calculus, predator-prey system

Question

The following system describes a predator prey system in which the prey has an Allee effect. What is the threshold of the prey to persist when alone? Find the nullclines and the steady states of the system. For which values of m is there a coexistence steady state? Draw the phase plane with direction arrows for $m = 0.9$. Sketch the solution curve starting at $(1.1, 0.1)$ and sketch each component of the solution as a function of time.

$$\frac{dx}{dt} =x(1−x)(x−0.5)−\frac{xy}{8}$$

$$\frac{dy}{dt} =−my+xy$$

Answer 1

I will map out the solution and you can fill in the missing details and parts of the question.

We are given the system:

$$\frac{dx}{dt} = x(1−x)\left(x−\dfrac{1}{2}\right)−\frac{xy}{8} \\ \frac{dy}{dt} = −m y + x y$$

To find the critical points, we want to find the points where we simultaneously have $x' = y'= 0$. This produces:

$$(x, y) = (0, 0), \left(\dfrac{1}{2}, 0\right), (1, 0), (m, -4(2m^2 - 3m + 1))$$

Draw a phase portrait for $m = \dfrac{9}{10}$ and include the initial point $(x, y) = \left(\dfrac{11}{10},\dfrac{1}{10}\right)$ (see red line in following phase portrait):

Sketch the solution curve starting at $\left(\dfrac{11}{10},\dfrac{1}{10}\right)$ and sketch each component of the solution as a function of time. This was done using numerical methods and here is a plot for $x(t), x'(t)$, $y(t), y'(t)$ as functions of time.

The following items are left for you to fill in:

• What is the threshold of the prey to persist when alone?
• Find the nullclines (below) and the steady states (included above) of the system.
• For which values of $m$ is there a coexistence steady state?

Update

Here is the phase portrait with the nullclines (purple and green) added in, along with the initial point (red).