Phase portrait (for solutions to coupled systems) & sketching trajectories: notes and practice questions

The populations of two interacting species, A and B, in a controlled environment are modelled by the following system of linear differential equations:

$\dot{x} = x + 2y$

$\dot{y} = 3x + 2y$

where $x(t)$ represents the population of species A and $y(t)$ represents the population of species B at time $t$. Find the general solution for this system.

Sketch the phase portrait for the system, clearly indicating the critical point, the equations of any asymptotes, and the general direction of trajectories. Briefly describe the long-term behaviour of the populations.

The concentration of a chemical, $C$ (in mol dm$^{-3}$), in a reaction vessel at time $t$ seconds is modelled by the differential equation

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 0$

(a) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to show that this equation can be written as

$\begin{pmatrix} \frac{\mathrm{d}C}{\mathrm{d}t} \\ \frac{\mathrm{d}V}{\mathrm{d}t} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix} \begin{pmatrix} C \\ V \end{pmatrix}$.

(ii) Find the eigenvalues for the matrix $\begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix}$.

(iii) Hence state the long-term rate of change of the chemical concentration.

The chemical reaction is now subjected to an external input, and the equation for the concentration is amended to

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 2t + 5$.

(b) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to write the differential equation as a system of coupled, first order differential equations. State the initial conditions given that, at $t = 0$, the concentration of chemical C is $1$ mol dm$^{-3}$ and its rate of change is $0$ mol dm$^{-3}$ s$^{-1}$.

(ii) Use Euler's method with a step length of $0.1$ to find the concentration of the chemical when $t = 1$ s. Give your answer to three significant figures.

(iii) Find the long-term rate of change of the chemical concentration.

A population model for two interacting species, $X$ and $Y$, is described by the following system of coupled differential equations, where $x(t)$ and $y(t)$ represent the deviations from their equilibrium populations. (a) Determine (with a worked reason) whether the populations of both species tend towards their equilibrium, move away from their equilibrium, or exhibit another behaviour.

$\frac{dx}{dt} = -3x + y$

$\frac{dy}{dt} = 2x - 4y$

(b) Determine (with a worked reason) whether the populations of both species tend towards their equilibrium, move away from their equilibrium, or exhibit another behaviour.

$\frac{dx}{dt} = x + 2y$

$\frac{dy}{dt} = 3x$

(c) Determine (with a worked reason) whether the populations of both species tend towards their equilibrium, move away from their equilibrium, or exhibit another behaviour.

$\frac{dx}{dt} = -x - 2y$

$\frac{dy}{dt} = 2x - y$

(d) Determine (with a worked reason) whether the populations of both species tend towards their equilibrium, move away from their equilibrium, or exhibit another behaviour.

$\frac{dx}{dt} = 2y$

$\frac{dy}{dt} = -2x$

The following question explores a possible method of drawing phase portraits for non-linear coupled systems, taking a predator-prey model as a particular example.

A forest ecosystem contains a population of rabbits ($x$, measured in hundreds), and a population of wolves ($y$, measured in hundreds).

Research indicates that the population dynamics of both rabbits and wolves can be modelled by the following differential equations, in which $t$ is measured in years.

$\frac{dx}{dt} = 3x - \frac{xy}{2}$

$\frac{dy}{dt} = -2y + \frac{2xy}{5}$

for $x, y \ge 0$

At a specific time, there are 400 rabbits and 400 wolves, represented here by the coordinate pair (4, 4). At this time, determine the rate of change of

rabbits.

wolves.

There are two equilibrium points for the populations: A(0, 0) and B ($p, q$).

Explain why A is an equilibrium point.

Find the value of $p$ and the value of $q$.

At points close to A(0, 0), we can ignore the $xy$ terms, so that the system can be approximated by:

$\frac{dx}{dt} = 3x$

$\frac{dy}{dt} = -2y$

for $x, y \ge 0$.

By solving these two differential equations,

find an expression for $x$ in terms of $t$.

find an expression for $y$ in terms of $t$.

Using your answers from part (c), show that phase portrait trajectories close to A may be given by the equation $x^2y^3 = k$, where $k$ is a positive constant.

Hence sketch, on a phase portrait, one possible trajectory for small values of $x$ and $y$.

Now consider points $(x, y)$ close to B on the phase plane. These coordinates can be rewritten as $x = p + X$ and $y = q + Y$, where $p$ and $q$ are the values from part (b)(ii).

By substituting into the original model, show that, for small values of $X$ and $Y$:

$\dot{X} \approx -\frac{5Y}{2}$

Similarly, it can be shown that $\dot{Y} \approx \frac{12X}{5}$.

Given that $\begin{pmatrix} \dot{X} \\ \dot{Y} \end{pmatrix} = M \begin{pmatrix} X \\ Y \end{pmatrix}$, where $M$ is a square matrix, write down $M$.

By finding the eigenvalues of $M$, describe the path of the trajectories close to point B.

Hence sketch a complete set of trajectories in the phase plane for the original model, clearly indicating both equilibrium points.

In this forest ecosystem, at a specific time, there are 400 rabbits and 400 wolves.

Based on the values found in part (a), the wildlife keeper is worried and assumes that the wolves will quickly die out. Suggest whether this assumption is supported by the model. Justify your answer.

The populations of two interacting bacterial strains, Alpha and Beta, in a controlled environment are modelled by a system of coupled differential equations. Let $A$ be the population of strain Alpha (in thousands) and $B$ be the population of strain Beta (in thousands).

The system is given by:

$\frac{dA}{dt} = 1.5A - 2.0B$

$\frac{dB}{dt} = 2.5A - 1.5B$

At time $t = 0$ hours, the population of strain Alpha is $A = 6$ thousand and the population of strain Beta is $B = 3$ thousand.

Find the value of $\frac{dB}{dA}$ at $t = 0$.

The eigenvalues for this system are $\pm 4i$.

On the following phase portrait, sketch the trajectory that passes through the point $(6, 3)$. Clearly indicate the direction of this trajectory.

A microscopic particle is observed moving in a fluid. Its position $(x, y)$ metres, relative to a fixed origin O, at time $t$ seconds ($t \ge 0$), is modelled by the system of differential equations:

$\frac{dx}{dt} = -0.1x + 3y$

$\frac{dy}{dt} = -3x - 0.1y$

Find the eigenvalues of the matrix $A = \begin{pmatrix} -0.1 & 3 \\ -3 & -0.1 \end{pmatrix}$, giving your answers in the form $a + bi$, where $a \ne 0, b \ne 0$.

State what $a \ne 0$ indicates about the path of the particle.

State what the sign of $a$ indicates about the path of the particle.

At time $t = 0$, the particle is at position $(10, 0)$.

At time $t = 0$, find the value of $\frac{dy}{dt}$.

Find the value of $\frac{dy}{dx}$ at time $t = 0$.

Use your answers to parts (b) and (c) to sketch the path of the particle.