal number of
classmates.\
2. 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
3. Simulate and plot using Maple or any other platform
a. Logistic and Allee, Holling Type III growth, (
continuous model)
b. Logistic, Logistic with delay (discrete model)
4. Show that in Holling Type III functional response (2.26).
[p 24]
point N=0 is always unstable.
Example of Maple code for plotting
solutions to DE and discrete systems.
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);
HW2. Project.
Model population growth using Leslie model. Modify model
accounting for habitat limiting capacity.
Suggested sources:
https://www.populationpyramid.net/world/2016/
https://en.wikipedia.org/wiki/Leslie_matrix
http://www.math.utah.edu/~cherk/teach/12mathmodel/leslie-model.pdf
https://en.wikipedia.org/wiki/Euler%E2%80%93Lotka_equation
HW3. From
"Mathematical Introduction to Population Dynamics by Howie
Weiss
http://www.math.epn.edu.ec/emalca2010/files/material/Biomatematica/CUP101310.pdf
1. Normalize the Lotka-Volterra model by scaling population
sizes and time scale. Simulate dynamics of the normalized
system for several values of the remaining parameter, graph
the population density vs. time.
2. Consider the normalized Lotka-Volterra model as
in (3.41) (3.42)) with Holling-type responses
of type II and type III, see section 2.4.3. Find
equilibrium points and comment, simulate and graph the
population density vs. time.
3. Problem 25. p. 54. Consider the predator-prey
system with a simple prey refuge (eqs. (3.24)
. (3.25)) How the equilibrium point is
changed? Linearize the problem near the equilibrium
and comment on the change in the period depending on refuge
size. Simulate the dynamics for several values of V, graph the
population density vs. time.
HW4. Problems 17, 18, 19 p. 49