MATH 462: Homework 3
Winter 2006
Set: 3 Feb.
Due: 10 Feb.
1. Find the equilibrium points of the following differential equations. Study their
stability as the parameter varies, and draw a bifurcation diagram.
(i)dy
dt =k y (1 −y)
(ii)dy
dt =y2
−µ
(iii)dy
dt =y3+µ y
(iv)dy
dt =y2
−µ y
(v)dy
dt =y2
−ay + 4
2. Consider the constant yield harvesting model
dP
dt =k P (1 −
P
N)−h.
Pis the crop, and his the harvesting parameter (>0). Nondimensionalize
time, t, with respect to k−1(remember the dimensions of k) and Pwith respect
to Nto construct a new nondimensional equation with just one parameter.
(a) Solve the new equation analytically, if uis your new variable, let u(0) =
u0.
(b) State the equilibrium points and their stability.
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