Coupled systems and phase portraits Problem set A

Find the eigenvalues and eigenvectors of the matrix \(\begin{pmatrix}1&4\\2&3\end{pmatrix}\).

A system is \(\dot x=4x + 3y,\ \dot y=x + 2y\).

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system.
2. (b) Sketch the phase portrait, showing the directions of the eigenvectors, and name the type of equilibrium.

The numbers of two strains of bacteria in a culture, in thousands, are modelled by \(\dot x=2x + y,\ \dot y=3x + 4y\), where t is in hours. At \(t=0\) there are 6 thousand of strain X and 2 thousand of strain Y.

1. (a) Find the eigenvalues and eigenvectors of the matrix of the system.
2. (b) Write down the general solution, then find x(t) and y(t) for the starting values given.
3. (c) Comment on how suitable the model is in the long term.

The numbers of foxes (x) and rabbits (y) on an island, in hundreds, are modelled by \(\dot x=8x - 5y,\ \dot y=10x - 7y\), where t is in years. Initially there are 14 hundred foxes and 18 hundred rabbits.

1. (a) Find x(t) and y(t).
2. (b) Show the solution curve on a phase portrait for \(0\le x,y\le50\). Show the directions of the two eigenvectors.
3. (c) What is the ratio of foxes to rabbits in the long term?

The deviations of the numbers of two competing species, in hundreds, from their usual levels are modelled by \(\dot x=-3x - 4y,\ \dot y=2x + y\), where t is in years, with \(x(0)=5\) and \(y(0)=2\).

1. (a) Use technology to sketch the solution curve in the x-y plane.
2. (b) Sketch the graphs of x and y against t for \(0\le t\le8\) on separate diagrams.
3. (c) Describe how x and y change with t.

The diagram shows the direction field for a system of differential equations in x and y, and the solution curve starting at \((10,12)\).

1. (a) Estimate the maximum value of x along the solution curve.
2. (b) Estimate the long-term state of the system.

The diagram shows the direction field for a system of differential equations in x and y. Say whether each statement is true or false.

1. (a) Whatever the starting values, x approaches 9 and y approaches 7.
2. (b) x always decreases.
3. (c) If x is greater than 9, then x decreases.
4. (d) If y is greater than 7, then y decreases.