Math 5110/6830
Homework 7.1
1. Consider the system
x
ẋ = 2x 1 −
− xy
2
y
ẏ = 3y 1 −
− 2xy
3
(a) Find all the fixed points.
(b) Use linearization to classify the stability of each fixed point.
2. Qualitatively analyze each of the following systems, that is
• draw a phase plane with nullclines, fixed points and direction field arrows;
• find the fixed points analytically, and determine their stability.
a)
ẋ = y 2 − x2
ẏ = x − 1
b)
ẋ = x3 − y
ẏ = y 2 − x
3. In the general two species interactions model that we considered in class:
ẋ = αx + βxy
ẏ = γy + δxy
analyze the following cases:
a) Mutualism of two species that cannot survive alone
b) A competition model in which one of the populations is self-sustainable and
the other is not.
In each case I want to see nullclines with representative arrows on and between nullclines; fixed points and their stability analysis via linearization; phase
space (probably on a separte graph)that clearly shows representative trejectories in the phase space; a description, in words, for what will happen with either
population when they start at different initial conditions.
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