Ordinary Differential Equations

Assignment 1

The assignment is due on Monday, 13 May 2013, at the tutorial.

Problem 1. Models that are commonly used in fisheries are

dN

dt= g(N)N,

where g(N) is given by

g(N) =reN Ricker model

g(N) =r

+NBeverton-Holt,

where and are positive. Analyse the behaviour of the solutions to theseequations. This includes the following: solve the equations by the methodof separation of variables and write N as an explicit or implicit function oft, draw the solutions on the phase line, state the steady states (equilibriumpoints), their nature, and the long term fate of the populations.

Problem 2. Species may derive mutual benefit from their association; thistype of interaction is known as mutualism. May (1976) suggests the followingset of equations to describe a possible pair of mutualists:

dN1

dt= rN1

(1

N1

k1 + N2

)

dN2

dt= rN2

(1

N2

k2 + N1

),

where , , r, k1, and k2 are positive parameters, < 1, and Ni is thepopulation of the ith species.

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1. Explain why the equations describe a mutualistic interaction. (Hint:What are the carrying capacities of each species and how are theyaffected by the other population?)

2. Find the steady states of the system.

Problem 3. Consider the differential equation x = x+ cos t.

a) Find the general solution to this equation.

b) Prove that there is a unique periodic solution to this equation.

c) Compute the Poincare map p : {t = 0} {t = 2pi} for this equation anduse this to verify again that there is a unique periodic solution.

Problem 4. First-order differential equations need not have solutions that aredefined for all times.

a) Find the general solution of the equation x = x2.

b) Discuss domains over which the solution is defined.

c) Draw the graph of the solution which satisfies x(0) = 1. What curve isthis graph?

Problem 5. Find the general solution of the system

X =

(1 23 6

)X

and draw its phase portrait.

Problem 6. Describe all possible 2 2 matrices whose eigenvalues are 0 and1.

Problem 7. Find the general solution of the system

X =

(1 21 1

)X

and draw its phase portrait.

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