Math 591: Homework 1 (Due: Friday, Sept 14, 2012)
1. Consider the Maltusian population model
u = r(t)u u
u(0) = u0
where u = population, = constant death rate and r(t) is a
T -periodic growth rate meant to represent seasonal birth rate
changes. Let r deno
Math 591: Homework 2 (Due: October 10, 2012)
1. Each matrix A denes a linearized system z = Az . For each system
clearly
a)
b)
c)
d)
Dene all eigenspaces Ek
Dene two independent solutions xk (t)
Dene (and draw in part d) E s , E u and E c
Draw a phase por
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Math 591: Homework 3 (Due: Wednesday, Nov 7)
1. [20] A three species competition model is given by
dx1
=
dt
dx2
=
dt
dx3
=
dt
where a (0, 1) and b > 1.
x1 (1 x1 bx2 ax3 )
x2 (1 ax1 x2 bx3 )
x3 (1 bx1 ax2 x3 )
a) Find all equilibria including the coexisten
Math 591 (2012)
Assignment 4: Due: Wednesday, Dec 5, 2012
1. (15) Species may derive mutual benet from their association. This type
of interaction is known as mutualism. A model of mutualism for a pair of
species is
dx
dt
dy
dt
x
1 + y
y
1
2 + x
1
= rx
=
Math 591: Homework 7 (Due: April 12, 2013)
1. [60] Suppose that a chemical, ecological or electrical model of some system
is given by (in dimensional form)
Ut = DUxx + rU (U a)(2a U )
x IR
where D, r and a are positive parameters.
a) Show that by rescalin
Math 592 (2013)
Assignment 5
Due: Friday, February 1, 2013
+
1. (20) For some cell, calcium chloride CaCl2 dissociates into calcium S1
and two ions of chloride S2 . Electroneutrality in the intracellular and
extracellular regions then requires
ci
[Ca2+ ]
Math 591: Homework 6 (Due: March 6, 2013)
1. [20] The Mc Kean (1970) version of the FitzHugh-Nagumo model is
v
= f (v ) w
w
= v w v0
where > 0, v0 > 0, 0 <
1,
v
v
f (v ) =
1v
v < /2
/2 < v < (1 + )/2
v > (1 + )/2
and (0, 1). Here f (v ) is piecewise cont