Problem Bank | IB Math AIHL

[Maximum mark: 13]

Ecologists model small deviations x(t) and y(t) of prey and predator populations from equilibrium by the linear system

dtdx=−2x+y,dtdy=−x−2y.

Show that dxdy=−2x+y−x−2y.
[1]
Find dxdy at:
1. (2,1).
  [1]
2. (−2,−1).
  [1]
Find the eigenvalues and a corresponding eigenvector for the matrix

A=(−211−2).

[6]
Hence write down the general solution to the system (x(t),y(t)).
[2]
Determine whether the equilibrium (0,0) is stable or unstable, with justification.
[2]

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[Maximum mark: 13]

Ecologists model small deviations x(t) and y(t) of prey and predator populations from equilibrium by the linear system

dtdx=−2x+y,dtdy=−x−2y.

Show that dxdy=−2x+y−x−2y.
[1]
Find dxdy at:
1. (2,1).
  [1]
2. (−2,−1).
  [1]
Find the eigenvalues and a corresponding eigenvector for the matrix

A=(−211−2).

[6]
Hence write down the general solution to the system (x(t),y(t)).
[2]
Determine whether the equilibrium (0,0) is stable or unstable, with justification.
[2]

Ask Plex AI about this problem

Get hints, ask questions, and work through this problem step by step

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