1.15 Eigenvalues and eigenvectors questions for IB Maths AI

Show that the characteristic polynomial of $\mathit{M}$ is ${\lambda }^2-7\lambda +10$.

Hence find the eigenvalues of $\mathit{M}$.

Write down the matrix $M$ such that $\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=M\left(\begin{array}{c}x\\ y\end{array}\right)$.

Show that the eigenvalues, $\lambda$, of $M$ satisfy ${\lambda }^2+0.2\lambda +1.01=0$.

Find the eigenvalues of $M$, giving your answers in the form $a+b\mathrm{i}$.

Write the eigenvalue with positive imaginary part in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

State, giving a reason, what your answer to part (c) shows about the shape of the trajectories in the phase portrait.

State, giving a reason, what happens to the populations in the long term.

Find the direction of the trajectory at the point $(x,y)=(1,0)$, and hence state whether the trajectories spiral clockwise or anticlockwise.

A change in the habitat alters the model to ${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=kx-2y$, ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=0.5x+ky$, where $k$ is a constant. The eigenvalues are now $k\pm \mathrm{i}$.

Describe the long-term behaviour of the populations if $k>0$.

Write down the matrix $\mathit{L}$ such that $\left(\begin{array}{c}{J}_{n+1}\\ A_{n+1}\end{array}\right)=\mathit{L}\left(\begin{array}{c}{J}_n\\ A_n\end{array}\right)$.

(i) Find $J_1$ and $A_1$.

(ii) Find the total number of females in the reserve after 6 years, according to this model.

Show that the eigenvalues of $\mathit{L}$ are $1.1$ and $-0.3$.

Find an eigenvector of $\mathit{L}$ for each eigenvalue.

Show that $\left(\begin{array}{c}310\\ 130\end{array}\right)=50\left(\begin{array}{c}5\\ 3\end{array}\right)+20\left(\begin{array}{c}3\\ -1\end{array}\right)$.

Hence show that the total number of females after $n$ years is $T_n=400(1.1{)}^n+40(-0.3{)}^n$.

Find the first year in which the total number of females is predicted to exceed 3000.

Describe the long-term behaviour of the beaver population according to this model. Justify your answer.

The reserve cannot support an ever-growing population, so the ecologists plan to relocate some adult females each year. This changes only the proportion of adult females that remain as adults in the reserve, from 0.6 to $a$.

The population is stable in the long term when $\mathit{L}$ has an eigenvalue equal to 1.

Find the value of $a$ for which the population is stable.

The natural survival rate of adults remains 60 %, and the relocated females are chosen from the adults who would otherwise have remained.

Find the percentage of these adult females that must be relocated each year.

Explain what the model predicts if the ecologists relocate slightly more than this percentage every year.

State two limitations of this model.

Show that $\mathit{v}$ is an eigenvector of $\mathit{M}$, and state the corresponding eigenvalue.

Find the other eigenvalue of $\mathit{M}$.

Find an eigenvector corresponding to the eigenvalue in part (b).

Write down the transition matrix $\mathit{T}$ for one year, with the first row and column for FitZone.

Find the number of members of each gym at the start of 2027.

Find the eigenvalues of $\mathit{T}$ and a corresponding eigenvector for each.

Write down a matrix $\mathit{P}$ and a diagonal matrix $\mathit{D}$ such that .

Hence show that the number of FitZone members $n$ years after the start of 2025 is $2000+1000(0.7{)}^n$.

Write down the number of FitZone members in the long term.

Find the first year in which FitZone has fewer than 2100 members at the start of the year.

At one moment there are 6000 aphids and 100 ladybirds in the greenhouse.

Find the value of $\frac{\mathrm{d}x}{\mathrm{d}t}$ and the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at this moment, and interpret your answers in context.

Consider the case in which there are no ladybirds in the greenhouse.

Write down the differential equation satisfied by $x$.

Hence find the long-term number of aphids when there are no ladybirds, justifying your answer.

Show that $(4,2)$ is an equilibrium point of the system.

Write down the coordinates of the other two equilibrium points.

To investigate the behaviour close to $(4,2)$, let $x=4+p$ and $y=2+q$, where $p$ and $q$ are small. You may assume that terms in $p^2$, $q^2$ and $pq$ can be ignored.

Show that $\frac{\mathrm{d}p}{\mathrm{d}t}\approx -0.4p-1.6q$.

Find a similar approximation for $\frac{\mathrm{d}q}{\mathrm{d}t}$.

The approximations can be written as $\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\mathit{M}\left(\begin{array}{c}p\\ q\end{array}\right)$.

Find the eigenvalues of $\mathit{M}$.

Hence describe the behaviour of the aphid and ladybird populations when they start close to $(4,2)$.

At the start of one season there are 2000 aphids and 100 ladybirds in the greenhouse.

Use Euler's method with a step length of 0.25 weeks to estimate the number of aphids and the number of ladybirds after 1 week.

Sketch, on a set of axes with $x$ on the horizontal axis and $y$ on the vertical axis, the trajectory that starts at $(2,1)$. Show the direction of motion along the trajectory.

Selin considers spraying a pesticide that kills aphids and ladybirds at the same rate. She modifies her model to

Find the equilibrium point of the modified model at which both species are present.

Comment on whether spraying the pesticide is a sensible way to reduce the aphid population in the long term.